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NCERT Exercises

Question 1.
in an n-type silicon, which of the following statement is true?
(a) Electrons are majority carriers and trivalent atoms are the dopants.
(b) Electrons are minority carriers and pentavalent atoms are the dopants.
(c) Holes are minority carriers and pentavalent atoms are the dopants.
(d) Holes are majority carriers and trivalent atoms are the dopants.
Solution:
(c) For n-type silicon statement (c) is true.

Question 2.
Which of the statements given in previous question is true for p-type semiconductors?
Solution:
(d) For p-type semiconductors statement (d) is true.

Question 3.
Carbon, silicon and germanium have four valence electrons each.These are characterised by valence and conduction bands separated by energy band gap respectively equal to (E_g)_c’ (E_g)_si and (E_g)_Ge. Which of the following statements is true?

Solution:
(c) : The energy band gap is largest for carbon, less for silicon and least for germanium. So, the correct statement is (c).
(E_g)_c > (E_g)_si > (Eg)_Ge

Question 4.
In an unbiased p-n junction, holes diffuse from the p-region to n-region because
(a) free electrons in the n-region attract them.
(b) they move across the junction by the potential difference.
(c) hole concentration in p-region is more as compared to n-region.
(d) all of the above.
Solution:
(c) : In the unbiased p-n junction, holes diffuse from the p-region to n-region because holes concentration in the p-region is high as compared to n-region.

Question 5.
When a forward bias is applied to a p-n junction, it
(a) raises the potential barrier.
(b) reduces the majority carrier current to zero.
(c) lowers the potential barrier.
(d) none of the above
Solution:
(c) : Under forward biasing the move¬ment of majority charge carriers across the junction reduces the width of depletion layer or lowers the potential barrier.

Question 6.
For transistor action, which of the following statements is/are correct:
(a) Base, emitter and collector regions should have similar size and doping concentrations.
(b) The base region must be very thin and lightly doped.
(c) The emitter junction is forward biased and collector junction is reverse biased.
(d) Both the emitter junction as well as the collector junction are forward biased.
Solution:
(b, c) For a transistor circuit in action, statements (b) and (c) are correct.

Question 7.
For a transistor amplifier, the voltage gain
(a) remains constant for all frequencies.
(b) is high at high and low frequencies and constant in the middle frequency range.
(c) is low at high and low frequencies and constant at mid frequencies.
(d) none of the above.
Solution:
(c) : The voltage gain in a transistor as amplifier is low at high and low frequencies and constant at mid frequencies.

Question 8.
In half-wave rectification, what is the output frequency if the input frequency is 50 Hz. What is the output frequency of a full-wave- rectifier for the same input frequency?
Solution:
In half wave rectification, only one ripple is obtained per cycle in the output.

Output frequency of a half wave rectifier = input frequency = 50 Hz In full wave rectification, two ripples are obtained per cycle in the output.
Output frequency = 2 × input frequency = 2 × 50 = 100 Hz

Question 9.
For a CE-transistor amplifier, the audio signal voltage across the collector resistance of 2 kΩ is 2 V. Suppose the current amplification factor of the transistor is 100, find the input signal voltage and base current, if the base resistance is 1 kΩ
Solution:

Question 10.
Two amplifiers are connected one after the other in series (cascaded). The first amplifier has a voltage gain of 10 and the second has a voltage gain of 20. If the input signal is 0. 01 volt, calculate the output ac signal.

Solution:
Total voltage gain can be calculated as

Question 11.
A p-n photo diode is fabricated from a semiconductor with band gap of 2.8 eV. Can it detect a wavelength of 6000 nm?
Solution:
Energy of the incident photon with a band gap of 6000 nm.

The photo diode need an energy of 2.8 eV to give response to incident light. As E < E_g, the given photo diode cannot detect the radiation of wavelength 6000 nm.
Question 12.
The number of silicon atoms per m³ is 5 × 10²⁸. This is doped simultaneously with 5 × 10²² atoms per m³ of Arsenic and 5 × 10²⁰ per m³ atoms of Indium. Calculate the number of electrons and holes. Given that n, = 1.5 × 10¹⁶ m³. Is the material n-type or p-type?
Solution:
We know that for each atom doped of Arsenic, one free electron is received. Similarly, for each atom doped of indium, a vacancy is created. So, the number of free electrons introduced by pentavalent impurity added n_e = N_As = 5 × 10²² m^-3
The number of holes introduced by trivalent impurity added

Question 13.
In an intrinsic semiconductor the energy gap E_g is 1.2 eV. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at 600 K and that at 300 K? Assume that the temperature dependence of intrinsic carrier concentration n, is given by

Solution:
Conductivity is given by σ = e (n_e μ_e +n_h μ_h) For intrinsic semiconductor n_e = n_h = n_i Also mobility of holes (p,,) « mobility of electrons (pj So, conductivity a = enepe „ Temperature dependence of intrinsic carrier concentration


Conductivity increases rapidly with the rise of temperature.

Question 14.
In a p-n junction diode, the current I can be

is called the reverse saturation current, V is the voltage across the diode and is positive for forward bias and negative for reverse bias, and I is the current through the diode, kB is the Boltzmann constant (8.6 × 10^-5 eV/K) and T is the absolute temperature. If for a given diode I₀ = 5 × 10^-12 A and T= 300 K, then
(a) What will be the forward current at a forward voltage of 0.6 V?
(b) What will be the increase in the current if the voltage across the diode is increased to 0.7 V?
(c) What is the dynamic resistance?
(d) What will be the current if reverse bias voltage changes from 1 V to 2 V?
Solution:
The current I through a junction diode is given as

(a) When V = 0.6 V

(b) When V = 0.7 V,

(c)

(d) For both the voltages, the current I will be almost equal to I_0, showing almost infinite dynamic resistance in the reverse bias. I=-I₀= – 5 × 10^-12 A.

Question 15.
YOU are given the two circuits as shown in figure. Show that circuit (a) acts as OR gate while the circuit (b) acts as AND gate.

Solution:
Let us first find the Boolean expression for logic circuit (a)

By De-Morgan’s theorem we know

so, above logic circuit provides output as AND gate

Question 16.
Write the truth table for a NAND gate connected as given in following figure.

Hence identify the exact logic operation carried out by this circuit.
Solution:
The Boolean expression for NAND gate will be

Question 17.
You are given two circuits as shown in figure which consist of NAND gates. Identify the logic operation carried out by the two circuits.

Solution:
(a) Let us find Boolean expression for given logic circuit

(b) By De-Morgan’s theorem

so, the given logic circuit acts as OR gate.

Question 18.
Write the truth table for circuit given in figure below consisting of NOR gates and identify the logic operation (OR, AND, NOT) which this circuit is performing.

(Hint: A – 0, B = 1 then A and B inputs of second NOR gate will be 0 and hence Y = 1. Similarly work out the values of Yfor other combinations of A and B. Compare with the truth table of OR, AND, NOT gates and find the correct one.)
Solution:
Let us find the Boolean expression for the logic circuit.

Hence, the output of given logic circuit shows that logic circuit act as OR gate. Truth Table

Question 19.
Write the truth table for the circuits given in figure consisting of NOR gates only. Identify the logic operations (OR, AND, NOT) performed by the two circuits.

Solution:
Boolean expression for logic circuit (a)

Here the given NOR gate with short circuit input is acting as NOT gate. Its truth table is

Boolean expression for logic circuit (b)

Hence, the logic circuit acts like AND gate. Its truth table is

Question 1.
(a) Two stable isotopes of lithium  and  have respective abundances of 7.5% and 92.5%. These isotopes have masses 6.01512 u and 7.01600 u respectively. Find the atomic mass of lithium.
(b) Boron has two stable isotope.s,  and  Their respective masses are 10.01294 u and 11.00931 u, and the atomic mass of boron is
10.811 u. Find the abundances of  and .
Solution:
Abundance of  is 7.5% and abundance

(b) Let abundance of  x% than abundance of  will be (100 – x)%.

Question 2.
The three stable isotopes of neon : ,  and have respective abundances of 90.51%, 0.27% and 9.22%. The atomic masses of the three isotopes are 19.99 u, 20.99 u and 21.99 u, respectively. Obtain the average atomic mass of neon.
Solution:
Average atomic mass of neon with the given abundances,

Question 3.
Obtain the binding energy (in MeV) of a nitrogen nucleus , given m  = 14.00307 u
Solution:
The nucleus contains 7 protons and 

Question 4.
Obtain the binding energy of the nuclei  and  in units of MeV from the following data:
 = 55.934939 u
 = 208.980388 u
Solution:
Let us first find the binding energy of


Question 5.
A given coin has a mass of 3.0 g. Calculate the nuclear energy that would be required to separate all the neutrons and protons from each other. For simplicity assume that the
coin is entirely made of  atoms (of mass 62.92960 u).
Solution:
Let us first find the B.E. of each copper nucleus and then we can find binding energy

Question 6.
Write nuclear reaction equations for
(i) a-decay of 
(ii) a-decay of 
(iii) p-decay of 
(iv) p-decay of 
(v) p+-decay of 
(vi) p+-decay of 
(vii) Electron capture of 
Solution:

Question 7.
A radioactive isotope has a half-life of T years. How long will it take the activity to reduce to
(a) 3.125%
(b) 1% of its original value?
Required time, as cannot be solved by direct calculation as in part (a).
Solution:


Required time, as cannot be solved by direct calculation as in part (a)

Question 8.
The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive  present with the stable carbon isotope . When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of , and the measured activity, the age of the specimen can be approximately estimated. This is the principle of  dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilization.
Solution:
In order to estimate age, let us first find the activity ratio in form of time ‘t’. Given normal activity, R0 = 15 decays min^-1 Present activity, R = 9 decays min^-1, Tin = 5730 years Since activity is proportional to the number of radioactive atoms, therefore,

Question 9.
Obtain the amount of  necessary to provide a radioactive source of 8.0 mCi strength. The half-life of  is 5.3 years.
Solution:
Here rate of disintegration required


As 1 mole i.e., 60 g of cobalt contains 6.023 × 10²³ atoms, so, the mass of cobalt required for given rate of disintegration

Question 10.
The half-life of  is 28 years. What is the disintegration rate of 15 mg of this isotope?
Solution:

Question 11.
Obtain approximately the ratio of the nuclear radii of the gold isotope  and the silver isotope .
Solution:
We know the radius of nucleus depend upon mass number ‘A’

Question 12.
Find the Q-value and the kinetic energy of the emitted a-particle in the a-decay of
(a)  and (b) .
Given  = 226.02540 u,
 = 222.01750 u,
 = 220.01137 u,
 = 216.00189 u, and
m_x = 4.00260 u.
Solution:

Question 13.
The radionuclide “C decays according to  + e⁺ + v: T_1/2 = 20.3 min
The maximum energy of the emitted positron is 0.960 MeV.
Given the mass values:
 = 11.011434 u
 = 11.009305 u
Calculate Q and compare it with the maximum energy of the positron emitted.
Solution:
The given equation

As we know that different positrons comes out with different possible energies shared between daughter nucleus and positron.
So, the Q value of reaction is almost same as the maximum energy of positron emitted.

