Q. 153.9( 12 Votes )

A spherical conducting shell of inner radius r1 and outer radius r2 has a charge Q.

(a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell?

(b) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain.

Answer :

(a) Charge placed at the centre of the sphere is + q. Thus, charge induced on the inner surface be –q.

Surface charge density at the inner surface of the sphere,

σ1 = =

Charge placed at the centre of the sphere is + q. Thus, charge induced on the inner surface be + q and surface charge distribution is Q. Therefore Total charge = Q + q.

Surface charge density at the outer surface of the sphere,

σ2 = =

(b) Yes

Whatever the shape of the shell, the electric field intensity in the cavity is zero. Take a closed loop such that it is inside the cavity and in conductor. Net work done in carrying charge along closed loop by electric field is zero. Thus, electric field is zero, whatever the shape.

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Answer the following:

(a) The top of the atmosphere is at about 400 kV with respect to the surface of the earth, corresponding to an electric field that decreases with altitude. Near the surface of the earth, the field is about 100 Vm–1. Why then do we not get an electric shock as we step out of our house into the open? (Assume the house to be a steel cage so there is no field inside!)

(b) A man fixes outside his house one evening a two metre high insulating slab carrying on its top a large aluminium sheet of area 1m2. Will he get an electric shock if he touches the metal sheet next morning?

(c) The discharging current in the atmosphere due to the small conductivity of air is known to be 1800 A on an average over the globe. Why then does the atmosphere not discharge itself completely in due course and become electrically neutral? In other words, what keeps the atmosphere charged?

(d) What are the forms of energy into which the electrical energy of the atmosphere is dissipated during a lightning?

(Hint: The earth has an electric field of about 100 Vm–1 at its surface in the downward direction, corresponding to a surface charge density = –10–9 C m–2. Due to the slight conductivity of the atmosphere up to about 50 km (beyond which it is good conductor), about + 1800 C is pumped every second into the earth as a whole. The earth, however, does not get discharged since thunderstorms and lightning occurring continually all over the globe pump an equal amount of negative charge on the earth.)

NCERT - Physics Part-I

Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by

Where, is a unit vector normal to the surface at a point and σ is the surface charge density at that point. (The direction of is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is .

NCERT - Physics Part-I