Class 12 Physics CHAPTER 2 ELECTRIC CHARGES AND FIELDS
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NCERT Notes For Class 12 Physics CHAPTER 2 ELECTRIC CHARGES AND FIELDS
Class 12 Physics CHAPTER 2 ELECTRIC CHARGES AND FIELDS
ELECTROSTATIC POTENTIAL AND CAPACITANCE
Potential energy difference
 Electric potential energy difference between two points is the work required to be done by an external force in moving charge q from one point to another.
Electrostatic potential energy
 Potential energy of charge q at a point is the work done by the external force in bringing the charge q from infinity to that point.
ELECTROSTATIC POTENTIAL
 The electrostatic potential (V ) at any point is the work done in bringing a unit positive charge from infinity to that point.
 W – work done, q – charge.
 Also, W = qV
 It is a scalar quantity.
 Unit is J/C or volt (V)
POTENTIAL DUE TO A POINT CHARGE
 The force acting on a unit positive charge (+1 C) at A , is
 Thus the work done to move a unit positive charge from A to B through a displacement dx is
 The negative sign shows that the work is done against electrostatic force.
 Thus the total work done to bring unit charge from infinity to the point P is
 Integrating
 Therefore electrostatic potential is given by
Variation of potential V with r
POTENTIAL DUE TO AN ELECTRIC DIPOLE
The potential due to the dipole at P is the sum of potentials due to the charges q and –q
 Using cosine law
 For r >> a
 Neglecting the higher order terms we get
 Similarly
 Thusand
 Using the Binomial theorem and retaining terms up to the first order in a/r,
 Thus the potential is
 Using p=q x2a, we get
Special cases
 Potential at point on the axial line
At the axial point θ=0, therefore
 Potential at point on the equatorial line At the equatorial line θ=90^{0}, thus , V=0.
POTENTIAL DUE TO A SYSTEM OF CHARGES
 By the superposition principle, the potential at a point due to a system of charges is the algebraic sum of the potentials due to the individual charges.
 Thus V = V_{1 }V_{2} + …….V_{n}
Potential due to a uniformly charged spherical shell
 For a uniformly charged spherical shell, the electric field outside the shell is as if the entire charge is concentrated at the centre
 Thus potential at a distance r, from the shell is
 Where r R≥ , radius of the shell
 Inside the shell, the potential is a constant and has the same value as on its surface.
Equipotential surface
 Surface with constant value of potential at all points on the surface.
Properties of Equipotential surface
 Work done to move a charge on an equipotential surface is zero.
 Electric field is perpendicular to the surface.
 Two equipotential surfaces never intersect.
Equipotential surface of a single charge
Equipotential surfaces for a uniform electric field
Equipotential surfaces for a dipole
Equipotential surfaces for two identical positive charges
Relation between electric field and potential
 The work done to move a unit positive charge from B to A is
Work = Edr
 This work equals the potential difference V_{A}–V_{B}.
 Thus
 That is
POTENTIAL ENERGY OF A SYSTEM OF CHARGES
For a system of two charges
 As there is no external field work done in bringing q_{1} from infinity to r_{1} is zero.
 The potential due to the charge q_{1} is
 The work done in bringing charge q_{2} from infinity to the point r_{2} is
 This work gets stored in the form of potential energy of the system.
 Thus, the potential energy of a system of two charges q_{1} and q_{2} is
For a system of three charges
 The work done to bring q_{1} from infinity to the point is zero.
 The work done to bring q_{2 }is
 The charges q_{1} and q_{2} produce a potential, which at any point P is given by
 Work done in bringing q_{3} from infinity to the point r_{3} is
 The total work done in assembling the charges
POTENTIAL ENERGY IN AN EXTERNAL FIELD
Potential energy of a single charge
 Potential energy of q at r in an external field is

 where V(r) is the external potential at the point r.
Potential energy of a system of two charges in an external field
 Work done to bring q1 is
 Work done on q2 against the external field
 Work done on q2 against the field due to q1
 Thus the total Work done in bringing q_{2} to r_{2}
 Thus, Potential energy of the system = the total work done in assembling the configuration
Potential energy of a dipole in an external field
 The torque experienced by the dipole is
 The amount of work done by the external torque will be given by
 This work is stored as the potential energy of the system.
