Chapter 6 Electromagnetic Induction Class 12 Physics Hand Written Notes By Ashish Anand Sir

Class 12 Physics Revision Notes Chapter 6 Electromagnetic Induction

• Magnetic Flux: Magnetic flux through a plane of area dA placed in a uniform magnetic field B  where  is the angle between magnetic field lines and area vector of the surface.
• Dimensions of magnetic flux: 
• SI unit:- Weber (Wb)
• Faraday’s Law:
  a) First Law: whenever there is a change in the magnetic flux linked with a circuit with time, an induced emf is produced in the circuit which lasts as long as the change in magnetic flux continues.
  b) Second Law: The magnitude of the induced emf is directly proportional to the rate of change of magnetic flux linked with the closed circuit. 
• Lenz’s Law: The direction of the induced emf or current in the circuit is such that it opposes the cause due to which it is produced i.e. it opposes the change in magnetic flux, so that-
   Where N is the number of turns in the coil
  Lenz’s law is based on energy conservation.
• Induced EMF and Induced Current:  
  Charge depends only on net change in flux does not depends on time.

1. Induced EMF,
2. Induced current, 

• Induced Emf due to Linear Motion of a Conducting Rod in a Uniform Magnetic Field
  The induced emf, 
  If   are perpendicular to each other, then 
• Fleming’s Right Hand Rule is used to find the direction of induced current set up in the conductor.
• Induced EMF due to Rotation of a Conducting Rod in a Uniform Magnetic Field:The induced emf,  Where n is the frequency of rotation of the conducting rod.
• Induced EMF due to Rotation of a Metallic Disc in a Uniform Magnetic Field: 
• Induced EMF, Current and Energy Conservation in a Rectangular Loop Moving in a Non – Uniform Magnetic Field with a Constant Velocity:
• Energy supplied in this process appears in the form of heat energy in the circuit.

1. The net increase in flux crossing through the coil in time Δt is, 
2. Induced emf in the coil is, 
3. If the resistance of the coil is R, then the induced current in the coil is, 
4. Resultant force acting on the coil is 
5. The work done against the resultant force 
6. Energy supplied due to flow of current I in time Δt is, 
   
   Or H = W

• Rotation of Rectangular Coil in a Uniform Magnetic Field:

1. Magnetic flux linked with coil
2. Induced emf in the coil 
3. Induced current in the coil. 
4. Both Emf and current induced in the coil are alternating.

• Self-Induction and Self Inductance:

1. The phenomenon in which an induced emf is produced by changing the current in a coil is called self induction.
   
   
   where L is a constant, called self inductance or coefficient of self – induction.
2. S.I. Unit- Henry (H)
3. Dimension- [ML²T^-2A^-2]
4. Self inductance of a circular coil 
5. Self inductance of a solenoid 
6. Two coils of self – inductances L₁ and L₂, placed far away (i.e., without coupling) from each other.
7. For series combination: 
8. For parallel combination:

• Mutual Induction and Mutual Inductance:

1. On changing the current in one coil, if the magnetic flux linked with a second coil changes and induced emf is produced in that coil, then this phenomenon is called mutual induction.Or    Thefore, M₁₂ = M₂₁ = M
2. Mutual inductance two coaxial solenoids 
3. If two coils of self- inductance L₁ and L₂ are wound over each other, the mutual inductance is,  Where K is called coupling constant.
4. Mutual inductance for two coils wound in same direction and connected in series 
5. Mutual inductance for two coils wound in opposite direction and connected in series 
6. Mutual inductance for two coils in parallel ​

• Energy Stored in an Inductor: 
• Magnetic Energy Density: 
• Eddy Current: When a conductor is moved in a magnetic field, induced currents are generated in the whole volume of the conductor. These currents are called eddy currents.
• Transformer:

1. It is a device which changes the magnitude of alternating voltage or current.
2. For ideal transformer: 
3. In an ideal transformer: 
4. In step – up transformer:  
5. In step – down transformer:  
6. Efficiency 

• Generator or Dynamo: It is a device by which mechanical energy is converted into electrical energy. It is based on the principle of electromagnetic induction.
• Different Types of Generator:

1. AC Generator- It consists of field magnet, armature, slip rings and brushes.
2. DC Generator- It consists of field magnet, armature, commutator and brushes.

• Motor: It is a device which converts electrical energy into mechanical energy.
  Back emf 
  Current flowing in the coil, 
  Where R is the resistance of the coil.
  Out put Power =  
  Efficiency,

Ch-5 Magnetism and Matter Class 12 Physics Hand Written Notes By Ashish Anand Sir

Magnetism and Matter Class 12 notes Physics Chapter 5

Introduction

Magnetic phenomena are universal in nature. The science of magnetism grew from the observation that a certain ore could attract small pieces of iron and point in a certain direction when kept on the floating cork. The ore was originally found in the district of Magnesia in Asia Minor (now in western Turkey) and therefore named magnetite.

In this chapter, we shall study about magnetic properties and behaviour of matter.

Bar Magnet

When iron filings are sprinkled on a sheet of glass placed over a short bar magnet, a particular pattern is formed and the following conclusions are drawn

1. One pole is designated as the north pole and the other as the south pole.
2. When suspended freely, these poles point approximately towards the geographic north and south poles.
3. Like poles repel each other and unlike poles attract each other.
4. The poles of a magnet can never be separated.

(i) Magnetic Field Lines

1. A magnetic field line is an imaginary curve, the tangent to which at any point gives the direction of magnetic field B at that point.
2. The magnetic field lines of a magnet form a close-continuous loop.
3. Outside the body of the magnet, the direction of magnetic field lines are from the north pole to the south pole.
4. No two magnetic field lines can intersect each other. This is because, at the point of intersection, we can draw two tangents. This would mean two directions of the magnetic field at the same point, which is not possible.
5. The larger the number of field lines crossing per unit area, the stronger the magnitude of the magnetic field B.

(ii) Coulomb’s Law of Magnetism

Let pole strength of a monopole be q_m, then the magnetic force between two isolated poles kept at separation r is

F∝qm(1)×qm(2)r2F∝qm(1)×qm(2)r2

F=μ04πqm(1)×qm(2)r2F=μ04πqm(1)×qm(2)r2

This force will be attractive if one pole is North and the other is South and the force will be repulsive if both poles are of the same type (i.e. North-North or South-South).

(iii) Magnetic Field due to a Monopole

Magnetic field due to monopole at a point is equal to the magnetic force experienced by a unit pole strength if kept at that point.

B=μ04πmr2B=μ04πmr2

It is away from the pole if it is N-pole and it is towards the pole if it is S-pole.

(iv) Magnetic Dipole Moment of a Bar Magnet

It is equal to the product of any one pole strength and separation between two poles

M=m×2lM=m×2l

It is directed from South-pole to north-pole.

(v) Magnetic Dipole Moment of a Bar Magnet

(A) On axial position

Magnetic field due to N-pole at P.

B1=μ04πm(r−l)2……(1)B1=μ04πm(r-l)2……(1)

Magnetic field due to S-pole at point P.

B2=μ04πm(r+l)2……(2)B2=μ04πm(r+l)2……(2)

Net magnetic field at point P.

B=B1−B2B=B1-B2

B=μ04πm(r−l)2−μ04πm(r+l)2B=μ04πm(r-l)2-μ04πm(r+l)2

B=μ04πm[(r+l)2−(r−l)2(r2−l2)2]B=μ04πm[(r+l)2-(r-l)2(r2-l2)2]

B=μ04πm×4rl(r2−l2)2B=μ04πm×4rl(r2-l2)2

B=μ04π2(m×2l)r(r2−l2)2B=μ04π2(m×2l)r(r2-l2)2

B=μ04π2M×r(r2−l2)2B=μ04π2M×r(r2-l2)2

For short bar magnet r > > l

B=μ04π2Mr3B=μ04π2Mr3

It is along magnetic dipole moment.

(B) On normal bisector

Magnetic field due to N-pole.

B1=μ04πm(r2+l2)……(1)B1=μ04πm(r2+l2)……(1)

Magnetic field due to S-pole.

B2=μ04πm(r2+l2)……(2)B2=μ04πm(r2+l2)……(2)

From symmetry net field at P will be

B=B1cosθ+B2cosθB=B1cosθ+B2cosθ

B=2[μ04πm(r2+l2)]cosθB=2[μ04πm(r2+l2)]cosθ

B=μ04π2m(r2+l2)×l(r2+l2)12B=μ04π2m(r2+l2)×l(r2+l2)12

B=μ04πM(r2+l2)32B=μ04πM(r2+l2)32

For short bar magnet r > > l

B=μ04πMr3B=μ04πMr3

Torque on a Magnetic Dipole in Uniform Magnetic Field

We know torque of electric dipole in electric field E

→τ=→P×→Eτ→=P→×E→

replacing E by B and P by M, we get torque on magnetic dipole

→τ=→M×→Bτ→=M→×B→

→τ=MBsinθτ→=MBsinθ

Work done in Rotating a Magnetic Dipole in Uniform Magnetic Field

Let at an instant dipole is at θ from the magnetic field, then the torque acting on the dipole is

τ = MB sinθ

In order to rotate the dipole against this torque by dθ angle, work is done on it by some external source.

dw = τ dθ

dw = MB sinθ dθ

Work done on dipole to rotate it from initial orientation θ₁ to final orientation θ₂ is

w=MB∫θ2tη1sinθw=MB∫tη1θ2sinθ

w=MB[cosθ1−cosθ2]w=MB[cosθ1-cosθ2]

Gauss’s Law in Magnetism

This law states that “the surface integral of a magnetic field over a closed surface is zero i.e. the net magnetic flux through any closed surface is always zero”.