Question 14.
The nucleus  decays by β^– emission. Write t down the β^– decay equation and determine
r the maximum kinetic energy of the electrons emitted. Given that:
 = 22.994466 amu,
 = 22.989770 amu.
Solution:
The β^– decay of  may be explained as

As  is massive, the kinetic energy released is mainly shared by electron-positron pair. When the neutrino carries no energy, the electron has a maximum kinetic energy equal to 4.374 MeV.

Question 15.
The Q value of a nuclear reaction A + b—>C+d is defined by Q = [m_A + m_b-m_c– m_d] c², where the masses refer to the respective nuclei, Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic.
(i) 
(ii) 
Atomic masses are given to be

Solution:
(i) Let us find the Q value in given first equation,

Negative Q value shows that reaction is endothermic.
(ii) Q value in the given second equation

Positive Q shows that the reaction is exothermic.

Question 16.
Suppose, we think of fission of a  nucleus into two equal fragments, . IS the fission energetically possible? Argue by working out Q of the process.
Given,  = 55.93494 u
and  = 27.98191 u.
Solution:
The fission of Fe-56 into two fragments of

As the Q-value is negative, the fission is not possible energycally.

Question 17.
The fission properties of  are very similar to those of . The average energy released per fission is 180 MeV. How much energy, in MeV, is released if all the atoms in 1 kg of pure  undergo fission?
Solution:

Question 18.
A 1000 MW fission reactor consumes half of its fuel in 5.00 y. How much  did it contain initially? Assume that the reactor operates 80% of the time and that all the energy generated arises from the fission of and that this nuclide is consumed by the fission process.
Solution:
In the fission of one nucleus of , energy generated is 200 MeV.

Question 19.
How long can an electric lamp of 100 W be kept glowing by fusion of 2.0 kg of deuterium? Take the fusion reaction as

Solution:
Number of atoms present in 2 g of deuterium = 6.023 × 10²³ Total number of atoms present in 2000 g of deuterium

Energy released in the fusion of 2 deuterium atoms = 3.27 MeV

Question 20.
Calculate the height of potential barrier for a head-on collision of two deuterons. The effective radius of deuteron can be taken to be 2fm.
Solution:
For head on collision, distance between centers of two deuterons

This is a measure of height of coulomb barrier.

Question 21.
From the relation R = R₀ A^1/3, where R₂ is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e., independent of A).
Solution:

As R is constant, p is contact so, nuclear density is constant irrespective of mass number or size.

Question 22.
For the β⁺ (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the /(-shell, is captured by the nucleus and a neutrino is emitted).

Show that if β⁺ emission is energetically allowed, electron capture is necessarily allowed but not vice-versa.
Solution:
Let us first consider positron emission.

This mean if Q₁ > 0 then Q₂ > 0 but vice vesa is not necessarily allowed. So, electron capture is not necessary for positron emission.

Question 23.
In a periodic table the average atomic mass of magnesium is given as 24.312 u. The average value is based on their relative natural abundance on earth. The three isotopes and their masses are  (23.98504 u),  (24.98584 u) and  (25.98259 u).The natural abundance of  78.99% by mass. Calculate the abundances of the other two isotopes.
Solution:
Let the abundance of isotope  is

Question 24.
The neutron separation energy is defined as the energy required to remove a neutron from the nucleus. Obtain the neutron separation energies of the nuclei  and  from
the following data:

Solution:
Neutron separation of  can be obtained as E = Energy equivalent of total mass afterward – Energy equivalent of nucleus before

Question 25.
A source contains two phosphorus radio -nuclides  (T_1/2 = 14.3 days) and  (Tv2 = 25.3 days). Initially, 10% of the decays come from  How long one must wait until 90% do so?
Solution:
In the mixture of P-32 and P-33 initially 10% decay came from P-33. Hence initially 90% of the mixture is P-32 and 10% of the mixture is P-33. Let after time’t’ the mixture is left with 10% of P-32 and 90% of P-33. Half life of both P-32 and P-33 are given as 14.3 days and 25.3 days respectively. Let V be total mass undecayed initially and ‘y’ be total mass undecayed finally. Let initial number of P-32 nuclides = 0.9 x Final number of P-32 nuclides = 0.1 y Similarly, initial number of P-33 nuclides = 0.l x Final number of P-33 nuclides = 0.9 y For isotope P-32

Question 26.
Under certain circumstances, a nucleus can decay by emitting a particle more massive than an a-partide. Consider the following decay processes:

Calculate the Q-values for these decays and determine that both are energetically allowed.
Solution:
Let us calculate Q value for the given decay process. For first decay process

Since, Q value is positive in both the cases, hence decay process in both ways are possible

Question 27.
Consider the fission of  by fast neutrons. In one fission event, no neutrons are emitted and the final end products, after the beta decay of the primary fragments are  and . Calculate Qfor this fission process. The relevant atomic and particle masses are:

Solution:
The fission of U-238 by fast neutrons into fragments Ce-140 and Ru-99 with energy released Q can be written as

Question 28.
Consider the D-T reaction (deuterium – tritium fusion)

(a) Calculate the energy released in MeV in this reaction from the data
m  = 2.014102 u, m  = 3.016049 u
(b) Consider the radius of both deuterium and tritium to be approximately 2.0 fm. What is the kinetic energy needed to overcome the coulomb repulsion between the two nuclei? To what temperature must the gases be heated to initiate the reaction?
Solution:

Classically, K.E. atleast equal to this amount is required to overcome Coulomb repulsion. Using the relation

Question 29.
Obtain the maximum kinetic energy of β-particles and the radiation frequencies of y-decays in the decay scheme shown in figure. You are given that
 = 197.968233 u,  = 197.966760 u

Solution:
Energy corresponding to y₁

Question 30.
Calculate and compare the energy released by
(a) fusion of 1.0 kg of hydrogen deep within the Sun and
(b) the fission of 1.0 kg of ²³⁵U in a fission reactor.
Solution:
(a) In the fusion reactions taking place within core of sun, 4 hydrogen nuclei combines to form a helium nucleus with the release of 26 MeV of energy.

(b) Energy released per fission of U-235 is 200 MeV.

So the energy released in fusion of 1 kg of Hydrogen is nearly 8 times the energy released in fission of 1 kg of uranium-235.

Question 31.
Suppose India has a target of producing by 2020 AD, 200,000 MW of electric power, ten percent of which is to be obtained from nuclear power plants. Suppose we are given that, on an average, the efficiency of utilization (/.e., conversion to electric energy) of thermal energy produced in a reactor was 25%. How much amount of fissionable uranium would our country need per year? Take the heat energy per fission of ²³⁵U to be about 200 MeV.
Solution:
10% of total power 200,000 MW to be obtained from nuclear power plant by 2020 AD.


Hence mass of uranium needed per year = 3.08 × 10⁴ kg

NCERT Exercises

Question 1.
Choose the correct alternative from the clues given at the end of each statement:
(a) The size of the atom in Thomson’s model is _____ the atomic size in Rutherford’s model. (Much greater than / no different from / much less than).
(b) In the ground state of _____ electrons are in stable equilibrium, while in _____ electrons always experience a net force. (Thomson’s model / Rutherford’s model).
(c) A classical atom based on _____ is doomed to collapse. (Thomson’s model / Rutherford’s model)
(d) An atom has a nearly continuous mass distribution in a _____ but has a highly non-uniform mass distribution in _____ (Thomson’s model / Rutherford’s model).
(e) The positively charged part of the atom possesses most of the mass in _____ (Rutherford’s model / both the models).
Solution:
(a) not different from
(b) Thomson’s model, Rutherford’s model
(c) Rutherford’s model
(d) Thomson’s model, Rutherford’s model
(e) both the models

Question 2.
Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K). What results do you expect?
Solution:
The nucleus of a hydrogen atom is a proton. The mass of a proton is 1.67 × 10^-27 kg, whereas the mass of an incident a-particle is 6.64 × 10^-27 kg. Because the incident a-particles are more massive than the target nuclei (protons), the a-particle won’t bounce back even in a head on collision. It is similar to a football colliding with a tennis ball at rest. Thus, there would be no appreciable scattering.

Question 3.
What is the shortest wavelength present in the Paschen series of spectral lines?
Solution:
For shortest wavelength of Paschen series, n₁ = 3, n₂ = ∞

Question 4.
A difference of 2.3 eV separates two energy levels in an atom. What is the frequency of radiation emitted when the atom makes a transition from the upper level to the lower level?
Solution:

Question 5.
The ground state energy of hydrogen atom is -13.6 eV. What are the kinetic and potential energies of the electron in this state?
Solution:

Question 6.
A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of photon.
Solution:
Energy of an electron in the nth orbit of

Question 7.
(a) Using the Bohr’s model, calculate the speed of the electron in a hydrogen atom in the n = 1, 2 and 3 levels,
(b) Calculate the orbital period in each of these levels.
Solution:
(a) Speed of the electron in Bohr’s nth orbit is

(b) Orbital period of electron in Bohr’s first orbit is


Question 8.
The radius of the innermost electron orbit of a hydrogen atom is 5.3 × 10^-11 m. What are the radii of the n = 2andn = 3 orbits?
Solution:
Radius of innermost electron

Question 9.
A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
Solution:
In ground state, energy of gaseous hydrogen at room temperature=-13.6 eV. When it is bombarded with 12.5 eV electron beam, the energy becomes – 13.6 + 12.5 = – 1.1 eV. The electron would jump from n = 1 to n = 3, 13 6 where E₃ =  = -1.5 eV. On de-excitation the electron may jump from n = 3 to n = 2 giving rise to Balmer series. It may also jump from n = 3 to n = 1, giving rise to Lyman series.

Question 10.
ln accordance with the Bohr’s model, find the quantum number that characterises the earth’s revolution around the sun in an orbit of radius 1.5 × 10¹¹ m with orbital speed 3 × 10⁴ m s^-1. (Mass of earth = 6.0 × 102²⁴kg)
Solution:
According to Bohr’s quantization condition of angular momentum, Angular momentum of the earth around the sun,

Question 11.
Answer the following questions, which helpyou understand the difference between Thomson’s model and Rutherford’s model better.
(a) Is the average angle of deflection of a-particles by a thin gold foil predicted by Thomson’s model much less, about the same, or much greater than that predicted by Rutherford’s model?
(b) Is the probability of backward scattering (i.e., scattering of a-particles at angles greater than 90°) predicted by Thomson’s model much less, about the same, or much greater than that predicted by Rutherford’s model?
(c) Keeping other factors fixed, it is found experimentally that for small thickness t, the number of a-particles scattered at moderate angles is proportional to What clue does this linear dependence on f provide?
(d) In which model is it completely wrong to ignore multiple scattering for the calculation of average angle of scattering of a-particles by a thin foil?
Solution:
(a) Nearly the same. This is because we are considering the average angle of deflection.
(b) Much less, because there is no such massive core (nucleus) in Thomson’s model as in Rutherford’s model.
(c) This suggests that scattering is mainly due to a single collision, because the chance of a single collision increases linearly with the number of the target atoms, and hence linearly with the thickness of the foil.
(d) In Thomson model, positive charge is distributed uniformly in the atom. So single collision causes very little deflection. The observed average scattering angle can be explained only by considering multiple scattering. Hence, it is wrong to ignore multiple scattering in Thomson’s model.