 Thus
 Therefore
U=−pEcosθ,
 Where p dipole moment , E – electric field
Properties of conductors
 Inside a conductor, electrostatic field is zero
 A conductor has free electrons
 In the static situation, the free charges have so distributed themselves that the electric field is zero everywhere inside
 At the surface electric field is normal.
 If E were not normal to the surface, it would have some nonzero component along the surface.
 Free charges on the surface of the conductor would then experience force and move.
 In the static situation, therefore, E should have no tangential component.
 The interior of a conductor can have no excess charge in the static situation
 A neutral conductor has equal amounts of positive and negative charges in every small volume or surface element.
 When the conductor is charged, the excess charge can reside only on the surface in the static situation.
 Electric potential is constant throughout the volume of the conductor.
 Since E = 0 inside the conductor and has no tangential component on the surface, no work is done in moving a small test charge within the conductor and on its surface.
 That is, there is no potential difference between any two points inside or on the surface of the conductor.
 Electric field at the surface of a charged conductor
 where σ is the surface charge density and ˆn is a unit vector normal to the surface in the outward direction
Derivation
 choose a pill box (a short cylinder) as the Gaussian surface about any point P on the surface.
 The pill box is partly inside and partly outside the surface of the conductor.
 It has a small area of cross section δS and negligible height.
 The contribution to the total flux through the pill box comes only from the outside (circular) crosssection of the pill box.
 The charge enclosed by the pill box is σδS.
 By Gauss’s law
Electrostatic shielding
 The vanishing of electric field inside a charged conducting cavity is known as electrostatic shielding.
 The effect can be made use of in protecting sensitive instruments from outside electrical influence.
 Why it is safer to be inside a car during lightning?
 Due to Electrostatic shielding, E=0 inside the car.
DIELECTRICS
 Dielectrics are nonconducting substances.
 They have no (or negligible number of ) charge carriers.
Conductor in an external field
 In an external field the free charge carriers in the conductor move and an electric field which is equal and opposite to the external field is induced inside the conductor.
 The two fields cancel each other and the net electrostatic field in the conductor is zero.
Dielectric in an external field
 In a dielectric, the external field induces dipole moment by stretching or reorienting molecules of the dielectric.
 Thus a net electric field is induced inside the dielectric in the opposite direction.
 The induced field does not cancel the external field.
 Dielectric substances may be made of polar or non polar molecules.
Non polar molecule in an external field
 In an external electric field, the positive and negative charges of a nonpolar molecule are displaced in opposite directions.
 The nonpolar molecule thus develops an induced dipole moment.
Linear isotropic dielectrics
 When a dielectric substance is placed in an electric field, a net dipole moment is induced in it.
 When the induced dipole moment is in the direction of the field and is proportional to the field strength the substances are called linear isotropic dielectrics.
Polar molecule in external field
 In the absence of any external field, the different permanent dipoles are oriented randomly due to thermal agitation; so the total dipole moment is zero.
 When an external field is applied, the individual dipole moments tend to align with the field.
 A dielectric with polar molecules also develops a net dipole moment in an external field.
Polarisation
 The dipole moment per unit volume is called polarisation and is denoted by P.
 For linear isotropic dielectrics,
 where χ_{e} is the electric susceptibility of the dielectric medium.
A rectangular dielectric slab placed in a uniform external field
 The polarised dielectric is equivalent to two charged surfaces with induced surface charge densities, say σ_{p} and –σ_{p}
 That is a uniformly polarised dielectric amounts to induced surface charge density, but no volume charge density.
Capacitor
 It is a charge storing device.
 A capacitor is a system of two conductors separated by an insulator.
 A capacitor with large capacitance can hold large amount of charge Q at a relatively small V.
Capacitance
 The potential difference is proportional to the charge , Q.
 Thus
 The constant C is called the capacitance of the capacitor. C is independent of Q or V.
 The capacitance C depends only on the geometrical configuration (shape, size, separation) of the system of two conductors
 SI unit of capacitance is farad.
 Other units are, 1 μF = 10^{–6} F, 1 nF = 10^{–9} F, 1 pF = 10^{–12} F, etc.