∮→B.−→dS=0∮B→.dS→=0

Earth’s Magnetism

1. The earth’s magnetism was assumed to arise from a very large bar magnet placed deep inside the earth along its rotational axis but the main argument against the theory is that the interior of the earth is too hot to maintain any magnetism.
2. The pattern of the earth’s magnetic field varies with a position as well as time. This is most affected by the solar wind.
3. The magnetic field lines of the earth appear the same as a magnetic dipole located at the center of the earth.
4. The pole near the geographic north pole is called the north magnetic pole and the pole near the geographic south pole is called the south magnetic pole.
5. Geographic meridian: It is a vertical plane passing through the geographic north-south direction. It contains the longitude circle and axis of rotation of the earth.
6. Magnetic meridian: It is a vertical plane passing through an N–S line of the freely suspended magnet.

Magnetic Declination

It is the angle between the true geographic north-south direction and the north-south line shown by a compass needle at a place. Its value is more at higher latitudes and smaller near the equator. The declination in India is small.

Magnetic Inclination or Dip

It is the angle between the axis of the needle, (in magnetic meridian) that is free to move about a horizontal axis and horizontal. Thus dip is an angle that the total magnetic field of earth Be makes with the surface of the earth. The angle of dip is maximum δ = 90º at poles. It is zero at the magnetic equator.

Horizontal and Vertical component of Earth Magnetic Field

The component of the earth’s magnetic field B_e along horizontal is called horizontal component B_H.

BH=Becosδ…..(1)BH=Becosδ…..(1)

The component of earth’s magnetic field along vertical is called vertical component B_v.

BV=Besinδ…..(2)BV=Besinδ…..(2)

Relation Between Horizontal and Vertical Component

Squaring and adding equation (1) and (2), we get

B2H+B2V=B2e(cos2δ+sin2δ)BH2+BV2=Be2(cos2δ+sin2δ)

Be=√B2H+B2VBe=BH2+BV2

Dividing equation (2) by (1)

BVBH=tanδBVBH=tanδ

Magnetisation and Magnetic Intensity

(i) Magnetisation

When a magnetic material is placed in a magnetic field, the induced dipole moment develops in the material. The induced dipole moment per unit volume in the magnetic material is called the intensity of magnetisation or magnetisation density). It is denoted by →II→ and is a vector quantity. Its direction is the same as the direction of induced dipole moment in the material.

→I=→MNetVI→=M→NetV

Where →MNet=M→Net=Net induced dipole moment in the material. The unit of intensity of magnetisation →II→ is A/m.

(ii) Magnetic Intensity

In order to magnetize a magnetic material, it is kept in an external field B₀. The ratio of magnetising field to the permeability of free space is called magnetic intensity H.

→H=→B0μ0H→=B→0μ0

The unit of magnetic intensity is the same as that of the intensity of magnetisation i.e. A/m.

(iii) Magnetic Susceptibility (χmχm)

It is defined as the “ratio of the magnitude of the intensity of magnetisation |→II→| to that of magnetic intensity |→HH→|”. It is a scalar quantity with no dimension, no unit. The physical significance of magnetic susceptibility is that it is the degree of ease with which a magnetic material can be magnetised. A material with a higher value X_m can easily be magnetised.

χm=IHχm=IH

(iv) Magnetic Permeability (μμ)

It is defined as the ratio of the magnitude of total magnetic field (B) inside material to that of the magnitude of magnetic intensity (H)”. It is a scalar quantity. Its unit is Wb/A-m. The physical significance of magnetic permeability is that it measures the extent to which a magnetising field can penetrate or permeate a given magnetic material.

μ=BHμ=BH

Relative Permeability (μrμr)

It is the ratio of permeability of a medium to that of permeability of free space.

μr=μμ0μr=μμ0

Magnetic Properties of Materials

Curie and Faraday observed that almost all substances have certain magnetic properties. On the basis of the magnetic behaviour of different materials, they divided them into three categories : (i) Diamagnetism; (ii) Paramagnetism; (iii) Ferromagnetism.

(i) Diamagnetism

The substances which have a tendency to move from stronger to weaker part of the external magnetic field is called Diamagnetic substance and this phenomenon is known as Diamagnetism. They develop this tendency because they are feebly magnetized in a direction opposite to that of the external magnetizing field. Ex- bismuth, copper, lead, silicon, nitrogen, water and sodium chloride.

Properties of Diamagnetism

1. The magnetic field lines are expelled by these substances.
2. The magnetic field inside diamagnetic substance (B) is less than in free space B₀.
3. The relative permeability of diamagnetic substances is less than one.
4. X_m is negative for diamagnetic material.
5. Magnetic susceptibility X_m of diamagnetic substance is independent of temperature.

(ii) Paramagnetism

The substances which get feebly magnetized in the direction of the applied external magnetic field is called paramagnetic substance and this phenomenon is known as Paramagnetism. Therefore they have a tendency to move from the region of the weak magnetic field to a strong magnetic field i.e. they get weakly attracted to a magnet. Ex- Aluminium, sodium, calcium, oxygen and copper chloride.

Properties of Paramagnetism

1. Magnetic field lines tend to pass through these substances therefore magnetic field inside the substance is more than the outside.
2. The magnetic field inside paramagnetism substance (B) is greater than in free space B₀.
3. The relative permeability of paramagnetism substance is greater than one.
4. The magnetic susceptibility of a paramagnetic substance is small and positive.

(iii) Ferromagnetism

The substances which get strongly magnetised when placed in an external magnetic field is called ferromagnetic substance, and this phenomenon is known as ferromagnetism, so they have a strong tendency to move from a region of the weak magnetic field to a strong magnetic field. They get strongly attracted to the magnet. Ex- Iron, cobalt, nickel, alloys like alnico, etc.

Properties of Ferromagnetism

1. Magnetic field lines tend to crowd into ferromagnetic material.
2. The permeability of ferromagnetic materials is very large, of the order of hundreds and thousands.
3. Magnetic susceptibility X_m of ferromagnetic substances is very high, therefore, they can be magnetized easily and strongly.
4. With the rise in temperature, the susceptibility of ferromagnetic materials decreases. At a certain temperature ferromagnetic substance is converted into paramagnetic substance. This transition temperature is called Curie temperature or Curie point T_C.

Curie’s Law

The magnetic susceptibility of a paramagnetic substance is inversely proportional to absolute temperature T.

χm∝1Tχm∝1T

χm=CTχm=CT

The constant C is called Curie’s constant.

Curie-Weiss law

At temperature above the Curie temperature, a ferromagnetic substance becomes an ordinary paramagnetic substance whose magnetic susceptibility obeys the Curie-Weiss law according to which

χm=CT−Tcχm=CT-Tc

Hysteresis

1. When the intensity of magnetisation (I) of ferromagnetic substances is plotted against magnetic intensity for a complete cycle of magnetisation and demagnetisation the resulting loop is called hysteresis loop.
2. When the intensity of magnetising field (H) is increased, the intensity of magnetisation increases because more and more domains are aligned in the direction of the applied field.
3. When all domains are aligned, the material is magnetically saturated. Beyond this, if the intensity of magnetizing field (H) is increased, the intensity of magnetisation (I) does not increase.
4. The value of the intensity of magnetisation (I) left in the material at H = 0, is called retentivity or remanence.
5. Now if magnetizing field is applied in the reverse direction and its intensity H is increased, the material starts de-magnetising. The value of magnetising field needed to reduce magnetisation to zero is called coercivity (OC).
6. As the reverse magnetising field is increased further, the material again becomes saturated. Now, if the magnetising field is reduced after attaining the reverse saturation, the cycle repeats itself.
7. The area enclosed by the loop represents a loss of energy during a cycle of magnetisation and demagnetisation.

Hard and Soft Magnets

(i) Hard Magnets

The ferromagnetic material which retains magnetisation for a long period of time is called hard magnetic material or hard ferromagnets. Some hard magnetic materials are Alnico (an alloy of iron, aluminum, nickel, cobalt, and copper) and naturally occurring lodestone. They are used for permanent magnets. Permanent magnet material should have high retentivity and high coercivity.

(ii) Soft Magnets

The ferromagnetic material which retains magnetisation as long as the external field persists is called soft magnetic materials or soft ferromagnets. Soft ferromagnets are soft iron. Such material is used for making electromagnets. For electromagnets, the material should have low retentivity and low coercivity. Electromagnets are used in electric bells, loudspeakers, and telephone diaphragms.

Permanent Magnets and Electromagnets

(i) Permanent Magnets

The substances which at room temperature retain their magnetisation for a long period of time are called Permanent magnets. Permanent magnets should have (a) high retentivity and (b) high coercivity. As the material, in this case, is never put to cyclic changes of magnetization, hence hysteresis is immaterial. From the viewpoint of these facts, steel is more suitable for the construction of permanent magnets than soft iron. The fact that the retentivity of iron is little greater than that of steel is outweighed by the much smaller value of its coercivity.

(ii) Electromagnets

An electromagnet is a temporary strong magnet and is just a solenoid with its winding on a soft iron core which has high permeability and low retentivity.