Question 12.
The gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about 10^-40. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.
Solution:
The radius of the first orbit in Bohr’s model is calculated by considering the electrostatic force between electrons and proton in nucleus.

This radius is much greater than the estimated size of the whole universe.

Question 13.
Obtain an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n – 1). For large n, show that this frequency equals the classical frequency of revolution of the electron in the orbit.
Solution:
Let us first find the frequency of revolution of electron in the orbit classically. In Bohr’s model velocity of electron in nth

Now, let us find the frequency of radiation emitted when a hydrogen atom de-excites from level h to level (n – 1).

Equation (i) and (ii) are equal, hence for large value of n, the classical frequency of revolution of electron in nth orbit is same as frequency of radiation when electron de-excite from level n to(n – l).

Question 14.
Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom 10 10 m)
(a) Construct a quantity with the dimensions of length from the fundamental constants e, m_e, and c. Determine its numerical value.
(b) You will find that the length obtained in (a) is many orders of magnitude smaller than the atomic dimensions. Further, it involves c. But energies of atoms are mostly in non-relativistic domain where c is not expected to play any role. This is what may have suggested Bohrto discard c and look for ‘something else’ to get the right atomic size. Now, the Planck’s constant h had already made its appearance elsewhere. Bohr’s great insight lay in recognising that h, m_e, and e will yield the right atomic size. Construct a quantity with the dimension of length from h, m_e, and e and confirm that its numerical value has indeed the correct order of magnitude.
Solution:

(b) A quantity with the dimension of length from h, m_e and e

The length is of the order of atomic size (10^-10m)
Question 15.
The total energy of an electron in the first excited state of the hydrogen atom is about – 3.4 eV.
(a) What is the kinetic energy of the electron in this state?
(b) What is the potential energy of the electron in this state?
(c) Which of the answers above would change if the choice of the zero of potential energy is changed?
Solution:
Kinetic energy of an electron in an orbit,

(a) Kinetic energy of electron in this state E = -K
(b) Potential energy E = U/2, U = 2E = 2 (-3.4) = -6.8 eV
(c) If thg zero of the potential energy is chosen differently, the kinetic energy remain the same. Although potential energy and hence total energy changes.

Question 16.
If Bohr’s quantisation postulate (angular momentum = nh/2n) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why then do we never speak of quantisation of orbits of planets around the sun?
Solution:
Angular momentum mvr = n  associated with planetary motion are incomparably large relative to h. For example angular momentum of earth in its orbital motion is of the order of 10⁷⁰ .
For such large value of n, the difference in successive energies and angular momenta of the quantised levels of the Bohr model are so small that one can predict the energy level continuous.

Question 17.
Obtain the first Bohr’s radius and the ground state energy of a muonic hydrogen atom [/.e., an atom in which a negatively charged muon (μ) of mass about 207 me orbits around a proton].
Solution:
In Bohr’s model, the radius of nth orbit,

In the given muonic hydrogen atom, a negatively charged muon (μ^–) of mass 207 me revolve around a proton. Therefore radius of electron and muon can be written as

NCERT Exercises

Question 1.
Find the:
(a) maximum frequency, and
(b) minimum wavelength of X-rays produced by 30 kV electrons.
Solution:
(a) Maximum energy of X-ray photon = Maximum energy of an accelerated electron

Question 2.
The work function of caesium metal is 2.14 eV. When light of frequency 6 × 10¹⁴ Hz is incident on the metal surface, photo emission of electrons occurs. What is the:
(a) maximum kinetic energy of the emitted electrons,
(b) stopping potential, and
(c) maximum speed of the emitted photo-electrons?
Solution:


Question 3.
The photoelectric cut-off voltage in a certain experiment is 1.5 V. What is the maximum kinetic energy of photo electrons emitted?
Solution:

Question 4.
Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. The power emitted is 9.42 mW.
(a) Find the energy and momentum of each photon in the light beam.
(b) How many photons per second, on the average, arrive at a target irradiated by this beam? (Assume the beam to have uniform cross-section which is less than the target area)
(c) How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?
Solution:

(a) Energy of each photon,

(b) Number of photons arriving per second at the target,

(c) As mv = p
∴ Velocity, 
= 0.63 m s^-1

Question 5.
The energy flux of sunlight reaching the surface of the earth is 1.388 × 10³. How many photons (nearly) per square metre are incident on the earth per second? Assume that the photons in the sunlight have an average wavelength of 550 nm.
Solution:
Energy of each photon,

Number of photons incident on earth’s surface per second per square metre

Question 6.
In an experiment on photoelectric effect, the slope of the cut-off voltage versus frequency of incident light is found to be 4.12 × 10^-15 V s. Calculate the value of Planck’s constant.
Solution:

Question 7.
A 100 W sodium lamp radiates energy uniformly in all directions. The lamp is located at the centre of a large sphere that absorbs all the sodium light which is incident on it. The wavelength of the sodium light is 589 nm.
(a) What is the energy per photon associated with the sodium light?
(b) At what rate are the photons delivered to the sphere?
Solution:
Here P = 100 W, λ = 589 nm = 589 × 10^-9 m
(a) Energy of each photon,

(b) Rate at which photons are delivered to sphere,

Question 8.
The threshold frequency for a certain metal is 3.3 × 1014 Hz. If light of frequency 8.2 x 10¹⁴ Hz is incident on the metal, predict the cut off voltage for the photoelectric emission.
Solution:
According to Einstein’s relation

Question 9.
The work function for a certain metal is 4.2 eV. Will this metal give photoelectric emission for incident radiation of wavelength 330 nm?
Solution:
Let us calculate the energy associated with photons incident

Since, energy of incident photons i.e., 3.77 eV is less than work function, hence no emission will take place.

Question 10.
Light of frequency 7.21 × 10¹⁴ Hz is incident on a,metal surface. Electrons with a maximum speed of 6.0 × 10⁵ m s^-1 are ejected from the surface. What is the threshold frequency for photo emission of electrons?
Solution:
According to Einstein’s equation

Question 11.
Light of wavelength 488 nm is produced by an argon laser which is used in the photoelectric r effect. When light from this spectral line is incident on the emitter, the stopping (cut-off) potential of photo electrons is 0.38 V. Find the work function of the material from which the r emitter is made,
Solution:
Energy of incident radiation

Question 12.
Calculate the
(a) momentum, and
(b) de Broglie wavelength of the electrons accelerated through a potential difference of 56 V.
Solution:
An electron which is accelerated through a potential difference of 56 V will have kinetic energy K = 56 eV
(a) Momentum associated with accelerated electron

(b) Wavelength of electron accelerated

Question 13.
What is the
(a) momentum,
(b) speed, and
(c) de Broglie wavelength of an electron with kinetic energy of 120 eV.
Solution:

(a) Momentum of electron,

(b) Speed of electron

(c) de-Broglie wavelength associated with electron

Question 14.
The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which
(a) an electron, and
(b) a neutron, would have the same de Broglie wavelength.
Solution:
(a) Kinetic energy required by electron to have de-Broglie wavelength of 589 nm

(b) Kinetic energy of neutron to have de- Broglie wavelength of 589 nm

Question 15.
What is the de Broglie wavelength of
(a) a bullet of mass 0.040 kg travelling at the speed of 1.0 km s^-1
(b) a ball of mass 0.060 kg moving at a speed of 1.0 ms^-1, and
(c) a dust particle of mass 1.0 × 10^-9 kg drifting with a speed of 2.2 m s^-1?
Solution:
de Broglie wavelength 
(a) Wavelength associated with bullet

(b) Wavelength associated with ball

(c) Wavelength associated with dust particle

Question 16.
An electron and a photon each have a wavelength of 1.00 nm. Find
(a) their momenta,
(b) the energy of the photon, and
(c) the kinetic energy of electron.
Solution:
(a) Momentum for both electron and photon will be same for same wavelength.

Question 17.
(a) For what kinetic energy of a neutron will the associated de Broglie wavelength be 1.40 × 10^-10 m?
(b) Also find the de Broglie wavelength of a neutron, in thermal equilibrium with matter having an average kinetic energy
Solution:

Question 18.
Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).
Solution:
For a photon, de Broglie wavelength, 
For an electromagnetic radiation of frequency u and wavelength λ'(= c/v), Momentum,

Thus the wavelength of the electromagnetic radiation is the same as the de Broglie wavelength of the photon.

Question 19.
What is the de Broglie wavelength of a nitrogen molecule in air at 300 K? Assume that the molecule is moving with the root-mean-square speed of molecules at this temperature. (Atomic mass of nitrogen = 14.0076 u, K = t. 38 × 10^-23 J K^-1).
Solution:
Let us first calculate mass of each N₂ molecule.

Question 20.
(a) Estimate the speed with which electrons emitted from a heated cathode of an evacuated tube impinge on the anode maintained at a potential difference of 500 V with respect to the cathode. Ignore the small initial speeds of the electrons. The ‘specific charge’ of the electron i.e., its e/m is given to be 1.76 × 10¹¹ C kg^-1.
(b) Use the same formula you employ in (a) to obtain electron speed for an anode potential of 10 MV. Do you see what is wrong? In what way is the formula to be modified?
Solution:
(a) Energy of accelerated electron

This speed of electron is impossible. Since nothing can move with a speed greater than speed of light (c = 3 × 1o⁸ m s^-1 The formula for kinetic energy E = 1/2 mv² valid only for v « c. For the situation when speed v is comparable to speed of light c, we use relativistic formula. The relativistic mass is given by

Substituting the values

Question 21.
(a) A monoenergetic electron beam with electron speed of 5.20 × 10⁶ m s^-1 is subjected to a magnetic field of 1.30 × 10^-4 T, normal to the beam velocity. What is the radius of the circle traced by the beam, given e/m for electron equals 1.76 × 10¹¹ C
kg^-1.
(b) Is the formula you employ in (a) valid for calculating radius of the path of a 20 MeV electron beam? If not, in what way is it modified?
Solution:

The normal magnetic field provides necessary centripetal force to the electron beam so that it can follow a circular path. Thus Force on an electron = Centripetal force due to magnetic field on an electron

The formula for radius of circular path is not valid at very high energies because such high energy

Question 22.
An electron gun with its collector at a potential of 100 V fires out electrons in a spherical bulb containing hydrogen gas at low pressure (10^-2 mm of Hg). A magnetic field of 2.83 × 10^-4 T curves the path of the electrons in a circular orbit of radius 12.0 cm. Determine e/m from the data.
Solution:

Question 23.
(a) An X-ray tube produces a continuous spectrum of radiation with its short wavelength end at 0.45 A. What is the maximum energy of a photon in the radiation?
(b) From your answer to (a), guess what order of accelerating voltage (for electrons) is required in such a tube ?
Solution:
(a) Minimum wavelength corresponds to maximum energy photons.