Symbol of capacitor
Fixed capacitance Variable capacitance
Dielectric strength
 The maximum electric field that a dielectric medium can withstand without breakdown is called its dielectric strength.
 The dielectric strength of air is about 3 × 10^{6 }Vm^{–1}.
THE PARALLEL PLATE CAPACITOR
 A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance
Capacitance of parallel plate capacitor
 Let A be the area of each plate and d the separation between them.
 The two plates have charges Q and –Q.
 Plate 1 has surface charge density σ = Q/A and plate 2 has a surface charge density –σ.
At the region I and II, E=0
 At the inner region
 The direction of electric field is from the positive to the negative plate.
 For a uniform electric field the potential difference is
 The capacitance C of the parallel plate capacitor is then
 Thus
Effect of dielectric on capacitance
 When dielectric medium is placed capacitance increases.
 When there is vacuum between the plates,
 The capacitance C_{0} in this case is
 When a dielectric is introduced
 so that the potential difference across the plates is
 For linear dielectrics, σ_{p} is proportional to E_{0}, and hence to σ.
 Thus, (σ – σ_{p}) is proportional to σ and we can write
 where K is the dielectric constant.
 Thus
 The capacitance C, with dielectric between the plates, is then
 That is
 The product ε_{0}K is called the permittivity of the medium and is denoted by ε.
 For vacuum K = 1 and ε = ε_{0}; ε_{0} is called the permittivity of the vacuum.
 Thus the dielectric constant of the substance is
 Also
 C_{0} – capacitance in vacuum, C capacitance in dielectric medium.
Combination of capacitors Capacitors in series
 In series charge is same and potential is different on each capacitors.
 The total potential drop V across the combination is
 Considering the combination as an effective capacitor with charge Q and potential difference V, we get
 Therefore, effective capacitance is
 For n capacitors in series
Capacitors in parallel
 In parallel the charge is different, potential is same on each capacitor.
 The charge on the equivalent capacitor is
 Thus
 Therefore
 In general , for n capacitors
Energy stored in a capacitor
 Energy stored in a capacitor is the electric potential energy.
 Charges are transferred from conductor 2 to conductor 1 bit by bit, so that at the end, conductor 1 gets charge Q.
 Work done to move a charge dq from conductor 2 to conductor 1, is dW = Potential ×Charge
 That is
 Since potential at conductor 1 is , q/C.
 Thus the total work done to attain a charge Q on conductor 1, is
 On integration we get,
 This work is stored in the form of potential energy of the system.
 Thus energy stored in the capacitor is
 Also or
Alternate method
 We have the Q – V graph of a capacitor,
 Energy = area under the graph
Energy Density of a capacitor
 Energy density is the energy stored per unit volume.
 We have
 Thus we get
 Thus energy per unit volume is given by
 That is the energy density of the capacitor is
VAN DE GRAAFF GENERATOR
Uses
 Used to generate high potential about million volts and electric field close to 3 × 10^{6} V/m.
 High potential generated is used to accelerate charged particles to high energies.
Principle
 When a small conducting sphere is kept inside a large sphere, potential is high at the small sphere.
 Potential due to small sphere of radius r carrying charge q
 Taking both charges q and Q, we have
 Thus the potential at the smaller sphere is higher than the larger sphere.
 If the two spheres are connected with a wire , charge will flow from smaller sphere to larger one.
Construction
 A large spherical conducting shell is supported at a height several meters above the ground on an insulating column.
 A long narrow endless belt insulating material, like rubber or silk, is wound around two pulleys – one at ground level, one at the centre of the shell.
Working
 The belt is kept continuously moving by a motor driving the lower pulley.
 It continuously carries positive charge, sprayed on to it by a brush at ground level, to the top.
 At the top, the belt transfers its positive charge to another conducting brush connected to the large shell.
 Thus positive charge is transferred to the shell, where it spreads out uniformly on the outer surface.
 In this way, voltage differences of as much as 6 or 8 million volts (with respect to ground) can be built up.
 The voltage produced by an openair Van de Graaff machine is limited by corona discharge to about 5 megavolts.
 Most modern industrial machines are enclosed in a pressurized tank of insulating gas these can achieve potentials of as much as about 25 megavolts.