Chapter 4 Moving Charges and Magnetism Class 12 Physics Hand Written Notes By Ashish Anand Sir

Class 12 Physics Revision Notes Chapter 4 Moving Charges and Magnetism

Biot – Savart law: The magnitude of Magnetic Field  is proportional to the steady current I due to an element  at a point P and inversely proportional to the distance r from the current element is,

• Magnetic field due to long straight current carrying conductor:
• If conductor is infinitely long, 
• Right-hand rule is used to find the direction of magnetic field due to straight current carrying conductor.
• Force on a Straight Conductor: Force F on a straight conductor of length  and carrying a steady current I placed in a uniform external magnetic field B, 
• Lorentz Force: Force on a charge q moving with velocity v in the presence of magnetic and electric fields B and E.
• Magnetic Force: The magnetic force  is normal to  and work done by it, is zero.
• Cyclotron: A charge q executes a circular orbit in a plane normal with frequency called the cyclotron frequency given by, 
  This cyclotron frequency is independent of the particle’s speed and radius.
• Magnetic Field due to Circular current carrying Coil: Magnetic field due to circular coil of radius ‘a’ carrying a current I at an axial distance r from the centre-
  
  At the centre of the coil, 
• Ampere’s Circuital Law: For an open surface S bounded by a loop C, then the Ampere’s law states that  where I refers to the current passing through S.
• If B is directed along the tangent to every point on the perimeter, then  Where Ie is the net current enclosed by the closed circuit.
• Magnetic Field: Magnetic field at a distance R from a long, straight wire carrying a current I is given by, 
  The field lines are circles concentric with the wire.
• Magnetic field B inside a long Solenoid carrying a current I:  Where n is the number of turns per unit length.
• For a toroid,  Where N is the total numbers of turns and r is the average radius.
• Magnetic Moment of a Planar Loop: Magnetic moment m of a planar loop carrying a current I, having N closely wound turns, and an area A, is 
• Direction of  is given by the Right – Hand Thumb Rule: Curl and palm of your right hand along the loop with the fingers pointing in the direction of the current, the thumb sticking out gives the direction of
• Loop placed in a Uniform Magnetic Field:

1. When this loop is placed in a uniform magnetic field B, Then, the force F on it is, F = 0
   And the torque on it is 
   In a moving coil galvanometer, this torque is balanced by a counter torque due to a spring, yielding.
    Where  is the equilibrium deflection and k is the torsion constant of the spring.

• Magnetic Moment in an Electron: An electron moving around the central nucleus has a magnetic moment , given by
   Where  is the magnitude of the angular momentum of circulating electron about the central nucleus.
• Bohr Magneton: The smallest value of  is called the Bohr magneton μBOr

Chapter-3 Current Electricity Class 12 Physics Hand Written Notes By Ashish Anand Sir

Current Electricity Class 12 notes Physics Chapter 3

Introduction

We considered all charges whether free or bound to be at rest in the previous two chapters. Charges in motion constitute an electric current. Lightning is one of the natural phenomena in which charges flow from clouds to earth through the atmosphere.

In this chapter, we will study some basic laws concerning steady electric current and their applications.

Electric Current

The rate of flow of electric charge through any cross-section of a conductor is known as electric current. If ΔQ amount of charge flows through any cross-section of the conductor in the interval t to (t + Δt), then it is defined as

i=ΔQΔti=ΔQΔt

The direction of current is taken as the direction of motion of positively charged particles and opposite to the direction of negatively charged particles. SI unit of current is ampere (A). It is a scalar quantity.

Current Density

The current density at any point in a conductor is the ratio of the current at that point in the conductor to the area of the cross-section of the conductor. It is a vector quantity and denoted by →JJ→.

→J=ΔiΔAJ→=ΔiΔA

The SI unit of current density is A/m².

Drift Speed

Drift Velocity is defined as the average velocity with which the free electrons move towards the positive end of a conductor under the influence of an external electric field applied. It is denoted by v_d.

vd=eEmτvd=eEmτ

Where, τ = relaxation time, E = electric field, m = mass, e = charge of electron.

Relation between Current Density and Drift Speed

Let, cross-sectional area of any conductor be A, the number of electrons per unit area be n, drift velocity be v_d, then the number of total moving electrons in t second will be

N = (nAv_dt)

So, moving charge in t second Q = (nAv_dt).e

Hence, electric current in t second =QtQt

i=nAvdteti=nAvdtet

i = neAv_d

We know J=iAJ=iA

Putting i = neAv_d in above equation

→JJ→=nev_d

Ohm’s Law

According to this law, “At constant temperature, the potential difference V across the ends of a given metallic wire (conductor) in a circuit (electric) is directly proportional to the current flowing through it”. i.e.,

V ∝ i

V = i.R

where, R = resistance of conductor

Mobility

Mobility is defined as the magnitude of the drift velocity per unit of the electric field. It is denoted by μ,

μ=vdEμ=vdE

Its SI unit is m²V^-1s^-1.

Resistance

Resistance is the ratio of the potential difference applied across the ends of the conductor to the current flowing through it.

R=ViR=Vi

The SI unit of R is ohm (Ω).

Resistivity

Resistivity is defined as the ratio of the electric field applied at the conductor to the current density of the conductor. It is denoted by ρ

ρ=EJρ=EJ ……(1)

If the length of the conductor be ‘l’, the cross-sectional area be ‘A’, the potential difference at the end of the conductor be ‘V’ and the electric current be ‘i’, then →EE→ and →JJ→ given by

→E=VlE→=Vl ……(2)

→J=iAJ→=iA …….(3)

Putting the value of E and J, from equations (2) and (3) into (1), we get

ρ=VliAρ=VliA

ρ=Vi.Alρ=Vi.Al

ρ=RAlρ=RAl

The constant of proportionality ρ depends on the material of the conductor but not on its dimensions. ρ is known as resistivity or specific resistance.

Conductivity

Conductivity is defined as the reciprocal resistivity of a conductor. It is expressed as,

σ=1ρσ=1ρ

SI unit is mho per metre (Ω^-1 m^-1).

Superconductivity

The resistivity of certain metals or alloys drops to zero when they are cooled below a certain temperature is called superconductivity.

Electrical Energy

When an electric current is moved in an electric circuit, then the energy of work done by taking a charge from one point to another point is called electric energy.

If a charge q at potential difference V is moved from one point to another point, then doing work will be

W = V . q …..(1)

Putting q = i.t in equation (1), we get

W =Vit

Putting V = i.R in equation (1), we get

W = i²Rt

Putting i=VRi=VR in equation (1), we get

W=V2RtW=V2Rt

Power

Electric power is the rate of doing work by electric charge. It is measured in watt and represented by P.

P=WtP=Wt [∵ 1HP = 746 watt]

Hence, P = Vi = i²R = V2RV2R

Resistor Colour Codes

A carbon resistor has a set of coaxial coloured rings in them, whose significance is listed in the above table. First two bands formed; the First two significant figures of the resistance in ohm. Third band; Decimal multiplier as shown in the table. Last band; Tolerance or possible variation in percentage as per the indicated value. For Gold ±5%, for silver ±10% and for No colour ± 20%.

Combination of Resistors

There are two types of resistance combinations.

(i) Series Combination

In Series Combination, different resistances are connected end to end. Equivalent resistance can be obtained as the formula,

R = R₁ + R₂ + R₃

NOTE: The total resistance in the series combination is more than the greatest resistance in the circuit.

(ii) Parallel Combination

In Parallel combination, the first end of all the resistances is connected to one point and the last end of all the resistances is connected to another point. Equivalent resistance can be obtained by the formula

1R=1R1+1R2+1R31R=1R1+1R2+1R3

NOTE: The total resistance in parallel combination is less than the least resistance of the circuit.

Cells, EMF, Internal Resistance

(i) Cells

An electrolytic cell consisting of two electrodes, called positive (P) and negative (N) immersed in an electrolytic solution as shown in the figure.

Electrodes exchange charges with the electrolyte. Positive electrode P has a potential difference V₊ between itself and electrolyte solution A immediately adjacent to it. Negative electrode N has a potential difference (V_–) relative to electrolyte B adjacent to it.

ε = V₊ – V_–

(ii) EMF

It is the difference in chemical potentials of electrodes used. It is also defined as the difference of potential across the electrodes of the cell when the electrodes are in an open loop.

ε = V₊ – V_–

(iii) Internal Resistance

It is the opposition offered by the electrolyte of the cell to the flow of current through itself. It is represented by r and given by

r=vir=vi

Cells in Series and Parallel

Kirchhoff’s Laws

Kirchhoff’s two rules are used for analysing electric circuits consisting of a number of resistors and cells interconnected in a complicated way.

Kirchhoff’s first rule: Junction rule

At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction.

Σ i = 0

Kirchhoff’s second rule: the Loop rule

The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero.

Σ iR = Σ E

Wheatstone Bridge

It is an application of Kirchhoff’s rules. The bridge is consisting of four resistances R₁, R₂, R₃ and R₄ as four sides of a square ABCD as shown in the figure.

Across the diagonally opposite points between A and C, battery E is connected. This is called the battery arm. To the remaining two diagonally opposite points B and D, a galvanometer G is connected to detect current. This line is known as the galvanometer arm.

Currents through all resistances and galvanometer are as shown in figure. In balanced Wheatstone bridge we consider the special case I_g = 0. Applying junction rule to junction B and D, we have

I₂ = I₄ and I₁ = I₃ 

Applying loop rule to loop ABDA

I₂R₂ + 0 – I₁R₁ = 0

I1I2=R2R1I1I2=R2R1 …..(i)

Applying loop rule to loop BCDB

I₄R₄ – I₃R₃ + 0 = 0

I₂R₄ – I₁R₃ = 0 (Using I₄ = I₂ and I₃ = I₁)

I1I2=R4R3I1I2=R4R3 …..(ii)

Using equation (i) and (ii), we have

R2R1=R4R3R2R1=R4R3 …..(iii)

The equation (iii) relating the four resistors is called the balance condition for the galvanometer to give zero or null deflection.

Meter Bridge

It is the practical application of the Wheatstone bridge. A standard wire AC of length one metre and of uniform cross-sectional area is stretched and clamped between two thick metal strips bent at right angles as shown in the figure.