(b) In order to emit photons from the surface of energy 27.6 keV, the incident electrons striking on surface should have higher energy i.e., of the order of 30 keV.

Question 24.
In an accelerator experiment on high energy collisions of electrons with positrons, a certain event is interpreted as annihilation of an electron positron pair of total energy 10.2 BeV into two y-rays of equal energy. What is the wavelength associated with each y-ray? (1 BeV = 10⁹ eV).
Solution:
In the process of annihilation, the total energy of electron-positron pair is shared equally by both y ray photons produced. Energy of two y-rays = Energy of electron- positron pair = 10.2 BeV = 10.2 × 10⁹ eV Energy of each y-ray photon is E = 5.1 × 109 eV = 5.1 x 10⁹ x 1.6 x 10^-19 J = 5.1 × 1.6 x 10^-10 J But E = hv =   Hence, wavelength associated with y-ray is

Question 25.
Estimating the following two numbers should be interesting. The first number will tell you why radio engineers do not need to worry much about ‘photons’. The second number tells you why our eye can never’eount photons’ even barely detectable light.
(a) The number of photons emitted per , second by a MW transmitter of 10 kW power emitting radio waves of length 500 m.
(b) The number of photons entering the pupil of our eye per second corresponding to the minimum intensity of white light that we humans can perceive (~ 10^-10 W m^-2). Take the area of the pupil ‘to be about 0.4 cm² and the average frequency of white light to be about 6 × 10¹⁴ Hz.
Solution:
(a) Here, power of tansmitter,

exceedingly small and the number of photons emitted per second in a radio beam is enormously large. Therefore, negligible error involved in ignoring the existence of a minimum quantum of energy (photon) and treating the total energy of a radio wave as continuous.
(b) Here, area of the pupil, A = 0.4 cm² = 0.4 × 10^-4 m², v = 6 × 10¹⁴ Hz Intensity = 10^-10 W m^-2 Energy of a photon is given by, E = hv = 6.63 × 10^-34 × 6 × 10¹⁴ J = 4 × 10^-19 J.
If n = number of photons falling per sec per unit area, the energy per unit area per sec due to these photons = total energy of n photons = n × 4 × 10^-19 J nr² Since, intensity = energy per unit area per second


∴ Number of photons entering the pupil per second = n x area of the pupil = 2.5 × 10⁸ × 0.4 × 10^-4 s^-1 = 10⁴ s^-1.
Though this number is not large as in part (a) above, it is large enough for us to ‘sense’ or ‘count’ the individual photons by our eye.

Question 26.
Ultraviolet light of wavelength 2271 A from a 100 W mercury source irradiates a photo-cell made of molybdenum metal. If the stopping potential is -1.3 V, estimate the work function of the metal. How would the photo-cell respond to a high intensity (~ 10⁵ W m²) red light of wavelength 6328 A produced by a He-Ne laser?
Solution:
Let us find energy of each photon of given ultraviolet light

Maximum kinetic energy of emitted electron can be judged by stopping potential of 1.3 volt.

Thus energy of red light photons is less than work function 4.17 eV, hence irrespective of any intensity, no emission will take place.

Question 27.
Monochromatic radiation of wavelength 640.2 nm (1 nm = 10^-9 m) from a neon lamp irradiates photosensitive material made of caesium on tungsten. The stopping voltage is measured to be 0.54 V.The source is replaced by an iron source and its 427.2 nm line irradiates the same photo-cell. Predict the new stopping voltage.
Solution:
From the first data, work function of given photosensitive material can be calculated.

Now the source is replaced by iron source which produce 427.2 nm wavelength.

Question 28.
A mercury lamp is a convenient source for studying frequency dependence of photoelectric emission, since it gives a number of spectral lines ranging from the UV to the red end of the visible spectrum. In our experiment with rubidium photo-cell, the following lines from a mercury source were used :

The stopping voltages, respectively, were measured to be:

Determine the value of Planck’s constant h, the threshold frequency and work function for the material.
Solution:
In order to calculate Planck’s constant ‘ll’ we need slope of the graph between cut off voltage and frequency. So, let us first calculate the frequency (u = c/λ) in each case and the table shows corresponding stopping potential.

V₀ versus n plot shows that the first four points lie nearly on a straight line which intercepts the x-axis at threshold frequency, u₀ = 5.0 × 10¹⁴ Hz. The fifth point u (= 4.3 × 10¹⁴ Hz) corresponds to u < u₀, so there is no photoelectric emission and no stopping voltage is required to stop the current.
Slope of, V₀ versus u graph is


(b) threshold frequency, v₀ = 5.0 × 10¹⁴ J

Question 29.
The work function for the following metals is given:
Na: 2.75 eV; K: 2.30 eV; Mo: 4.17 eV; Ni: 5.15 eV. Which of these metals will notgive photoelectric emission for a radiation of wavelength 3300 A from a He-Cd laser placed 1 m away from the photocell? What happens if the laser is brought nearer and placed 50 cm away?
Solution:
The distance between laser source and receiver does not affect the energy of each photon incident, hence does not affect the energy of emitted photo electrons. But the reduction in distance will increase the intensity of incident light and hence number of photons. This will increase the photoelectric current. where, wavelength of incident radiation is

Now, work function of Mo: 4.17 eV, Ni: 5.15 eV is more than energy of incident photon, hence these two metals will not give photoelectric emission.

Question 30.
Light of intensity 10^-5 W m^-2 falls on a sodium photo-cell of surface area 2 cm². Assuming that the top 5 layers of sodium absorb the incident energy, estimate time required for photoelectric emission in the wave-picture of radiation. The work function for the metal is given to be about 2 eV. What is the implication of your answer?
Solution:
Wave picture of radiation state that incident energy is uniformly distributed among all the electrons continuously. Let us first calculate the total number of recipient electrons in-5 layers of sodium. Consider each sodium atom has one electron free as conduction electron. Effective atomic area = 10^-20 m² Number of conduction electrons in 5 layers

Where experimental observation shows that emission of photo electrons is instantaneous = 10^-9 sec Thus wave picture fails to explain photoelectric effect.

Question 31.
Crystal diffraction experiments can be perfor- med using X-rays, or electrons accelerated through appropriate voltage. Which probe has greater energy? An X-ray photon or the electron? (For quantitative comparison, take the wavelength of the probe equal to 1 A, which is of the order of inter-atomic spacing in the lattice) (m_e = 9.11 × 10^-31 kg).
Solution:
Order of interatomic spacing is 1 A in the crystal lattice. So, for diffraction to take place the wavelength should be of the same . order. For X-ray photon (energy for wavelength of 1 A)


Thus in order to produce same wavelength X-ray photon should have higher energy than electron.

Question 32.
(a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV. An electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equally suitable? Explain. (m_n = 1.675 × 10^-27 kg)
(b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature (27°C). Hence explain why a fast neutron beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments.
Solution:
(a) Let us first calculate the wavelength of matter wave associated with neutron of kinetic energy 150 eV.

As the inter atomic spacing (1 A= 10^-10 m) is about hundred times greater than this wavelength, so a neutron beam of 150 eV energy is not suitable for diffraction experiments.

(b) Average kinetic energy of a neutron at absolute temperature T is


As this wavelength is comparable to inter atomic spacing (= 1 A) in a crystal, so thermal neutrons can be used for diffraction experiments. So high energy neutron beam should be first thermalised before using it for diffraction.

Question 33.
An electron microscope uses electrons accelerated by a voltage of 50 kV. Determine the de-Broglie wavelength associated with the electrons. If other factors (such as numerical aperture etc.) are taken to be roughly the same, how does the resolving power of an electron microscope compare with that of an optical microscope which uses yellow light?
Solution:
K.E. of an electron, accelerated by voltage of 50 kV.

Thus, the resolving power of an electron microscope is about 103 times greater than that of an optical microscope.

Question 34.
The wavelength of a probe is roughly a measure of the size of a structure that it can probe in some detail. The quark structure of protons and neutrons appears at the minute length scale of 10^-15 m or less. This structure was first probed in early 1970’s using high energy electron beam produced by a linear accelerator at Stand ford, USA. Guess what might have been the order of energy of these electron beams. (Rest mass energy of electron = 0.511 M eV).
Solution:
Momentum of electron associated with given wavelength

Second term showing rest mass energy is negligible. Energy, E = 1.989 × 10^-10 J = 1.24 BeV Thus, energies acquired by electron from the given accelerator must have been of the order of a few BeV.

Question 35.
Find the typical de-Broglie wavelength associated with a He atom in helium gas at room temperature (27°C) and 1 atm pressure; and compare it with the mean separation between two atoms under these conditions.
Solution:
Let us first find mass ‘in’ of each helium atom.

Wavelength of the wave associated with He atom at room.temperature

Now let us find mean separation between He atoms. mean separation,

Comparing the wavelength ‘X’ with mean separation V, it can be observed that separa¬tion is larger than

Question 36.
Compute the typical de Broglie wavelength of an electron in a metal at 27° C and compare it with the mean separation between two electrons in a metal which is given to be about 2 × 10^-10 m.
Solution:
Considering free electrons as gas. Kinetic energy at temperature T

Given that mean separation between two electrons is about 2 × 10^-10 m.

So, de-Broglie wavelength is much greater than the electron separation.

Question 37.
Answer the following questions:
(a) Quarks inside protons and neutrons are thought to carry fractional charges [(+ 2/3)e ; {—1/3)e]. Why do they not show up in Millikan’s oil-drop experiment?
(b) What is so special about the combination e/m? Why do we not simply talk of e and m separately?
(c) Why should gases be insulators at ordinary pressures and start conducting at very low pressure?
(d) Every metal has a definite work function. Why do all photo electrons not come out with the same energy if incident radiation is monochromatic? Why is there an energy distribution of photo electrons?
(e) The energy and momentum of an electron are related to the frequency and wavelength of the associated matter wave by the relations: E = hv, p = h/λ
But while the value of X is physically significant, the value of u (and therefore, the value of phase speed uλ) has no physical significance. Why?
Solution:
(a) The quarks having fractional charges are thought to be confined within a proton and a neutron. These quarks are bound by forces. These forces become stronger when the quarks are tried to be pulled apart. That is why, the quarks always remain together. It is due to this reason that though fractional charges do exist in nature but the observable charges are always integral multiple of charge of electron.

(b) The motion of electron in the electric and magnetic field is related with the basic equations

All these equations involve e and m together, i.e., there is no equation in which e or m occurring alone. As a result of it, we study e/m of electron and do not talk of e and m separately for an electron.

(c) At ordinary pressures a few positive ions and electrons produced by the ionisation of the gas molecules by energetic rays (like X-rays, y-rays, cosmic rays etc. coming from outer space and entering the earth’s atmosphere) are not able to reach their respective electrodes, even at high voltages, due to their frequent collisions with gas molecules and recombinations. That is why the gases at ordinary pressures are insulators.
At low pressures, the density of the gas decreases, the mean free path of the gas molecules become large. Now under the effect of external high voltage, the ions acquire sufficient energy before they collide with molecules causing further ionisation. Due to it, the number of ions in the gas increases and it becomes a conductor.

