Work and Energy Electrostatic force is conservative Work done by an external agency to move a charge :  Work done is path independent Electrostatic force is conservative Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Work done to bring a charge from infinity to : Potential is the work required to create the system (potential energy) per unit charge Dr. Champak B. Das (BITS, Pilani)

Energy of a Point Charge Distribution Ex: Case of assembling three point charges q2 rs12 q1 Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) q3 rs23 rs13 q1 q2 rs12 Dr. Champak B. Das (BITS, Pilani)

Energy of a Point Charge Distribution Work necessary to assemble n number of point charges Dr. Champak B. Das (BITS, Pilani)

Energy of a Continuous Charge Distribution Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) ELECTROSTATIC ENERGY (can be +ve/-ve) (always +ve) Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) ELECTROSTATIC ENERGY Energy of a point charge is infinite ! Energy is stored in the field/charge ? Doesn’t obey superposition principle ! Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Prob. 2.32 (a) : Find the energy stored in a uniformly charged solid sphere of radius R and charge q using: Ans (a): Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Prob. 2.32 (b) : Find the energy stored in a uniformly charged solid sphere of radius R and charge q using: Ans (b): Dr. Champak B. Das (BITS, Pilani)

CONDUCTORS Conductor: charges free to move within the material. Electrostatic Equilibrium: there is no net motion of charge within the conductor. Dr. Champak B. Das (BITS, Pilani)

When an external field is applied ? E = 0 inside a conductor. The existence of electrostatic equilibrium is consistent only with a zero field in the conductor. When an external field is applied ? Dr. Champak B. Das (BITS, Pilani)

A conductor in an electric field: Electrons move upward in response to applied field. Dr. Champak B. Das (BITS, Pilani)

A conductor in an electric field: (contd.) Electrons accumulate on top surface. Induced charges set up a field E in the interior. Dr. Champak B. Das (BITS, Pilani)

A conductor in an electric field: (contd.) Two surfaces of a conductor: sheets of charge Dr. Champak B. Das (BITS, Pilani)

A conductor in an electric field: (contd.) Field of induced charges tends to cancel off the original field  E0 must move enough electrons to the surface such that, E = E0 Dr. Champak B. Das (BITS, Pilani)

In the interior of the conductor NET FIELD IS ZERO. The process is Instantaneous Dr. Champak B. Das (BITS, Pilani)

same amount of positive and negative charges  = 0 inside a conductor. same amount of positive and negative charges NET CHARGE DENSITY IS ZERO. Dr. Champak B. Das (BITS, Pilani)

Any net charge resides on the surface Dr. Champak B. Das (BITS, Pilani)

A conductor is an equipotential. For any two points, a and b: E R r V R r Dr. Champak B. Das (BITS, Pilani)

E is  to the surface, outside a conductor. Else, the tangential component would cause charges to move Dr. Champak B. Das (BITS, Pilani)

A justification for surface distribution of charges in a conductor :  go for a configuration to minimize the potential energy Example : Solid sphere carrying charge q Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Induced Charges Conductor +q Induced charges Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) A cavity in a conductor +q Gaussian surface If +q is placed in the cavity, -q is induced on the surface of the cavity. Dr. Champak B. Das (BITS, Pilani)

Prob. 2.35: A metal sphere of radius R, carrying charge q is surrounded by a thick concentric metal shell. The shell carries no net charge. (a) Find the surface charge density at R, a and b Answer: a b R q Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Prob. 2.35(b): Find the potential at the centre, using infinity as the reference point. a b R q Answer: Dr. Champak B. Das (BITS, Pilani)

Surface charge on a conductor Recall electrostatic boundary condition: => Field outside a conductor: Dr. Champak B. Das (BITS, Pilani)

The surface charge density : OR Knowledge of E or V just outside the conductor  Surface charge on a conductor Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Force on a conductor Dr. Champak B. Das (BITS, Pilani)

Forces on charge distributions Force on a charge element dq placed in an external field E(e) : On a volume charge distribution : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Prob. 2.43: Find the net force that the southern hemisphere of a uniformly charged sphere exerts on the northern hemisphere. Z Ans: R  r Y Q X Dr. Champak B. Das (BITS, Pilani)

Forces on charge distributions Force on a charge element dq placed in an external field E(e) : On a volume charge distribution : On a surface charge distribution : Dr. Champak B. Das (BITS, Pilani)

Forces on surface charge distributions “ E is discontinuous across the distribution ” below above The force per unit area : Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Force on a conductor Force (per unit area) on the conductor surface: Outward Pressure on the conductor surface : The direction of the force is “outward” or “into the field”….. whether  is positive or negative Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) Prob. 2.38: A metal sphere of radius R carries a total charge Q. What is the force of repulsion between the northern hemisphere and the southern hemisphere? Z X Y R Q Ans:  Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) CAPACITORS Potential difference between two conductors carrying +Q and –Q charge: Dr. Champak B. Das (BITS, Pilani)

Different possible geometries: Capacitance : Is a geometrical property Units: Farad (= coulomb/volt) Different possible geometries: Planer Spherical Cylindrical Dr. Champak B. Das (BITS, Pilani)

Plates are very large and very close Dr. Champak B. Das (BITS, Pilani)

Dr. Champak B. Das (BITS, Pilani) A Spherical capacitor Dr. Champak B. Das (BITS, Pilani)

Cross section of a spherical capacitor Dr. Champak B. Das (BITS, Pilani)

A cylindrical capacitor Dr. Champak B. Das (BITS, Pilani)

Capacitance per unit length of a cylindrical capacitor Cross section of a cylindrical capacitor Prob 2.39 : Capacitance per unit length of a cylindrical capacitor Dr. Champak B. Das (BITS, Pilani)

Work done to charge a capacitor At any instant, Dr. Champak B. Das (BITS, Pilani)