The endpoints, where the wire is clamped are connected to a cell ε through a key K₁. The metal strip has two gaps across which resistors can be connected. One end of the galvanometer is connected to the mid-point of the metal strip between the gaps. The other end of the galvanometer is connected to a jockey, which can slide over AC to make electrical connections by its knife edge. R is the unknown resistance to be determined. S is the standard known resistance from a resistance box.

Let the jockey be in contact with point D. Length of portion AD of wire be l₁. Resistance of the portion AD is R_AD = R_ml₁ and resistance of the portion DC is R_DC = R_m(100 – l₁), where R_m is the resistance per centimetre of the wire. Now R, S, R_AD and R_DC represent four resistances of the Wheatstone bridge.

RS=RADRDC=Rml1Rm(100−l1)RS=RADRDC=Rml1Rm(100-l1)

Unknown resistance R in terms of standard known resistance S is given by

R=S(l1100−l1)R=S(l1100-l1)

When the galvanometer shows zero deflection then length AD = l₁. The balance condition gives

The percentage error in R can be minimised by adjusting the balance point near the middle of the bridge (i.e., l₁ is closed to 50 cm) by making a suitable choice of S.

Potentiometer

It is a versatile instrument consisting of a long piece of uniform wire AC across which a standard cell B is connected. more

Summary

• Electric current: The rate of flow of charge normally through any cross-section is known as current. It is a scalar quantity and its SI unit is ampere.
• Ohm’s law: When the physical conditions of the conductor remain the same, the current through a conductor is directly proportional to the potential difference across its ends. I ∝ V
• Current density (ˆjj^): At a point, it is a vector having a magnitude equal to current for the unit normal area surrounding that point and normal to the direction of charge flow and direction in which current passes through the point.
• Drift velocity (→vv→_d): It is the average uniform velocity acquired by free electrons inside a metal by the application of an electric field, which is responsible for current through it.
• Mobility (μ): It is defined as drift velocity per unit of the electric field.
• Relaxation time: Average time interval between two successive collisions of electrons.
• Resistance (R): It is the property of substance by virtue of which it opposes the flow of current through it.
• Conductance (G): It is the reciprocal of resistance G = 1/R.
• Resistivity or specific resistance (ρ): It is the resistance of the conductor per unit length per unit area of cross-section.
• Conductivity or specific conductance (σ): Reciprocal resistivity is called conductivity (σ = 1/ρ).
• Electrochemical cell: It is a device which converts chemical energy into electrical energy to maintain the flow of charge in a circuit.
• Electromotive force or emf of a cell (ε): It is the difference in chemical potentials of electrodes used.
• Internal resistance (r): It is the opposition offered by the electrolyte of the cell to the flow of current through itself.
• Kirchhoff’s law (i) Junction rule: At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction.(ii) Loop rule: The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero.
• Colour code for resistor value: Correct sequence can be remembered by B.B. ROY of Great Britain has Very Good Wife. Capital letters correspond to colours in the correct sequence having powers 0 to 9. Tolerance for the gold strip is 5%, Silver is 10% and no colour is 20%.
• Kirchhoff’s junction rule is based on the conservation of charge. Kirchhoff’s loop rule is based on the conservation of energy.

Ch-2 Electrostatic Potential and Capacitance Class 12 Physics Hand ritten Notes By Ashish Anand Sir

Electrostatic Potential and Capacitance Notes Class 12 Physics Chapter 2

→ The S.I. unit of electric potential and a potential difference is volt.

→ 1 V = 1 J C^-1.

→ Electric potential due to a + ve source charge is + ve and – ve due to a – ve charge.

→ The change in potential per unit distance is called a potential gradient.

→ The electric potential at a point on the equatorial line of an electric dipole is zero.

→ Potential is the same at every point of the equipotential surface.

→ The electric potential of the earth is arbitrarily assumed to be zero.

→ Electric potential is a scalar quantity.

→ The electric potential inside the charged conductor is the same as that on its surface. This is true irrespective of the shape of the conductor.

→ The surface of a charged conductor is equipotential irrespective of its shape.

→ The potential of a conductor varies directly as the charge on it. i.e., V ∝ lA

→ Potential varies inversely as the area of the charged conductor i.e.

→ S.I. unit of capacitance is Farad (F).

→ The aspherical capacitor consists of two concentric spheres.

→ A cylindrical capacitor consists of two co-axial cylinders.

→ Series combination is useful when a single capacitor is not able to tolerate a high potential drop.

→ Work done in moving a test charge around a closed path is always zero.

→ The equivalent capacitance of series combination of n capacitors each of capacitance C is
C_s = Cn

→ C_s is lesser than the least capacitance in the series combination.

→ The parallel combination is useful when we require large capacitance and a large charge is accumulated on the combination.

→ If two charged conductors are connected to each other, then energy is lost due to sharing of charges, unless initially, both the conductors are at the same potentials.

→ The capacitance of the capacitor increases with the dielectric constant of the medium between the plates.

→ The charge on each capacitor remains the same but the potential difference is different when the capacitors are connected in series.

→ P. D. across each capacitor remains the same but the charge stored across each is different during the parallel combination of capacitors.

→ P.E. of the electric dipole is minimum when θ = 0 and maximum when θ = 180°

→ θ = 0° corresponds to the position of stable equilibrium and θ = π to the position of unstable equilibrium.

→ The energy supplied by a battery to a capacitor is CE² but energy stored
in the capacitor is 12 CE².

→ A suitable material for use as a dielectric in a capacitor must have a high dielectric constant and high dielectric strength.

→ Van-de Graaf generator works on the principle of electrostatic. induction and action of sharp points on a charged conductor.

→ The potential difference between the two points is said to be 1 V if 1 J of work is done in moving 1 C test charge from one point to the another.

→ The electric potential at a point in E→: It is defined as the amount of work done in moving a unit + ve test charge front infinity to that point.

→ Electric potential energy: It is defined as the amount of work is done in bringing the charges constituting a system from infinity to their respective locations.

→ 1 Farad: The capacitance of a capacitor is said to be 1 Farad if 1 C charge given to it raises its potential by 1 V

→ Dielectric: It is defined as an insulator that doesn’t conduct electricity but the induced charges are produced on its faces when placed in a uniform electric field.

→ Dielectric Constant: It is defined as the ratio of the capacitance of the capacitor with a medium between the plates to its capacitance with air between the plates

→ Polarisation: It is defined as the induced dipole moment per unit volume of the dielectric slab.

→ The energy density of the parallel plate capacitor is defined as the energy per unit volume of the capacitor.

→ Electrical Capacitance: It is defined as the ability of the conductor to store electric charge.

Important Formulae

→ Electric potential at a point A is
V_A = W∞Aq0

→ V = 14πε0.qr

→ Electric field is related to potential gradient as:
E = – dVdr

→Electric potential at point on the axial line of an electric dipole is:
V = 14πε0⋅qr2

→ Electric P.E. of a system of point charges is given
υ = 14πε0∑ni=1∑nj=1j≠iqiajrij

→ V due to a charged circular ring on its axis is given by:
V = 14πε0⋅q(R2+r2)1/2

→ V at the centre of ring of radius R is given by
V = 14πε0⋅qR

→ The work done in moviag a test large from one point A to another point B having positions vectors rA→ and rA→ respectively w.r.t. q is given by
WAB = 14πε0⋅q⋅(1rB−1rA)

→ Line integral of electric field between points A and B is given by.
∫AB E→ dl→ = 14πε0⋅q(1rA−1rB)

→ Electric potential energy of an electric dipole is
U = – p→. E→

→ Capacitance of the capacitor is given by
C = qV

→ P.E. of a charged capacitor is:
U = 12 qV = 12 CV² = q22C

→ C of a parallel plate capacitor with air between the plates is:
C₀ = ε0⋅Ad
C₀ = ε0KAd

→ C of a parallel plate capacitor with a dielectric medium between the plates is:
C = CmC0=E0E

→ Common potential as
V = C1V1+C2V2C1+C2

→ loss of electrical energy = 12(C1C2C1+C2)(V1−V2)

→ Energy supplied by battery is CE2 and energy stored in the capacitor is 12 CE².

→ The equivalent capacitance of series combination of three capacitor is given by
1Cs=1C1+1C2+1C3

→ The equivalent capacitance of parallel grouping of three capacitors is
C_p = C₁ + C₂ + C₃

→ Capacitance of spherical capacitor is
C = 4πε₀ abb−a
a, b are radii of inner and outer spheres.

→ Capacitance of a cylindrical capacitor is given by:
C = 2πε0loge(ba)
when b, a are radii of outer and inner cylinder.

→ Capacitance of a capacitor in presence of conducting slab between the plates is .
C = C01−td = ∞ if t = d.

→Capacitances of a capacitor with a dielectric medium between the plates is given by
C = C0[1−td(1−1R)]
C = K C₀ If t = d

→ Reduced value of electric field in a dielectric slab is given by
E = E₀ – Pε0
where P = σ_p = induced charge density.

→ Capacitance of an isolated sphere is given by
C = 4πε₀ r .
C = 4πε₀ Kr

Chapter-1 Electric Charges and Fields Class 12 Physics Handwritten Notes By Ashish Anand Sir

Electric Charges and Fields Class 12 Notes Physics Chapter 1

▶ Introduction

The study of static charges is called electrostatics and this complete electrostatic will be discussed in two chapters. In this chapter, we begin with a discussion of electric charge, some properties of charged bodies, and the fundamental electric force between two charged bodies.

▶ What is Electric Charge?

Electric Charge is a fundamental property of a matter which is responsible for electric forces between the bodies. Two electrons placed at a small separation are found to repeal each other, this repulsive force (Electric force) is only because of the electric charge on electrons.

