Question

If the sum of charge outside and inside the conducting shell is $K\times Q$. Find the magnitude of $K$?

Answer 1

Given: Solid insulating sphere, have net positive charge = 3Q uniformly distributed. Is concentric with conducting spherical shell of inner radius b and outer radius c and net charge -Q. Sum of charge inside and outside the conducting shell = $K\times Q$

To find: the magnitude of K

Sol: Since all points within this region are located inside conducting material, the electric field inside the shell must be zero, therefore E=0.

Now suppose we draw a Gaussian surface inside the shell (b < r < c). By Gauss's Law, this surface must enclose zero charges to ensure the electric vanishes. From this, we conclude there must be $Q_{in}=-3Q$ distributed on the inner surface. Now conservation of charge, demands

$Q_{tot}=Q_{in}+Q_{out}=-Q⟹Q_{out}=-Q-Q_{in}=-Q-(-3Q)\therefore Q_{out}=2Q$

But given sum of charge outside and inside conducting shell

= $K\times Q⟹K=-1$