Question

Using the method of images, discuss the problem of a point charge q inside a hollow, grounded, conducting sphere of inner radius a. Find,

a) the potential inside the sphere;

b) the induced surface-charge density;

c) the magnitude and direction of the force acting on q.

d) Is there any change in the solution if the sphere is kept at a fixed potential V? If the sphere has a total charge Q on its inner and outer surfaces?

Using the method of images, discuss the problem of a point charge q inside a hollow, grounded, conducting sphere of inner radius a. Find (a) the potential inside the sphere; (b) the induced surface-charge density; (c) the magnitude and direction of the force acting on q- (d Is there any change in the solution if the sphere is kept at a fixed potential V? pt at a fined potential v? If the sphere has a total charge Q on its inner and outer surfaces?

Answer 1

(a) Phi_+ (x^bar) = 1/ 4 pi epsilon_0 q/ Vertical-bar x^bar - y^bar Vertical-bar phi_- (x^bar) = 1/ 4 pi epsilon_0 q^1/ Vertical-bar x^bar - y^bar 1 Vertical-bar implies phi (x^bar) = 1/ 4 pi epsilon_0 (q/ Vertical-bar x^bar - y^bar Vertical-bar + q^1/ Vertical-bar x^bar - y^bar 1 Vertical-bar ) y^1 = a^2/ y, q^1 = -q a/ y phi (x) = q/ 4 pi epsilon_0 [1/ Vertical-bar x^bar - y^bar Vertical-bar - 1/ Vertical-bar y/ a x^bar - a/ y y^bar Vertical-bar ] (b) sigma = -epsilon_0 partial differential phi/ partial differential x Vertical-bar _x = a = epsilon_0/4 pi epsilon_0 partial differential/partial differential x (q/ (x^2 + y^2 - 2 xy cos r)^2 + q/ (x^2 + y^12 - x y^1 cos r)^2) = -q/ 4 pi a (1 - y^2/ a^2)/ (a^2 + y^2 - 2 ay cos r)^3/2 (c) Vertical-bar F^rightarrow Vertical-bar = 1/4 pi