When a glass rod is rubbed with silk, the rod acquires one kind of charge and the silk acquires the second kind of charge. This is true for any pair of objects that are rubbed to be electrified. Now if the electrified glass rod is brought in contact with silk, with which it was rubbed, they no longer attract each other.

Types of Electric Charge

There are two types of charges that exist in our nature.

• Positive Charge
• Negative Charge

If any object loses its electrons then they get a positive charge. It is denoted by (+q) sign. If any object gain electrons from another object then it gets a negative charge. It is denoted by (-q) sign. The charges were named as positive and negative by the American scientist Benjamin Franklin. If an object possesses an electric charge, it is said to be electrified or charged. When it has no charge it is said to be neutral.

▶ Basic Properties of Electric Charge

The important properties and characteristics of electric charge are given below.

(i) Attraction and Repulsion:- Like charges repel each other while unlike charges attract each other.

(ii) Electric Induction:- When a charged object brings to contact with another uncharged, it gets the opposite charge of the charged object. It is called charging by induction.

(iii) Charge is Quantized:- An object that is electrically charged has an excess or deficiency of some whole number of electrons. Since electrons cannot be divided into fractions of electrons, it means that the charge of an object is a whole-number multiple of the charge of an electron. For example, it cannot have a charge equal to the charge of 0.5 or 1000.5 electrons.

Mathematically q = ± ne, Here n = 1, 2, 3 and e = 1.6 × 10^–19 coulomb.

(iv) Electric Charge is Conserved:- According to this property, “An electric charge neither can be created nor can be destroyed” i.e. total net charge of an isolated system is always conserved. Thus, when a glass rod is rubbed with silk cloth, both glass rod, and silk cloth acquire opposite charges in the same quantity. Thus, the total amount of charge remains the same before rubbing as well as after rubbing.

▶ Conductors and Insulators

Some substances easily allow the passage of electricity through them while others do not. Substances that allow electricity to pass through them easily are called ‘conductors’. They have electrons that are free to move inside the material. Metals, human and animal bodies, earth, etc. are examples of conductors. Non-metals e.g., glass, plastic, and wood are ‘insulators’ because they do not easily allow the passage of electricity through them.

Most substances are either conductors or insulators. There is a third category called ‘semiconductors’ which are intermediate between conductors and insulators because they partially allow movement of charges through them.

▶ Charging by Induction

Now as we know that two oppositely charged bodies attract each other. But it also has been our observation that a charged body attracts a neutral body as well. This is explained on the basis of charging by induction. In the induction process, two bodies (at least one body must be charged) are brought very close, but they never touch each other.

Let us examine how a charged body attracts an uncharged body. Imagine a conducting or partially conducting body (sphere here) is kept on an insulating stand and a charged rod (positive, for example) is brought very close to it. It will attract electrons to its side and the farther end of the sphere will become positively charged as it is deficient in electrons.

▶ Coulomb’s Law

In 1785 Charles Coulomb (1736-1806) experimentally established the fundamental law of electric force between two stationary charged particles. He observed that An electric force between two charged particles has the following properties:

• It is directed along a line joining the two particles and is inversely proportional to the square of the separation distance r, between them.
• It is proportional to the product of the magnitudes of the charges, |q₁| and |q₂|, of the two particles.
• It is attractive if the charges are of opposite sign and repulsive if the charges have the same sign.

From these observations, Coulomb proposed the following mathematical form for the electric force between two charges. The magnitude of the electric force F between charges q₁ and q₂ separated by a distance r is given by

F=k|q1||q1|r2F=k|q1||q1|r2

where k is a constant called the Coulomb constant. The proportionality constant k in Coulomb’s law is similar to G in Newton’s law of gravitation. Instead of being a very small number like G (6.67 × 10^–11), the electrical proportionality constant k is a very large number. It is approximately

k = 8.9875 × 10⁹ N-m²C^–2

The constant k is often written in terms of another constant, ε₀, called the permittivity of free space. It is related to k by

k=14πεok=14πεo

∴F=14πεo|q1||q1|r2∴F=14πεo|q1||q1|r2

εo=14πk=8.85×10−12εo=14πk=8.85×10-12 C² / Nm²

▶ Electric Field

A charge produces something called an electric field in the space around it and this electric field exerts a force on any charge (except the source charge itself) placed in it. The electric field has its own existence and is present even if there is no additional charge to experience the force.

▶ Intensity of Electric Field

The intensity of the electric field due to a charge configuration at a point is defined as the force acting on a unit positive charge at this point. Hence if a charge q experiences an electric force F at a point then the intensity of the electric field at this point is given as

E = F / q

It has S.I. units of newtons per coulomb (N/C).

▶ Electric Field due to a Point Charge

To determine the direction of an electric field, consider a point charge q as a source charge. This charge creates an electric field at all points in the space surrounding it. A test charge q₀ is placed at point P, a distance r from the source charge. According to Coulomb’s law, the force exerted by q on the test charge is

F=14πε0qq0r2F=14πε0qq0r2

This force is directed away from the source charge q, since the electric field at P, the position of the test charge, is defined by

E=Fq0E=Fq0

we find that at P, the electric field created by q is

E=14πε0qr2E=14πε0qr2

▶ Electric Field Lines

Electric field lines are a way of pictorially mapping the electric field around a configuration of charges. An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point. The field lines follow some important general properties:

• The tangent to electric field lines at any point gives the direction of the electric field at that point.
• In free space, they are continuous curves that emerge from a positive charge and terminate at a negative charge
• They do not intersect each other. If they do so, then it would mean two directions of the electric field at the point of intersection, which is not possible.
• Electrostatic field lines do not form any closed loops. This follows from the conservative nature of the electric field

▶ Electric Dipole

A configuration of two charges of the same magnitude q, but of opposite sign, separated by a small distance (say 2a) is called an electric dipole.

The dipole moment for an electric dipole is a vector quantity directed from the negative charge to the positive charge and its magnitude is p = q × 2a (charge × separation). The SI unit of dipole moment is C-m (coulomb-metre).

Electric Dipole

▶ Electric Field Strangth due to Electric Dipole

▶(i) At Axial Position

The net electric field at P

E=−→E1+−→E2E=E1→+E2→, where E₁ and E₂ are fields due to +q and –q respectively

E1=14πε0q(r−a)2E1=14πε0q(r-a)2 …..(1)

E2=14πε0−q(r+a)2E2=14πε0-q(r+a)2 …..(2)

E=E1+E2E=E1+E2

From eq. (1) and (2)

E=q4πε0[1(r−a)2−1(r+a)2]E=q4πε0[1(r-a)2-1(r+a)2]

E=q4πε0[(r+a)2−(r−a)2(r2−a2)2]E=q4πε0[(r+a)2-(r-a)2(r2-a2)2]

E=q4πε04ar(r2−a2)2E=q4πε04ar(r2-a2)2

E=14πε02(2aq)r(r2−a2)2E=14πε02(2aq)r(r2-a2)2

E=14πε02Pr(r2−a2)2E=14πε02Pr(r2-a2)2

For a short dipole, a is very small in comparison

∴r2−a2≈r2∴r2-a2≈r2

E=14πε02Pr3E=14πε02Pr3

▶ (ii) At Broad side on Position

Electric field vector due to +q is

E1=14πε0q(r2+a2)E1=14πε0q(r2+a2)

Electric field vector due to –q is

E2=14πε0q(r2+a2)E2=14πε0q(r2+a2)

Resultant field at P is,

E=E1cosθ+E2cosθE=E1cosθ+E2cosθ

E=14πε0q(r2+a2)cosθ+14πε0q(r2+a2)cosθE=14πε0q(r2+a2)cosθ+14πε0q(r2+a2)cosθ

E=14πε02q(r2+a2)cosθE=14πε02q(r2+a2)cosθ

E=14πε02q(r2+a2)a√r2+a2E=14πε02q(r2+a2)ar2+a2

E=14πε02aq(r2+a2)23E=14πε02aq(r2+a2)23

E=14πε0P(r2+a2)23E=14πε0P(r2+a2)23

For a short dipole

∴r2+a2≈r2∴r2+a2≈r2

E=14πε0P(r2+a2)3E=14πε0P(r2+a2)3

▶ Torque of Electric Dipole in uniform Electric Field

An electric dipole is placed in a uniform electric field E

The force experienced by the dipole is

F = qE

The two forces form a couple and it tries to turn the dipole. The torque due to the couple is given by

τ = either force × perpendicular distance between the forces

τ = qE × (2a sin θ)

τ = (2aq) E sin θ

τ = pE sin θ

τ = p × E

▶ Electric Flux

The number of electric field lines that are passing perpendicular through the unit surface of any plane is called electric flux. Consider an electric field that is uniform in both magnitude and direction, as in the figure.

We can write this as N ∝ EA, which means that the number of field lines is proportional to the product of E and A. This is a measure of electric flux and is represented by the symbol φ. In the above case, φ = EA cosθ. The SI unit of electric flux is N-m²/C or V-m(volt-metre).