(d) By work function of a metal, we mean the minimum energy required for the electron
in the highest level of conduction band to get out of the metal. Since all the electrons in the metal do not belong to that level but they occupy a continuous band of levels, therefore, for the given incident radiation, electrons knocked off from different levels come out with different energies.

(e) de broglie wavelength associated with the moving particle is

Energy of the wave is E= hv =  Energy of moving particle

NCERT Exercises

Question 1.
Monochromatic light of wavelength 589 nm is incident from air on a water surface. What are the wavelength, frequency and speed of
(a) reflected, and
(b) refracted light? Refractive index of water is 1.33.
Solution:
(a) In the process of reflection wavelength, frequency and speed of incident light remain unchanged. So, speed of reflected light = speed of incident light

(b) In the process of refraction wavelength and speed changes but the frequency remain the same. Speed of light in water

Question 2.
What is the shape of the wave front in each of the following cases:
(a) Light diverging from a point source.
(b) Light emerging out of a convex lens when a point source is placed at its focus.
(c) The portion of the wave front of light from a distant star intercepted by the Earth.
Solution:
(a) Spherical wave front : All particles vibrating in same phase will lie on a sphere.
(b) Plane wave front: Light will be a parallel beam after passing through the convex lens.
(c) Plane wave front : Light rays from a distant star are nearly parallel as a small portion of a huge spherical wave front is nearly plane.

Question 3.
(a) The refractive index of glass is 1.5. What is the speed of light in glass? (Speed of light in vacuum is 3.0 × 108 m s^-1)
(b) Is the speed of light in glass independent of the colour of light? If not, which of the two colours, red and violet, travels slower in a glass prism?
Solution:
(a) Speed of light in glass,

(b) Yes, speed of light in glass depends upon the colour of light (i.e., λ). Thus, speed of light is different for red and violet colours. As µ_v > µ_R so, λ_v < λ_R hence, υ_v < υ_R Speed of red colour is more than violet colour light in glass.

Question 4.
In a Young’s double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment.
Solution:
Here d = 0.28 mm, D = 1.4 m Distance of fourth bright fringe from center

Question 5.
In Young’s double-slit experiment using monochromatic light of wavelength X, the intensity of light at a point on the screen where path difference is X, is K units. What is the intensity of light at a point where path difference is λ/3?
Solution:
In Young’s double-slit experiment net intensity of light at a point on screen is

Question 6.
A beam of light consisting of two wavelengths, 650 nm and 520 nm is used to obtain interference fringes in a Young’s double-slit experiment.
(a) Find the distance of the third bright fringe on the screen from the central maximum for wavelength 650 nm.
(b) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide?
Solution:

(a) Distance of third bright fringe from the central maximum for the wavelength 650 nm.

(b) Let at linear distance ‘y’ from center of screen the bright fringes due to both wavelength coincides. Let n₁ number of bright fringe with wavelength λ₁ coincides with n2 number of bright fringe with wavelength λ₂. We can write

Also at first position of coincide, the nth bright fringe of one will coincide with (n + l) th bright fringe of other.

So, the fourth bright fringe of wavelength 520 nm coincides with 5th bright fringe of wavelength 650 nm.

Question 7.
In a double-slit experiment the angular width of a fringe is found to be 0.2° on a screen placed 1 m away. The wavelength of light used is 600 nm. What will be the angular width of the fringe if the entire experimental apparatus is immersed in water? Take refractive index of water to be 4/3.
Solution:


Question 8.
What is the Brewster angle for air to glass transition? (p for glass is 1.5)
Solution:

Question 9.
Light of wavelength 5000 A falls on a plane reflecting surface. What are the wavelength and frequency of the reflected light? For what angle of incidence is the reflected ray normal to the incident ray?
Solution:
In the reflected light the wavelength and frequency remain the same as that of incident light. Wavelength of reflected light = 5000 A, frequency of reflected light

For an angle of incidence 45° the reflected ray is normal to incident ray.

Question 10.
Estimate the distance for which ray optics is good approximation for an aperture of 4 mm and wavelength 400 nm.
Solution:
Fresnel distance required for a sufficient spreading of central bright fringe, so that diffraction is appreciable

So, for distance less than 40 m between slit and screen, ray optics is a good approximation as within this distance, the spreading is negligible.

Question 11.
The 6563 A Ha line emitted by hydrogen in a star is found to be red shifted by 15 A. Estimate the speed with which the star is receding from the earth.
Solution:

Since the star is receding away, hence its velocity v is negative (i.e. if AA is positive, v is negative)

Here, negative sign shows recession of star.

Question 12.
Explain how corpuscular theory predicts that the speed of light in a medium, say, water, is greater than the speed of light in vacuum. Is the prediction confirmed by experimental determination of the speed of light in water? If not, which alternative picture of light is consistent with experiment?
Solution:
In Newton’s corpuscular (particle) picture of refraction, particles of light incident from a rarer to a denser medium experience a force of attraction normal to the surface. This results in an increase in the normal component of velocity but the component along the surface remains unchanged.

Considering a ray of light going from a rarer medium (air) to a denser medium (water). Let c = speed of light in vacuum (or air), v = speed of light in water, i = angle of incidence, and r = angle of refraction Then according to Newton’s corpuscular theory,

Component of velocity c along surface of separation = Component of velocity v along the surface of separation

So, according to Newton’s corpuscular theory the speed of light in medium is larger than speed of light in air. υ > c but in fact the experimental observation shows that speed of light is smaller in denser medium as compared to rare medium υ < c.

Question 13.
You have learnt in the text how Huygens principle leads to the laws of reflection and refraction. Use the same principle to deduce directly that a point object placed in front of a plane mirror produces a virtual image whose distance from the mirror is equal to the distance of the object from the mirror.
Solution:

In figure, P is a point object placed at a distance . r from a plane mirror M₁M₂. With P as centre and PO = r as radius, draw a spherical arc; AB. This is the spherical wave front from the object, incident on M₁M₂ If mirrors were not present, the position of wave front AB would be A’B’ where PP’ = 2r. In the presence of the mirror, wave front A B would appear as A”PB”, according to Huygen’s construction. As it is clear from the figure A’B’ and A”B” are two spherical arcs located symmetrically on either side of M₁M₂. , Therefore, A’P’B’ can be treated as reflected image of A”PB”. From simple geometry, we find OP = OP’, which was to be proved.

Question 14.
Let us list some of the factors, which could possibly influence the speed of wave propagation:
(i) nature of the source.
(ii) direction of propagation.
(iii) motion of the sources and/or observer.
(iv) wavelength.
(v) intensity of the wave. On which of these factors, if any, does
(a) the speed of light in vacuum,
(b) the speed of light in a medium (say glass or water) depend?
Solution:
(a) Speed of light in vacuum is independent of all the factors listed above. It is also independent of relative motion between source and observer.
(b) Dependence of speed of light in a medium.
(i) The speed of light in a medium does not depend on the nature of the source. Although speed is determined by the properties of the medium of propagation.
(ii) The speed of light in a medium is independent of the direction of propagation for an isotropic media.
(iii) The speed of light is independent of the motion of the source relative to the medium but it depends upon the motion of the observer relative to the medium.
(iv) The speed of light in a medium depends on wavelength of light i.e., v ∝ λ..
(v) The speed of light in a medium is independent of intensity.

Question 15.
For sound waves, the Doppler formula for frequency shift differs slightly between the two situations :
(i) source at rest; observer moving, and
(ii) source moving; observer at rest.
The exact Doppler formulas for the case of light waves in vacuum are, however, strictly identical for these situations. Explain why this should be so. Would you expect the formulas to be strictly identical for the two situations in case of light travelling in a medium?
Solution:
Sound waves require a medium for propagation. The Doppler formula for frequency shift differs slightly in two situations
(i) Source at rest, observer moving

(ii) Observer at rest, source moving

The two formulas are different because motion of the observer relative to the medium is different in the two situations for light waves in vacuum. No such relative relation of observer and medium exist. Hence only the relative motion between the source and the observer counts and the

Question 16.
In double-slit experiment using light of wavelength 600 nm, the angular width of a fringe formed on a distant screen is 0.1°. What is the spacing between the two slits?
Solution:

Question 17.
Answer the following questions:
(a) In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band?
(b) In what way is diffraction from each slit related to the interference pattern in a double-slit experiment?
(c) When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the shadow of the obstacle. Explain why?
(d) Two students are separated by a 7 m partition wall in a room 10 m high. If both light and sound waves can bend around obstacles, how is it that the students are unable to see each other even though they can converse easily.
(e) Ray optics is based on the assumption that light travels in a straight line. Diffraction effects (observed when light propagates through small apertures/slits or around small obstacles) disprove this assumption. Yet the ray optics assumption is so commonly used in understanding location and several other properties of images in optical instruments. What is the justification?
Solution:
(a) Linear width of central maximum

On doubling the slit width ‘d’, the size of central diffraction band is halved.
Because the width of central maximum is halved. Its area become 1/4 times and hence the intensity become 4 times the initial intensity.
(b) In double slit experiment, an interference pattern is observed by waves from two slits but as each slit provide a diffraction pattern of its own, thus the intensity of interference pattern in Young’s double slit experiment is modified by diffraction pattern of each slit.
(c) Waves from the distant source are diffracted by the edge of the circular obstacle and these waves superimpose constructively at the centre of obstacle’s shadow producing a bright spot.
(d) We know for diffraction to take place, size of the obstacle/aperture should be of the order of wavelength. Wavelength of sound waves is of the order of few meters that is why sound waves can bend through the aperture in partition wall but wavelength of light waves is of the order of micrometer, hence light waves can not bend through same big aperture. That is why the two students can hear each other but cannot see each other.

(e) In optical instruments, the sizes of apertures are much larger as compared to wavelength of light. So the diffraction effects are negligibly small. Hence, the assumption that light travels in straight lines is used in the optical instruments.

Question 18.
Two towers on top of two hills are 40 km apart. The line joining them passes 50 m above a hill halfway between the towers. What is the longest wavelength of radio waves, which can be sent between the towers without appreciable diffraction effects?
Solution:

For diffraction of radio waves not to occur the distance of middle hill should be less than fresnel distance for a slit width ‘a’ of 50 m.

Distance between one of the towers and the hill halfway in between

Longest wavelength of radio wave which can be sent without appreciable diffraction effect

Thus wavelength of radio waves longer than 12.5 cm will bend due to the hill in the middle of towers.

Question 19.
A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum is at a distance of 2.5 mm from the centre of the screen. Find the width of the slit.
Solution:

First minimum is observed at a distance 2.5 mm from centre of the screen.