▶Solid Angle

A solid angle is defined as an angle that is made at a point in place by an area. The SI unit of solid angle is steradian, and it is expressed as ‘sr’.

sr=dAcosθr2sr=dAcosθr2

▶Gauss’s Law

It states that the electric flux Φ through any closed surface is equal to (1/ε_o) times the net charged q enclosed by the surface. That is

ϕ=∫E.dS=qεoϕ=∫E.dS=qεo

Proof: Consider a point charge q surrounded by a spherical surface of radius r centered on the charge. The magnitude of the electric field everywhere on the surface of the sphere is

E=14πε0qr2E=14πε0qr2 ……(1)

The electric field is perpendicular to the spherical surface at all points on the surface. The electric flux through the surface is

ϕE=∫EAcosθϕE=∫EAcosθ …..(2)

Putting the value of E from eq. (1)

ϕE=∫14πε0qr2AcosθϕE=∫14πε0qr2Acosθ

ϕE=q4πε0∫Acosθr2ϕE=q4πε0∫Acosθr2 

ϕE=q4πε0∫dωϕE=q4πε0∫dω 

ϕE=q4πε04πϕE=q4πε04π 

ϕE=qε0ϕE=qε0

This result says that the electric flux through a sphere that surrounds a charge q is equal to the charge divided by the constant ε₀

Let us note some important points regarding this law:

• Gauss’s law is true for any closed surface, no matter what its shape or size.
• The term q on the right side of Gauss’s law, includes the sum of all charges enclosed by the surface.
• Gauss’s law is often useful for a much easier calculation of the electrostatic field when the system has some symmetry.
• Gauss’s law is based on the inverse square dependence on distance contained in Coulomb’s law.

▶ Application of Gauss’s Law

• It is used to calculate electric field due to an infinitely long straight uniformly charged wire.
• It is also used to calculate electric field due to a uniformly charged infinite plane sheet.
• It is also used to calculate electric field due to a uniformly charged thin spherical shell.

(i). Electric field strength due to an infinitely long straight uniformly charged wire

Let a charged wire of infinite length be +q charge and its linear charge density λ be. To calculate the electric field due to this wire, let us assume a cylindrical Gaussian surface of radius r. Let the area of this Gaussian surface be dS₁, dS₂, dS₃.

Hence the total electric flux passing through the first surface dS₁,

ϕ1=∮E.dS1cosθϕ1=∮E.dS1cosθ

ϕ1=∮E.dS1cos90°ϕ1=∮E.dS1cos90°

ϕ1=0ϕ1=0 …..(1)

Similarly, the total electric flux passing through the second surface dS₂,

ϕ2=∮E.dS2cosθϕ2=∮E.dS2cosθ

ϕ2=∮E.dS2cos0ϕ2=∮E.dS2cos0

ϕ2=∮E.dS2ϕ2=∮E.dS2

ϕ2=E∮dS2ϕ2=E∮dS2

ϕ2=E(2πrl)ϕ2=E(2πrl) …..(2)

Similarly, the total electric flux passing through the third surface dS₃,

ϕ3=∮E.dS3cos90°ϕ3=∮E.dS3cos90°

ϕ3=0ϕ3=0 …..(3)

Hence, the total electric flux passing through the Gaussian surface,

ϕ=ϕ1+ϕ2+ϕ3ϕ=ϕ1+ϕ2+ϕ3

ϕ=0+E.2πrl+0ϕ=0+E.2πrl+0

ϕ=E.2πrlϕ=E.2πrl …..(4)

Putting ϕ=qε0ϕ=qε0 from Gauss’s theorem,

qε0=E.2πrlqε0=E.2πrl

E=ql2πrε0E=ql2πrε0

E=λ2πrε0E=λ2πrε0

▶Deduction of Coulomb’s law from Gauss’ Law

Consider a charge +q in place at origin in a vacuum. We want to calculate the electric field due to this charge at a distance r from the charge. Imagine that the charge is surrounded by an imaginary sphere of radius r as shown in the figure below. This sphere is called the Gaussian sphere.

Consider a small area element dS on the Gaussian sphere. We can calculate the flux through this area element due to charge as follows:

∮→E.−→dS=E∫dS∮E→.dS→=E∫dS

∮→E.−→dS=E(4πr2)∮E→.dS→=E(4πr2)

Using this in Gauss theorem we get

E(4πr2)=qε0E(4πr2)=qε0

E=14πε0qr2E=14πε0qr2

We know that

F=Eq0F=Eq0

F=14πε0qq0r2F=14πε0qq0r2

This is the required Coulomb’s law obtained from the Gauss theorem.

Class 12 Maths Notes Chapter 13 Probability

Event: A subset of the sample space associated with a random experiment is called an event or a case.
e.g. In tossing a coin, getting either head or tail is an event.

Equally Likely Events: The given events are said to be equally likely if none of them is expected to occur in preference to the other.
e.g. In throwing an unbiased die, all the six faces are equally likely to come.

Mutually Exclusive Events: A set of events is said to be mutually exclusive, if the happening of one excludes the happening of the other, i.e. if A and B are mutually exclusive, then (A ∩ B) = Φ
e.g. In throwing a die, all the 6 faces numbered 1 to 6 are mutually exclusive, since if any one of these faces comes, then the possibility of others in the same trial is ruled out.

Exhaustive Events: A set of events is said to be exhaustive if the performance of the experiment always results in the occurrence of at least one of them.
If E₁, E₂, …, E_n are exhaustive events, then E₁ ∪ E₂ ∪……∪ E_n = S.
e.g. In throwing of two dice, the exhaustive number of cases is 6² = 36. Since any of the numbers 1 to 6 on the first die can be associated with any of the 6 numbers on the other die.

Complement of an Event: Let A be an event in a sample space S, then the complement of A is the set of all sample points of the space other than the sample point in A and it is denoted by A’or A¯.
i.e. A’ = {n : n ∈ S, n ∉ A]

Note:
(i) An operation which results in some well-defined outcomes is called an experiment.
(ii) An experiment in which the outcomes may not be the same even if the experiment is performed in an identical condition is called a random experiment.

Probability of an Event
If a trial result is n exhaustive, mutually exclusive and equally likely cases and m of them are favourable to the happening of an event A, then the probability of happening of A is given by

Note:
(i) 0 ≤ P(A) ≤ 1
(ii) Probability of an impossible event is zero.
(iii) Probability of certain event (possible event) is 1.
(iv) P(A ∪ A’) = P(S)
(v) P(A ∩ A’) = P(Φ)
(vi) P(A’)’ = P(A)
(vii) P(A ∪ B) = P(A) + P(B) – P(A ∩ S)

Conditional Probability: Let E and F be two events associated with the same sample space of a random experiment. Then, probability of occurrence of event E, when the event F has already occurred, is called a conditional probability of event E over F and is denoted by P(E/F).

Similarly, conditional probability of event F over E is given as

Properties of Conditional Probability: If E and E are two events of sample space S and G is an event of S which has already occurred such that P(G) ≠ 0, then
(i) P[(E ∪ F)/G] = P(F/G) + P(F/G) – P[(F ∩ F)/G], P(G) ≠ 0
(ii) P[(E ∪ F)/G] = P(F/G) + P(F/G), if E and F are disjoint events.
(iii) P(F’/G) = 1 – P(F/G)
(iv) P(S/E) = P(E/E) = 1

Multiplication Theorem: If E and F are two events associated with a sample space S, then the probability of simultaneous occurrence of the events E and F is
P(E ∩ F) = P(E) . P(F/E), where P(F) ≠ 0
or
P(E ∩ F) = P(F) . P(F/F), where P(F) ≠ 0
This result is known as multiplication rule of probability.

Multiplication Theorem for More than Two Events: If F, F and G are three events of sample space, then

Independent Events: Two events E and F are said to be independent, if probability of occurrence or non-occurrence of one of the events is not affected by that of the other. For any two independent events E and F, we have the relation
(i) P(E ∩ F) = P(F) . P(F)
(ii) P(F/F) = P(F), P(F) ≠ 0
(iii) P(F/F) = P(F), P(F) ≠ 0
Also, their complements are independent events,
i.e. P(E¯ ∩ F¯) = P(E¯) . P(F¯)
Note: If E and F are dependent events, then P(E ∩ F) ≠ P(F) . P(F).

Three events E, F and G are said to be mutually independent, if
(i) P(E ∩ F) = P(E) . P(F)
(ii) P(F ∩ G) = P(F) . P(G)
(iii) P(E ∩ G) = P(E) . P(G)
(iv)P(E ∩ F ∩ G) = P(E) . P(F) . P(G)
If atleast one of the above is not true for three given events, then we say that the events are not independent.
Note: Independent and mutually exclusive events do not have the same meaning.

Baye’s Theorem and Probability Distributions
Partition of Sample Space: A set of events E₁, E₂,…,E_n is said to represent a partition of the sample space S, if it satisfies the following conditions:
(i) E_i ∩ E_j = Φ; i ≠ j; i, j = 1, 2, …….. n
(ii) E₁ ∪ E₂ ∪ …… ∪ E_n = S
(iii) P(E_i) > 0, ∀ i = 1, 2,…, n

Theorem of Total Probability: Let events E₁, E₂, …, E_n form a partition of the sample space S of an experiment.If A is any event associated with sample space S, then

Baye’s Theorem: If E₁, E₂,…,E_n are n non-empty events which constitute a partition of sample space S, i.e. E₁, E₂,…, E_n are pairwise disjoint E₁ ∪ E₂ ∪ ……. ∪ E_n = S and P(E_i) > 0, for all i = 1, 2, ….. n Also, let A be any non-zero event, the probability

Random Variable: A random variable is a real-valued function, whose domain is the sample space of a random experiment. Generally, it is denoted by capital letter X.
Note: More than one random variables can be defined in the same sample space.

Probability Distributions: The system in which the values of a random variable are given along with their corresponding probabilities is called probability distribution.
Let X be a random variable which can take n values x₁, x₂,…, x_n.
Let p₁, p₂,…, p_n be the respective probabilities.
Then, a probability distribution table is given as follows:

such that P₁ + p₂ + P₃ +… + p_n = 1
Note: If x_i is one of the possible values of a random variable X, then statement X = x_i is true only at some point(s) of the sample space. Hence ,the probability that X takes value x, is always non-zero, i.e. P(X = x_i) ≠ 0

Mean and Variance of a Probability Distribution: Mean of a probability distribution is

Bernoulli Trial: Trials of a random experiment are called Bernoulli trials if they satisfy the following conditions:
(i) There should be a finite number of trials.
(ii) The trials should be independent.
(iii) Each trial has exactly two outcomes, success or failure.
(iv) The probability of success remains the same in each trial.