Question 20.
Answer the following questions:
(a) When a low-flying aircraft passes overhead, we sometimes notice a slight shaking of the picture on our TV screen. Suggest a possible explanation.
(b) As you have learnt in the text, the principle of linear superposition of wave displacements is basic to understanding intensity distributions in diffraction and interference patterns. What is the justification of this principle?
Solution:
(a) The low flying aircraft reflects the TV signals. Due to superposition between the direct signal received by the antenna and the reflected signals from aircraft. We sometimes notice slight shaking of the picture on the TV screen.
(b) Superposition principle states how to explain the formation of resultant wave by combination of two or more waves. Let and y₂ represent instantaneous displacement of two superimposing waves, then resultant waves instantaneous displacement is given by

Question 21.
In deriving the single slit diffraction pattern, it was stated that the intensity is zero at angles of nλ/a. Justify this by suitably dividing the slit to bring out the cancellation.
Solution:
Let the single-slit of width a be divided into n smaller slits. If a’ is the width of each one of the smaller slits, a’ = a/n. For the single¬slit to produce zero intensity, each one of the smaller slits should also produce zero intensity? This is possible if

Question 1.
A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Describe the nature and size of the image. If the candle is moved closer to the mirror, how would the screen have to be moved?
Solution:
The object is kept between ƒ and C. So the image should be real, inverted and beyond C. To locate the sharp image, screen should be placed at the position of image.

So, the image is inverted and magnified. Thus in order to locate the sharp image, the screen should be kept 54 cm in front of concave mirror and image on the screen will be observed real, inverted and magnified. If the candle is moved closer to the mirror, the real image will move away from the mirror, hence screen has to be shifted away from the mirror to locate the sharp image.

Question 2.
A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.
Solution:
A convex mirror always form virtual image, which is erect and small in size of an object kept in front of it. Focal length of convex mirror ƒ = + 15 cm Object distance u = – 12 cm Using mirror formula

Therefore, image is virtual, formed at 6.67 cm at the back of the mirror.

It shows the image is erect, small in size and virtual. When the needle is moved farther from mirror the image also move towards focus and decreasing in size. As u approaches v approaches focus but never beyond ƒ.

Question 3.
A tank is filled with water to a height of 12.5 cm. The apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be 9.4 cm. What is the refractive index of water? If water is replaced by a liquid of refractive index 1.63 upto the same height, by what distance would the microscope have to be moved to focus on the needle again?
Solution:
We know the relation

Now if the water is replaced by other liquid, the apparent depth will change and microscope will have to be further moved to focus the image. With new liquid

Now the microscope will have to shift from its initial position to focus on the needle again which is at 7.67 cm depth. Shift distance = 9.4 – 7.67 = 1.73 cm.

Question 4.
Figures (a) and (b) show refraction of a ray in air incident at 60° with the normal to a glass- air and water-air interface, respectively. Predict the angle of refraction in glass when the angle of incidence in water is 45° with the normal to a water-glass interface (figure c).

Solution:
(a) Applying Snell’s law for the refraction from air to glass. Refractive index of glass w.r.t. air

(b) Now Snell’s law for the refraction from air to water

(c) Now the light beam is incident at an angle 45° from water to glass

Question 5.
A small bulb is placed at the bottom of a tank containing water to a depth of 80 cm. What is the area of the surface of water through which light from the bulb can emerge out? Refractive index of water is 1.33. (Consider the bulb to be a point source)
Solution:
As shown in the figure all those light rays which are incident on the surface at angle of incidence more than critical angle, does total internal reflection and are reflected back in water only. All those light rays which are incident below critical angle emerges out of surface bending away from normal. All those light beams which are incident at critical angle grazes the surface of water.

Question 6.
A prism is made of glass of unknown refractive index. A parallel beam of light is incident on a face of the prism. The angle of minimum deviation is measured to be 40°. What is the refractive index of the material of the prism? The refracting angle of the prism is 60°. If the prism is placed in water (refractive index 1.33), predict the new angle of minimum deviation of a parallel beam of light.
Solution:
When the light beam is incident from air on to the glass prism, the angle of minimum deviation is 40°. Refractive index of glass w.r.t. air.


Now the prism is placed in water, new angle of minimum deviation can be calculated.

Question 7.
Double-convex lenses are to be manufactured from a glass of refractive index 1.55, with both faces of the same radius of curvature. What is the radius of curvature required if the focal length is to be 20 cm?
Solution:
Both faces should be of same radius of curvature

So, the radius of curvature should be 22 cm for each face of lens.

Question 8.
A beam of light converges at a point P. Now a lens is placed in the path of the convergent beam 12 cm from P. At what point does the beam converge if the lens is
(a) a convex lens of focal length 20 cm, and
(b) a concave lens of focal length 16 cm?
Solution:
(a) The convex lens is placed in the path of convergent beam.

The image 1 is formed by further converging beams at a distance of 7.5 cm from lens.
(b) A concave lens is placed in the path of convergent’ beam, the concave lens further diverge the light.

The image I is formed by diverged rays at 48 cm away from concave lens.

Question 9.
An object of size 3.0 cm is placed 14 cm in front of a concave lens of focal length 21 cm. Describe the image produced by the lens. What happens if the object is moved further away from the lens?
Solution:
Object of size 3 cm is placed 14 cm in front of concave lens.


So, the image is virtual, erect, of the size 1.8 cm and is located 8.4 cm from the lens on the same side as object. As the object is moved away from the lens, the virtual image moves towards the focus of the lens but never beyond it. The image also reduces in size as shift towards focus.

Question 10.
What is the focal length of a convex lens of focal length 30 cm in contact with a concave lens of focal length 20 cm? Is the system a converging or a diverging lens? Ignore thickness of the lenses.
Solution:
Equivalent focal length of the combination

Hence, system will behave as a diverging lens of focal length 60 cm.

Question 11.
A compound microscope consists of an objective lens of focal length 2.0 cm and an eyepiece of focal length 6.25 cm separated by a distance of 15 cm. How far from the objective should an object be placed in order to obtain the final image at (a) the least distance of distinct vision (25 cm), and (b) at infinity? What is the magnifying power of the microscope in each case?
Solution:
(a) We want the final image at least distance of distinct vision. Let the object in front of objective is at distance υ₀.

Now we can get required position of object in point of objective.

(b) We want the final image at infinity. Let us again assume the object in front of objective at distance υ_0.

The object distance υ_e for the eyepiece should be equal to ƒ_e = 6.25 cm to obtain final image at ∞. So, image distance of objective lens

Question 12.
A person with a normal near point (25 cm) using a compound microscope with objective of focal length 8.0 mm and an eyepiece of focal length 2.5 cm can bring an object placed at 9.0 mm from the objective in sharp focus. What is the separation between the two lenses? Calculate the magnifying power of the microscope.
Solution:
The image is formed at least distance of distinct vision for sharp focus. The separation between two lenses will be υ₀ + |υ_e|

Let us find first υ₀ the image distance for objective lens.

Also we can find object distance for eyepiece υ_e as we know

Question 13.
A small telescope has an objective lens of focal length 144 cm and an eyepiece of focal length 6.0 cm. What is the magnifying power of the telescope? What is the separation between the objective and the eyepiece?
Solution:

Question 14.
(a) A giant refracting telescope at an observatory has an objective lens of focal length 15 m. If an eyepiece of focal length 1.0 cm is used, what is the angular magnification of the telescope?
(b) If this telescope is used to view the moon, what is the diameter of the image of the moon formed by the objective lens? The diameter of the moon is 3.48 × 10⁶ m and the radius of lunar orbit is 3.8 × 10⁸ sm.
Solution:
(a) ƒ_o = 15 m and ƒ_e = 1.0 cm angular magnification by the telescope normal adjustment

(b) The image of the moon by the objective at lens is formed on its focus only as the moon is nearly infinite distance as compared to focal length.

Distance of object i.e., Radius of lunar orbit, R_o = 3.8 × 10⁸ cm Distance of image for objective lens i.e., focal length of objective lens ƒ_o = 15 m Radius of image of moon by objective lens can be calculated.

Question 15.
Use the mirror equation to deduce that:
(a) An object placed between f and 2f of a concave mirror produces a real image beyond 2 f.
(b) A convex mirror always produces a virtual image independent of the location of the object.
(c) The virtual image produced by a convex mirror is always diminished in size and is located Between the focus and the pole.
(d) An object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.
Solution:
(a) We know for a concave mirror ƒ < 0 [negative] and u < 0 [negative]

(b) For a convex mirror, ƒ > 0, always positive and object distance u < 0, always negative.

So, whatever be the value of u, a convex mirror always forms a virtual image.
(c) In convex mirror focal length is positive hence ƒ > 0 and for an object distance from mirror with negative sign (u < 0)

hence the image is located between pole and focus of the mirror

So, the image is virtual and diminished.
(d) In concave mirror, ƒ < 0 for object placed between focus and pole of concave mirror

Question 16.
A small pin fixed on a table top is viewed from above from a distance of 50 cm. By what distance would the pin appear to be raised if it is viewed from the same point through a 15 cm thick glass slab held parallel to the table? Refractive index of glass = 1.5. Does the answer depend on the location of the slab?
Solution:
The shift in the image by the thick glass slab can be calculated. Here, shift only depend upon thickness of glass slab and refractive index of glass.
Shift = Real thickness – Apparent of thickness

The answer does not depend on the location of the slab.

Question 17.
(a) Figure shows a cross-section of’light pipe’ made of a glass fiber of refractive index 1.68. The outer covering of the pipe is made of a material of refractive index 1.44. What is the range of the angles of the incident rays with the axis of the pipe for which total reflections inside the pipe take place, as shown in the figure.
(b) What is the answer if there is no outer covering of the pipe?

Solution:

(a) Let us first derive the condition for total internal reflection. Critical angle for the interface of medium 1 and medium 2.

Condition for total internal reflection from core to cladding

Now, for refraction at first surface air to core.

Thus all incident rays which makes angle of incidence between 0° and 60° will suffer total internal reflection in the optical fiber.
(b) When there is no outer covering critical angle from core to surface.

Thus all incident rays at first surface 0° to 90° will suffer total internal reflection inside core.

Question 18.
Answer the following questions:
(a) You have learnt that plane and convex mirrors produce virtual images of objects. Can they produce real images under some circumstances? Explain.
(b) A virtual image, we always say, cannot be caught on a screen. Yet when we’see’a virtual image, we are obviously bringing it on to the ‘screen’ (i.e., the retina) of our eye. Is there a contradiction?
(c) A diver under water, looks obliquely at a fisherman standing on the bank of a lake.
(d) Does the apparent depth of a tank of water change if viewed obliquely? if so, does the apparent depth increase of decrease?
(e) The refractive index of diamond is much greater than that of ordinary glass. is this fact of some use to a diamond cutter?
Solution:
(a) In this situation when rays are convergent behind the mirror, both plane mirror and convex mirror can form real images of virtual objects.

(b) Here, the retina is working as a screen, where the rays are converging, but this screen is not at the position of formed virtual image, in fact the reflected divergent rays are converged by the eye lens at retina. Thus, there is no contradiction.

(c) An observer in denser medium will observe the fisherman taller than actual height, due to refraction from rare to denser medium.

(d) Apparent depth decreases if viewed obliquely as compared to when observed near normally.
(e) As hence, sin   refractive

index of diamond is much greater than that of ordinary glass, hence critical angle C for diamond is much smaller (24°) as compared to that of glass (42°).
A skilled diamond cutter thus can take the advantage of such large range of angle of incidence available for total internal reflection 24° to 90°. The diamond can be cut with so many faces, to ensure that light entering the diamond does multiple total internal reflections before coming out. This behavior produce brilliance i.e., sparkling effect in the diamond.

Question 19.
The image of a small electric bulb fixed on the wall of a room is to be obtained on the opposite wall 3 m away by means of a large convex lens. What is the maximum possible focal length of the lens required for the purpose?
Solution:
Let the object be placed x m in front of lens the distance of image from the lens is (3 – x) m.

Condition for image to be obtained on the screen, i.e.m real image. 9 – 12ƒ > 0 or 9 > 12ƒ or f < 0.75 m. so, maximum focal length is  0.75 m.

Question 20.
A screen is placed 90 cm from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20 cm. Determine the focal length of the lens.
Solution:
The image of the object can be located on the screen for two positions of convex lens such that u and v are exchanged.
The separation between two positions of the lens is x = 20 cm. It can be observed from figure.

Question 21.
(a) Determine the ‘effective focal length of the combination of two lenses in question 10, if they are placed 8.0 cm apart with their principal axes coincident. Does the answer depend on which side of the combination a beam of parallel light is incident? Is the notion of effective focal length of this system useful at all?
(b) An object 1.5 cm in size is placed on the side of the convex lens in the above arrangement. The distance between the object and the convex lens is 40 cm. Determine the magnification produced by the two-lens system, and the size of the image.
Solution:
(a)
(i) Let a parallel beam of light incident first on convex lens, refraction at convex lens


The parallel beam of light appears to diverge from a point 216 cm from the center of the two lens system.
(ii) Now let a parallel beam of light incident first on concave lens.

The image I1 will act as real object for convex lens at 28 cm.

Thus the parallel incident beam appears to diverge from a point 420 -4 = 416 cm on the left of the center of the two lens system. Hence the answer depend upon which side of the lens system the parallel beam is made incident. Therefore the effective focal length is different in two situations.

(b) Now an object of 1.5 cm size is kept 40 cm in front of convex lens in the same system of lenses.

Question 22.
At what angle should a ray of light be incident on the face of a prism of refracting angle 60° so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524.
Solution:
The beam should be incident at critical angle or more than critical angle, for total internal reflection at second surface of the prism.

Let us first find critical angle for air glass interface.

Question 23.
You are given prisms made of crown glass and flint glass with a wide variety of angles. Suggest a combination of prisms which will
(a) deviate a pencil of white light without much dispersion.
(b) disperse (and displace) a pencil of white light without much deviation.
Solution:
By using two identical shape prism of crown glass and flint glass kept with their base on opposite sides, we can observe deviation without dispersion or dispersion without deviation.
(a) Deviation without dispersion A white beam incident on crown glass will suffer deviation 6, and angular dispersion Δθ, both

Now the light beam again suffer deviation and dispersion by flint glass prism.


Negative sign shows that two prisms must be placed with their base on opposite sides. Net deviation in this condition

(b) Dispersion without deviation

Here again negative sign shows that two prisms must be placed with their base on opposite sides. Net angular dispersion
Δθ = Δθ₁ + Δθ₂
Δθ = A₁ (μ_v1 – μ_R1) + A₂ (μ_v2 – μ_R2)

Question 24.
For a normal eye, the far points at infinity and the near point of distinct vision is about 25 cm in front of the eye. The cornea of the eye provides a converging power of about 40 dioptres, and the least converging power of the eye lens behind the cornea is about 20 dioptres. From this rough data estimate the range of accommodation (i.e., the range of converging power of the eye-lens) of a normal eye.
Solution:
Here the least converging power of eye lens is given as 20 dioptres behind the cornea. If we can calculate the maximum converging power, then we can get the range of accommodation of a normal eye.
(a) To see objects at infinity, the eye uses its least converging power = 40 + 20 = 60 dioptres
∴ Approximate distance between the retina and the cornea eyelens

(b) To focus object at the near point on the retina, we have

dioptres Power of the eye lens = 64 – 40 = 24 dioptres Thus the range of accommodation of the eye lens is approximately 20 to 24 dioptres.

Question 25.
Does short-sightedness (myopia) or long¬> sightedness (hypermetropia) imply necessarily that the eye has partially lost its ability of r accommodation? If not, what might cause these defects of vision?
Solution:
A person with normal ability of accommodation may be myopic or hypermetropic due to defective eye structure, j- When the eye ball from front to back gets too elongated the myopic defect occur, similarly when the eye ball from front to back gets too shortened the hypermetropia defect occur.
When the eye ball has normal length ‘ but the eye lens loses partially its ability of accommodation, the defect is called “presbyopia” and is corrected in the same I manner as myopia or hypermetropia.

Question 26.
A myopic person has been using spectacles of power – 1.0 dioptre for distant vision. During old age, he also needs to use separate reading glass of power + 2.0 dioptres. Explain what may have happened.
Solution:
Myopic person uses spectacles of power – 1.0 dioptre or concave lens of focal length ƒ =  = -100 cm in order to observe clearly object at infinity. Far point of the person can be calculated as

Similarly if the person uses spectacles of power + 2.0 dioptre then he must be using convex lens of power + 2.0 dioptre. Focal length and near point can be calculated as

Thus the person also has the defect of hypermetropia and has a near point 50 cm. So having both defects he needs different lenses for distant vision and to see closer objects.

Question 27.
A person looking at a person wearing a shirt with a pattern comprising vertical and horizontal lines is able to see the vertical lines more distinctly than the horizontal ones. What is this defect due to? How is such a defect of vision corrected?
Solution:
This defect is called astigmatism. It arises due to non spherical cornea. The eye lens is ideally spherical and has same curvature in different planes, but in an astigmatic eye due to non spherical cornea the curvature may be insufficient in different planes.
In the given situation the curvature in the vertical plane is enough, so vertical lines are visible distinctly. But the curvature is insufficient in the horizontal plane, hence horizontal lines appear blurred. The defect can be corrected by using a cylindrical lens with its axis along vertical. The parallel rays in the vertical plane will suffer no extra refraction but the parallel rays in the horizontal plane will be refracted largely and converges at the retina, according to the requirement to form the clear image of horizontal lines.

Question 28.
A man with normal near point (25 cm) reads a book with small print using a magnifying glass: a thin convex lens of focal length 5 cm.
(a) What is the closest and the farthest distance at which he should keep the lens from the page so that he can read the book when viewing through the magnifying glass?
(b) What is the maximum and the minimum angular magnification (magnifying power) possible using the above simple microscope?
Solution:
(a) At closest distance of the object the image is formed at least distance of distinct vision and eye is most strained.

At farthest distance of the object the image is formed at ∞ and eye is most relaxed.

(b) Maximum angular magnification

Question 29.
A card sheet divided into squares each of size 1 mm² is being viewed at a distance of 9 cm through a magnifying glass (a converging lens of focal length 10 cm) held close to the eye.
(a) What is the magnification produced by the lens? How much is the area of each square in the virtual image?
(b) What is the angular magnification (magnifying power) of the lens?
(c) Is the magnification in (a) equal to the magnifying power in (b)? Explain.
Solution:
(a) For magnification by the magnifying lens.

(b) Angular magnification,

(c) No, the linear magnification by a lens and magnifying power (angular magnification) of magnifying glass have different values. The linear magnification is calculated using

image of object at eye lens ‘p’ to the angle subtended by object assumed to be at least distance at eye lens ‘a’.
The linear magnification and angular magnification in microscope have similar magnitude when image is at least distance of distinct vision i.e., 25 cm.

Question 30.
(a) At what distance should the lens be held from the figure in previous question in order to view the squares distinctly with the maximum possible magnifying power?
(b) What is the magnification in this case?
(c) Is the magnification equal to the magnifying power in this case? Explain.
Solution:
(a) For maximum magnifying power the image should be at least distance of distinct 1 vision i.e., 25 cm.

(b) Linear magnification in the situation of maximum magnifying power.

(c) Maximum magnifying power in the same situation

So, it can be observed that in the situation when image is least distance of distinct vision the angular magnification and linear magnification have similar values.

Question 31.
What should be the distance between the object in previous question and the magnifying glass if the virtual image of each square in the figure is to have an area of 6.25 mm². Would you be able to see the squares distinctly with your eyes very close to the magnifier?
Solution:
Now we want the area of square shaped virtual image as 6.25 mm². So, each side of image is I =  = 2.5 mm (Linear magnification) For the given magnifying lens of focal length 10 cm we can calculate the required position of object.

Thus the required virtual image is closer than normal near point. Thus the eye cannot observe the image distinctly.

Question 32.
Answer the following question:
(a) The angle subtended at the eye by an object is equal to the angle subtended at the eye by the virtual image produced by a magnifying glass. In what sense then does a magnifying glass provide angular magnification?
(b) In viewing through a magnifying glass, one usually positions one’s eyes very close to the lens. Does angular magnification change if the eye is moved back?
(c) Magnifying power of a simple microscope is inversely proportional to the focal length of the lens. What then stops us from using a convex lens of smaller and smaller focal length and achieving greater and greater magnifying power?
(d) Why must both the objective and the eyepiece of a compound microscope have short focal lengths?
(e) When viewing through a compound microscope, our eyes should be positioned not on the eyepiece but a short distance away from it for best viewing. Why? How much should be that short distance between the eye and eyepiece?
Solution:
(a) In magnifying glass the object is placed closer than 25 cm, which produces image at 25 cm. This closer object has larger angular size than the same object at 25 cm. In this way although the angle subtended by virtual image and object is same at eye but angular magnification is achieved.

(b) On moving the eye backward away from lens the angular magnification decreases slightly, as both the angle subtended by the
image at eye ‘a’ and by the object at eye ‘α’ decreases. Although the decrease in angle subtended by object a is relatively smaller.

(c) If we decrease focal length, the lens has to be thick with smaller radius of curvature. In a thick lens both the spherical aberrations and chromatic aberrations become pronounced. Further, grinding for small focal length is not easy. Practically we can not get magnifying power more than 3 with a simple convex lens.

(d) Magnifying power of a compound microscope is given by

(e) If we place our eye too close to the eyepiece, we shall not collect much of the light and also reduce our field of view. When we position our eye slightly away and the area of the pupil of our eye is greater, our eye will collect all the light refracted by the objective, and a clear image is observed by the eye.

Question 33.
An angular magnification (magnifying power) of 30 X is desired using an objective of focal length 1.25 cm and an eyepiece of focal length 5 cm. How will you set up the compound microscope?
Solution:
Here we want the distance between given objective and eye lens for the required magnification of 30. Let the final image is formed at least distance of distinct vision for eyepiece.

Question 34.
A small telescope has an objective lens of focal length 140 cm and an eyepiece of focal length 5.0 cm. What is the magnifying power of the telescope for viewing distant objects when:
(a) the telescope is in normal adjustment (i.e., when the final image is at infinity)?
(b) the final image is formed at the least distance of distinct vision (25 cm)?
Solution:
(a) In normal adjustment magnifying power

(b) For the image at least distance of distinct vision

Question 35.
(a) For the telescope described in previous question 34 (a), what is the separation between the objective and the eyepiece?
(b) If this telescope is used to view a 100 m tall tower 3 km away. What is the height of the image of the tower formed by the objective lens?
(c) What is the height of the final image of the tower if it is formed at 25 cm?
Solution:
(a) The separation between objective lens and the eyepiece can be calculated in both the conditions of most relaxed eye and most strained eye. Most relaxed eye L= ƒ₀ + ƒ_e= 140 + 5 = 145 cm
Most strained eye object distance ‘u_e‘ for eye lens

(b)

(c) Now we want to find the height of final image A”B” assuming it to be formed at 25 cm.

Question 36.
A Cassegrain telescope uses two mirrors as shown in figure. Such a telescope is built with the mirrors 20 mm apart. If the radius of curvature of the large mirror is 220 mm and the small mirror is 140 mm, where will the final image of an object at infinity be?

Solution:
Image formed by concave mirror acts as a virtual object for convex mirror. Here parallel rays coming from infinity will focus at 110 mm an axis away from concave mirror. Distance of virtual object for convex mirror

Hence image is formed at 315 mm from convex mirror.

Question 37.
Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards as shown in figure. A current in the coil produces a deflection of 3.5° of the mirror. What is the displacement of the reflected spot of light on a screen placed 1.5 m away?

Solution:
If the mirror deflect by 3.5°, the reflected light deflect by 7°, deflection of the spot d can be calculated.

Question 38.
Figure shows an equiconvex lens (of refractive index 1.50) in contact with a liquid layer on top of a plane mirror. A small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be 45.0 cm. The liquid is removed and the experiment is repeated. The new distance is measured to be 30.0 cm. What is the refractive index of the liquid?

Solution:
Let us first consider the situation when there is no liquid between lens and plane mirror and the image is formed at 30 cm i.e., at the position of object. As the image is formed on the object position itself, the object must be placed at focus of Biconvex lens.

ƒ₀ = 30 cm Radius of curvature of convex lens can be calculated

Now a liquid is filled between lens and plane mirror and the image is formed at position of object at 45 cm. The image is formed on the position of object itself, the object must be placed at focus of equivalent lens of Biconvex of glass and Plano convex lens of liquid

NCERT Exercises

Question 1.
Figure shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 mm. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15 A.

(a) Calculate the capacitance and the rate of change of potential difference between the plates.
(b) Obtain the displacement current across the plates.
(c) Is Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain.
Solution:

(a) Capacitance of the parallel plate capacitor

(b) The magnitude of displacement current is equal to conduction current. ∴ I_d = I = 0.15 A
(c) Yes, Kirchhoff’s first law is very much applicable to each plate of capacitor as l_d = I. So current is continuous and constant across each plate.

Question 2.
A parallel plate capacitor as shown in figure made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad s^-1.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of 6 at a point 3.0 cm from the axis between the plates.

Solution:
(a) Capacity of capacitor C = 100 pF


(b) Yes, the conduction current in wires is always equal to displacement current within plates.
(c)

To find magnetic field B of a point 3 cm from the axis within plates, let us assume a loop of radius 3 cm with center on axis. Now modified Ampere’s Law.

Question 3.
What physical quantity is same for X-rays of wavelength 10^-10 m, red light of wavelength 6800 A and radio waves of wavelength 500 m?
Solution:
Though they all have different wavelengths and frequencies, but they have same speed i.e., speed of light 3 × 108 m s^-1 in vacuum.

Question 4.
A plane electromagnetic wave travels in vacuum along z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is 30 MHz, what is its wavelength?
Solution:
In electromagnetic wave, the electric field vector E and magnetic field vector B show their variations perpendicular to the direction of propagation of wave as well as perpendicular to each other. As the electromagnetic wave is travelling along z-direction, hence E and B show their variation in x-y plane.

Question 5.
A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band?
Solution:
Maximum wavelength in the band is for lowest frequency

Minimum wavelength in the band is for highest frequency

Question 6.
A charged particle oscillates about its mean equilibrium position with a frequency of 109 Hz. What is the frequency of electromagnetic wave produced by the oscillator?
Solution:
The frequency of electromagnetic wave produced will be same as the frequency of oscillation of charged particles i.e., 109 Hz.

Question 7.
The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum is B₀ = 510 nT. What is the amplitude of the electric field part of the wave?
Solution:

Question 8.
Suppose that the electric field amplitude of an electromagnetic wave is E₀ = 120 N C^-1 and that its frequency is u = 50.0 MHz. (a) Determine B₀, to, k, and X.
(b) Find expressions for E and B.
Solution:
(a) By relation, 

(b) Let the wave is propagating along A’-direction, electric field E is along y – direction and magnetic field B along z-direction.

Question 9.
The terminology of different parts of the electromagnetic spectrum is given in the NCERT text. Use the formula E = hυ and obtain the photon energy in units of eV for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation?
Solution:
The photon of given energy are released during transition between energy levels in the atom and the emitted photon energy is equal to difference of energies of energy levels, among which the transition has taken place. For example photon energy for
λ = 1 m can be calculated

So, for emission of radio waves the energy difference between energy levels should be 12.425 × 10^-7 eV. Similarly, photon energy for other wavelengths can also be calculated

So, for emission of y -rays the energy difference among energy levels should be of the order of MeV, where for visible radiation it should be of the order of eV.

Question 10.
In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 10¹⁰ Hz and amplitude 48 Vm^-1.
(a) What is the wavelength of the wave?
(b) What is the amplitude of the oscillating magnetic field?
(c) Show that the average energy density of the E field equals the average energy density of the B field, [c = 3 × 10⁸ m s^-1.]
Solution:
(a) Wavelength of electromagnetic wave is

(b) Amplitude of magnetic field, 

(c) Energy density as electric field,

Question 11.
Suppose that the electric field part of an electromagnetic wave in vacuum is £ = [(3.1 N/C)] cos[(1.8 rad/m) y + (5.4 x 10⁸ rad/s)f)] i.
(a) What is the direction of propagation?
(b) What is the wavelength λ?
(c) What is the frequency υ?
(d) What is the amplitude of the magnetic field part of the wave?
(e) Write an expression for the magnetic field part of the wave.
Solution:
(a) Electromagnetic wave is propagating in -y direction.
(b) Propagation constant, k =  = 1.8

(c) angular frequency, ω = 2πυ = 5.4 × 10⁸

(d) Amplitude of magnetic field pat, 

(e) Expression for magnetic field pat of wave

Question 12.
About 5% of the power of a 100 W light bulb is converted to visible radiation. What is the average intensity of visible radiation
(a) at a distance of 1 m from the bulb?
(b) at a distance of 10 m?
Assume that the radiation is emitted is o tropically and neglect reflection.
Solution:
The bulb as a point source, radiate light in all direction uniformly and it is given that only 5% of power is converted to visible radiations.
(a) Let us assume a sphere of radius 1 m. Surface area A = 4πr² = 4π (l)² = 4 π m² Intensity on the sphere

(b) Let us assume a sphere of 10 m radius. Surface area A = 4π² = 4K (10)² = 400 π m² Intensity on the sphere

Question 13.
Use the formula λ_mT = 0.29 cm K to obtain the characteristic temperature ranges for different parts of the electromagnetic spectrum. What do the numbers that you obtain tell you?
Solution:
A black body at very high temperature produce a continuous spectrum. Using Wien’s displacement law we can calculate the wavelength corresponding to maximum intensity of radiation emitted Required absolute temperature

Using above formula, the temperature of black body required for various wavelength is calculated.

To produce electromagnetic radiations of different wavelength, we need temperature ranges. To produce visible ’ radiation of X = 5 × 10^-7 m, we need to have source at temperature of 6000 K. A source at lower temperature will produce this wavelength but not with maximum intensity.

Question 14.
Given below are some famous numbers associated with electromagnetic radiations in different contexts in Physics. State the part of the electromagnetic spectrum to which each belongs.
(a) 21 cm [wavelength emitted by atomic ‘hydrogen in interstellar space].
(b) 1057 MHz [frequency of radiation arising from two close energy levels in hydrogen; known as Lamb shift].
(c) 2.7 K [temperature associated with the isotropic radiation filling all space-thought to be a relic of the ‘big-bang’ origin of the universe].
(d) 5890 A – 5896 A [double lines of sodium]
(e) 14.4 keV [energy of a particular transition in ⁵⁷Fe nucleus associated with a famous high resolution spectroscopic method (Mossbauer spectroscopy)].
Solution:
(a) 21 cm wavelength corresponds to radio waves, at nearly 1445 MHz.
(b) Electromagnetic wave of 1057 MHz also corresponds to radio waves.
(c) Using Wien’s Law the wavelength of the radiation mostly emitted at given temperature of heavenly bodies can be calculated

This wavelength corresponds to microwave region of electromagnetic waves.
(d) This wavelength corresponds to visible region (yellow) of electromagnetic waves.
(e) The frequency of radiation can be calculated

This frequency corresponds to X-ray region of electromagnetic waves.

Question 15.
Answer the following questions:
(a) Long distance radio broadcasts use short¬wave bands. Why?
(b) It is necessary to use satellites for long distance TV transmission. Why?
(c) Optical and radio telescopes are built on the ground but X-ray astronomy is possible only from satellites orbiting the earth. Why?
(d) The small ozone layer on top of the stratosphere is crucial for human survival. Why?
(e) If the earth did not have an atmosphere, would its average surface temperature be higher or lower than what it is now?
(f) Some scientists have predicted that a global nuclear war on the earth would be followed by a severe ‘nuclear winter’ with a devastating effect on life on earth. What might be the basis of this prediction?
Solution:
(a) Electromagnetic waves in frequency range of shortwave band reflect from ionosphere where lower frequency radio waves i.e., medium waves are absorbed. So, short waves are suitable for long distance radio broadcast.
(b) Very high frequency (VHF) and ultra high frequency (UHF) electromagnetic waves used in T.V transmission after frequency modulation do not reflect from ionosphere and can only be transmitted straight from antenna. So, the range is highly limited. In order to send the transmission for long distances, satellite is used.
(c) Atmosphere absorb X-rays, while visible and radio waves can penetrate through it. Hence optical telescope can work on ground but X-ray astronomical telescopes only work above atmosphere, hence installed on the satellite orbiting around earth.
(d) The thin ozone layer absorbs harmful ultraviolet radiations, y-rays and cosmic radiations. All these high energy radiations can cause damage to the living cells.
Thus the thin ozone layer on top of the stratosphere is crucial for human survival.
(e) In the absence of atmosphere, the day temperature can rise to many fold and temperature at night will be much below 0°C. Although the average temperature will be reduced in the absence of green house effect.
(f) The thick clouds produced by global nuclear war will prevent solar radiation to reach down across the globe. This would cause severe ‘nuclear winter’.