Binomial Distribution: The probability distribution of numbers of successes in an experiment consisting of n Bernoulli trials obtained by the binomial expansion (p + q)n, is called binomial distribution.
Let X be a random variable which can take n values x₁, x₂,…, x_n. Then, by binomial distribution, we have P(X = r) = ⁿC_r p^rq^n-r
where,
n = Total number of trials in an experiment
p = Probability of success in one trial
q = Probability of failure in one trial
r = Number of success trial in an experiment
Also, p + q = 1
Binomial distribution of the number of successes X can be represented as

Mean and Variance of Binomial Distribution
(i) Mean(μ) = Σ x_ip_i = np
(ii) Variance(σ²) = Σ x_i² p_i – μ² = npq
(iii) Standard deviation (σ) = √Variance = √npq
Note: Mean > Variance

Class 12 Mathematics Revision Notes Chapter 12 Linear Programming

Linear Programming Problem: A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative.

• A few important linear programming problems are:

(i)       Diet problems

(ii)      Manufacturing problems

(iii)     Transportation problems

(iv)     Allocation problems

• The common region determined by all the constraints including the non-negative constraints  of a linear programming problem is called the feasible region (or solution region) for the problem.
• Points within and on the boundary of the feasible region represent feasible solutions of the constraints. Any point outside the feasible region is an infeasible solution.
• Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.
• The following Theorems are fundamental in solving linear programming problems:

Theorem 1 Let R be the feasible region (convex polygon) for a linear programming problem and let  be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.

Theorem 2 Let R be the feasible region for a linear programming problem, and let be the objective function.  If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.

• If the feasible region is unbounded, then a maximum or a minimum may not exist. However, if it exists, it must occur at a corner point of R.
• Corner point method : For solving a linear programming problem. The method comprises of the following steps:

(i)     Find the feasible region of the linear programming problem and determine its corner points (vertices).

(ii)   Evaluate the objective function at each corner point. Let M and m respectively be the largest and smallest values at these points.

(iii)     If the feasible region is bounded, M and m respectively are the maximum and minimum values of the objective function.

• If the feasible region is unbounded, then,

(i)   M is the maximum value of the objective function, if the open half plane determined by has no point in common with the feasible region. Otherwise, the objective function has no maximum value.

(ii)   m is the minimum value of the objective function, if the open half plane determined by  has no point in common with the feasible region. Otherwise, the objective function has no minimum value.

• If two corner points of the feasible region are both optimal solutions of the same type, i.e., both produce the same maximum or minimum, then any point on the line segment joining these two points is also an optimal solution of the same type.

Class 12 Maths Notes Chapter 11 Three Dimensional Geometry

Direction Cosines of a Line: If the directed line OP makes angles α, β, and γ with positive X-axis, Y-axis and Z-axis respectively, then cos α, cos β, and cos γ, are called direction cosines of a line. They are denoted by l, m, and n. Therefore, l = cos α, m = cos β and n = cos γ. Also, sum of squares of direction cosines of a line is always 1,
i.e. l² + m² + n² = 1 or cos² α + cos² β + cos² γ = 1
Note: Direction cosines of a directed line are unique.

Direction Ratios of a Line: Number proportional to the direction cosines of a line, are called direction ratios of a line.
(i) If a, b and c are direction ratios of a line, then la = mb = nc
(ii) If a, b and care direction ratios of a line, then its direction cosines are

(iii) Direction ratios of a line PQ passing through the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are x₂ – x₁, y₂ – y₁ and z₂ – z₁ and direction cosines are


Note:
(i) Direction ratios of two parallel lines are proportional.
(ii) Direction ratios of a line are not unique.

Straight line: A straight line is a curve, such that all the points on the line segment joining any two points of it lies on it.

Equation of a Line through a Given Point and parallel to a given vector b⃗ 
Vector form r⃗ =a⃗ +λb⃗ 
where, a⃗  = Position vector of a point through which the line is passing
b⃗  = A vector parallel to a given line

Cartesian form

where, (x₁, y₁, z₁) is the point through which the line is passing through and a, b, c are the direction ratios of the line.
If l, m, and n are the direction cosines of the line, then the equation of the line is

Remember point: Before we use the DR’s of a line, first we have to ensure that coefficients of x, y and z are unity with a positive sign.

Equation of Line Passing through Two Given Points
Vector form: r⃗ =a⃗ +λ(b⃗ −a⃗ ), λ ∈ R, where a and b are the position vectors of the points through which the line is passing.

Cartesian form

where, (x₁, y₁, z₁) and (x₂, y₂, z₂) are the points through which the line is passing.

Angle between Two Lines
Vector form: Angle between the lines r⃗ =a1→+λb1→ and r⃗ =a2→+μb2→ is given as

Condition of Perpendicularity: Two lines are said to be perpendicular, when in vector form b1→⋅b2→=0; in cartesian form a₁a₂ + b₁b₂ + c₁c₂ = 0
or l₁l₂ + m₁m₂ + n₁n₂ = 0 [direction cosine form]

Condition that Two Lines are Parallel: Two lines are parallel, when in vector form b1→⋅b2→=∣∣∣b1→∣∣∣∣∣∣b2→∣∣∣; in cartesian form a1a2=b1b2=c1c2
or
l1l2=m1m2=n1n2
[direction cosine form]

Shortest Distance between Two Lines: Two non-parallel and non-intersecting straight lines, are called skew lines.
For skew lines, the line of the shortest distance will be perpendicular to both the lines.
Vector form: If the lines are r⃗ =a1→+λb1→ and r⃗ =a2→+λb2→. Then, shortest distance

where a2→, a1→ are position vectors of point through which the line is passing and b1→, b2→ are the vectors in the direction of a line.

Cartesian form: If the lines are

Then, shortest distance,

Distance between two Parallel Lines: If two lines l₁ and l₂ are parallel, then they are coplanar. Let the lines be r⃗ =a1→+λb⃗  and r⃗ =a2→+μb⃗ , then the distance between parallel lines is

Note: If two lines are parallel, then they both have same DR’s.

Distance between Two Points: The distance between two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) is given by

Mid-point of a Line: The mid-point of a line joining points A (x₁, y₁, z₁) and B (x₂, y₂, z₂) is given by

Plane: A plane is a surface such that a line segment joining any two points of it lies wholly on it. A straight line which is perpendicular to every line lying on a plane is called a normal to the plane.

Equations of a Plane in Normal form
Vector form: The equation of plane in normal form is given by r⃗ ⋅n⃗ =d, where n⃗  is a vector which is normal to the plane.
Cartesian form: The equation of the plane is given by ax + by + cz = d, where a, b and c are the direction ratios of plane and d is the distance of the plane from origin.
Another equation of the plane is lx + my + nz = p, where l, m, and n are direction cosines of the perpendicular from origin and p is a distance of a plane from origin.
Note: If d is the distance from the origin and l, m and n are the direction cosines of the normal to the plane through the origin, then the foot of the perpendicular is (ld, md, nd).

Equation of a Plane Perpendicular to a given Vector and Passing Through a given Point
Vector form: Let a plane passes through a point A with position vector a⃗  and perpendicular to the vector n⃗ , then (r⃗ −a⃗ )⋅n⃗ =0
This is the vector equation of the plane.
Cartesian form: Equation of plane passing through point (x₁, y₁, z₁) is given by
a (x – x₁) + b (y – y₁) + c (z – z₁) = 0 where, a, b and c are the direction ratios of normal to the plane.

Equation of Plane Passing through Three Non-collinear Points
Vector form: If a⃗ , b⃗  and c⃗  are the position vectors of three given points, then equation of a plane passing through three non-collinear points is (r⃗ −a⃗ )⋅{(b⃗ −a⃗ )×(c⃗ −a⃗ )}=0.
Cartesian form: If (x₁, y₁, z₁) (x₂, y₂, z₂) and (x₃, y₃, z₃) are three non-collinear points, then equation of the plane is

If above points are collinear, then

Equation of Plane in Intercept Form: If a, b and c are x-intercept, y-intercept and z-intercept, respectively made by the plane on the coordinate axes, then equation of plane is xa+yb+zc=1

Equation of Plane Passing through the Line of Intersection of two given Planes
Vector form: If equation of the planes are r⃗ ⋅n1→=d1 and r⃗ ⋅n2→=d2, then equation of any plane passing through the intersection of planes is
r⃗ ⋅(n1→+λn2→)=d1+λd2
where, λ is a constant and calculated from given condition.
Cartesian form: If the equation of planes are a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂, then equation of any plane passing through the intersection of planes is a₁x + b₁y + c₁z – d₁ + λ (a₂x + b₂y + c₂z – d₂) = 0
where, λ is a constant and calculated from given condition.

Coplanarity of Two Lines
Vector form: If two lines r⃗ =a1→+λb1→ and r⃗ =a2→+μb2→ are coplanar, then
(a2→−a1→)⋅(b2→−b1→)=0

Angle between Two Planes: Let θ be the angle between two planes.
Vector form: If n1→ and n2→ are normals to the planes and θ be the angle between the planes r⃗ ⋅n1→=d1 and r⃗ ⋅n2→=d2, then θ is the angle between the normals to the planes drawn from some common points.

Note: The planes are perpendicular to each other, if n1→⋅n2→=0 and parallel, if n1→⋅n2→=∣∣n1→∣∣∣∣n2→∣∣
Cartesian form: If the two planes are a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂, then

Note: Planes are perpendicular to each other, if a₁a₂ + b₁b₂ + c₁c₂ = 0 and planes are parallel, if a1a2=b1b2=c1c2

Distance of a Point from a Plane
Vector form: The distance of a point whose position vector is a⃗  from the plane
r⃗ ⋅n^=dis|d−a⃗ n^|

Note:
(i) If the equation of the plane is in the form r⃗ ⋅n⃗ =d, where n⃗  is normal to the plane, then the perpendicular distance is ∣∣a⃗ ⋅n⃗ −d∣∣∣∣n⃗ ∣∣
(ii) The length of the perpendicular from origin O to the plane r⃗ ⋅n⃗ =dis|d|∣∣n⃗ ∣∣ [∵ a⃗  = 0]

Cartesian form: The distance of the point (x₁, y₁, z₁) from the plane Ax + By + Cz = D is

Angle between a Line and a Plane
Vector form: If the equation of line is r⃗ =a⃗ +λb⃗  and the equation of plane is r⃗ ⋅n⃗ =d, then the angle θ between the line and the normal to the plane is

and so the angle Φ between the line and the plane is given by 90° – θ,
i.e. sin(90° – θ) = cos θ

Cartesian form: If a, b and c are the DR’s of line and lx + my + nz + d = 0 be the equation of plane, then

If a line is parallel to the plane, then al + bm + cn = 0 and if line is perpendicular to the plane, then al=bm=cn

Remember Points
(i) If a line is parallel to the plane, then normal to the plane is perpendicular to the line. i.e. a₁a₂ + b₁b₂ + c₁c₂ = 0
(ii) If a line is perpendicular to the plane, then DR’s of line are proportional to the normal of the plane.
i.e. a1a2=b1b2=c1c2
where, a₁, b₁ and c₁ are the DR’s of a line and a₂, b₂ and c₂ are the DR’s of normal to the plane.

Class 12 Maths Notes Chapter 10 Vector Algebra

Vector: Those quantities which have magnitude, as well as direction, are called vector quantities or vectors.
Note: Those quantities which have only magnitude and no direction, are called scalar quantities.

Representation of Vector: A directed line segment has magnitude as well as direction, so it is called vector denoted as AB→ or simply as a⃗ . Here, the point A from where the vector AB→ starts is called its initial point and the point B where it ends is called its terminal point.

Magnitude of a Vector: The length of the vector AB→ or a⃗  is called magnitude of AB→ or a⃗  and it is represented by |AB→| or |a⃗ | or a.
Note: Since, the length is never negative, so the notation |a⃗ |< 0 has no meaning.

Position Vector: Let O(0, 0, 0) be the origin and P be a point in space having coordinates (x, y, z) with respect to the origin O. Then, the vector OP→ or r⃗  is called the position vector of the point P with respect to O. The magnitude of OP→ or r⃗  is given by

Direction Cosines: If α, β and γ are the angles which a directed line segment OP makes with the positive directions of the coordinate axes OX, OY and OZ respectively, then cos α, cos β and cos γ are known as the direction cosines of OP and are generally denoted by the letters l, m and n respectively.

i.e. l = cos α, m = cos β, n = cos γ Let l, m and n be the direction cosines of a line and a, b and c be three numbers, such that la=mb=nc=r⃗  Note: l² + m² + n² = 1

Types of Vectors
Null vector or zero vector: A vector, whose initial and terminal points coincide and magnitude is zero, is called a null vector and denoted as 0⃗ . Note: Zero vector cannot be assigned a definite direction or it may be regarded as having any direction. The vectors AA→ , BB→ represent the zero vector.

Unit vector: A vector of unit length is called unit vector. The unit vector in the direction of a⃗  is a^=a⃗ ∣∣a⃗ ∣∣

Collinear vectors: Two or more vectors are said to be collinear, if they are parallel to the same line, irrespective of their magnitudes and directions, e.g. a⃗  and b⃗  are collinear, when a⃗ =±λb⃗  or |a|→=λ|b|→

Coinitial vectors: Two or more vectors having the same initial point are called coinitial vectors.

Equal vectors: Two vectors are said to be equal, if they have equal magnitudes and same direction regardless of the position of their initial points. Note: If a⃗  = b⃗ , then |a|→=|b|→ but converse may not be true.

Negative vector: Vector having the same magnitude but opposite in direction of the given vector, is called the negative vector e.g. Vector BA→ is negative of the vector AB→ and written as BA→ = – AB→.
Note: The vectors defined above are such that any of them may be subject to its parallel displacement without changing its magnitude and direction. Such vectors are called ‘free vectors’.

To Find a Vector when its Position Vectors of End Points are Given: Let a and b be the position vectors of end points A and B respectively of a line segment AB. Then, AB→ = Position vector of B⃗  – Positron vector of A⃗ 
= OB→ – OA→ = b⃗  – a⃗ 

Addition of Vectors
Triangle law of vector addition: If two vectors are represented along two sides of a triangle taken in order, then their resultant is represented by the third side taken in opposite direction, i.e. in ∆ABC, by triangle law of vector addition, we have BC→ + CA→ = BA→ Note: The vector sum of three sides of a triangle taken in order is 0⃗ .

Parallelogram law of vector addition: If two vectors are represented along the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the sides. If the sides OA and OC of parallelogram OABC represent OA→ and OC→ respectively, then we get
OA→ + OC→ = OB→

Note: Both laws of vector addition are equivalent to each other.

Properties of vector addition
Commutative: For vectors a⃗  and b⃗ , we have a⃗ +b⃗ =b⃗ +a⃗ 

Associative: For vectors a⃗ , b⃗  and c⃗ , we have a⃗ +(b⃗ +c⃗ )=(a⃗ +b⃗ )+c⃗ 
Note: The associative property of vector addition enables us to write the sum of three vectors a⃗ , b⃗  and c⃗  as a⃗ +b⃗ +c⃗  without using brackets.

Additive identity: For any vector a⃗ , a zero vector 0⃗  is its additive identity as a⃗ +0⃗ =a⃗ 

Additive inverse: For a vector a⃗ , a negative vector of a⃗  is its additive inverse as a⃗ +(−a→)=0⃗ 

Multiplication of a Vector by a Scalar: Let a⃗  be a given vector and λ be a scalar, then multiplication of vector a⃗  by scalar λ, denoted as λ a⃗ , is also a vector, collinear to the vector a⃗  whose magnitude is |λ| times that of vector a⃗  and direction is same as a⃗ , if λ > 0, opposite of a⃗ , if λ < 0 and zero vector, if λ = 0.
Note: For any scalar λ, λ . 0⃗  = 0⃗ .

Properties of Scalar Multiplication: For vectors a⃗ , b⃗  and scalars p, q, we have
(i) p(a⃗  + a⃗ ) = p a⃗  + p a⃗ 
(ii) (p + q) a⃗  = p a⃗  + q a⃗ 
(iii) p(q a⃗ ) = (pq) a⃗ 
Note: To prove a⃗  is parallel to b⃗ , we need to show that a⃗  = λ a⃗ , where λ is a scalar.

Components of a Vector: Let the position vector of P with reference to O is OP→=r⃗ =xi^+yj^+zk^, this form of any vector is-called its component form. Here, x, y and z are called the scalar components of r⃗  and xi^, yj^ and zk^ are called the vector components of r⃗  along the respective axes.

Two dimensions: If a point P in a plane has coordinates (x, y), then OP→=xi^+yj^, where i^ and j^ are unit vectors along OX and OY-axes, respectively.
Then, ∣∣∣OP→∣∣∣=x2+y2−−−−−−√

Three dimensions: If a point P in a plane has coordinates (x, y, z), then OP→=xi^+yj^+zk^, where i^, j^ and k^ are unit vectors along OX, OY and OZ-axes, respectively. Then, ∣∣∣OP→∣∣∣=x2+y2+z2−−−−−−−−−−√

Vector Joining of Two Points: If P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) are any two points, then the vector joining P₁ and P₂ is the vector P1P2→

Section Formula: Position vector OR→ of point R, which divides the line segment joining the points A and B with position vectors a⃗  and b⃗  respectively, internally in the ratio m : n is given by

For external division,

Note: Position vector of mid-point of the line segment joining end points A(a⃗ ) and B(b⃗ ) is given by OR→=a⃗ +b⃗ 2

Dot Product of Two Vectors: If θ is the angle between two vectors a⃗  and b⃗ , then the scalar or dot product denoted by a⃗  . b⃗  is given by a⃗ ⋅b⃗ =|a⃗ |∣∣b⃗ ∣∣cosθ, where 0 ≤ θ ≤ π.
Note:
(i) a⃗ ⋅b⃗  is a real number
(ii) If either a⃗ =0⃗  or b⃗ =0⃗ , then θ is not defined.

Properties of dot product of two vectors a⃗  and b⃗  are as follows:

Vector (or Cross) Product of Vectors: If θ is the angle between two non-zero, non-parallel vectors a⃗  and b⃗ , then the cross product of vectors, denoted by a⃗ ×b⃗  is given by

where, n^ is a unit vector perpendicular to both a⃗  and b⃗ , such that a⃗ , b⃗  and n^ form a right handed system.
Note
(i) a⃗ ×b⃗  is a vector quantity, whose magnitude is ∣∣a⃗ ×b^∣∣=|a⃗ ||b|→sinθ
(ii) If either a⃗ =0⃗  or b⃗ =0⃗ , then0is not defined.

Properties of cross product of two vectors a⃗  and b⃗  are as follows: