Question

Two concentric spheres of radii R and r have positive charges $q_1$ and $q_2$ with equal surface charge densities. What is the electric potential at their common centre?

• A. $\frac{\sigma }{{ϵ}_0}(R+r)$
• B. $\frac{\sigma }{{ϵ}_0}(R-r)$
• C. $\frac{\sigma }{{ϵ}_0}\left(\frac{1}{R}+\frac{1}{r}\right)$
• D. $\frac{\sigma }{{ϵ}_0}\left(\frac{R}{r}\right)$

Answer 1

The correct option is A $\frac{\sigma }{{ϵ}_0}(R+r)$

It is given the two spheres have equal charge densities.

Hence, $\sigma =\frac{q_1}{4\pi R^2}=\frac{q_2}{4\pi r^2}$

Potential inside a sphere with charge distributed on surface only is constant inside the sphere.

Potential inside sphere of radius $R$ is:

$V_R=\frac{q_1}{4\pi {\epsilon }_{o}R}=\frac{\sigma R}{{\epsilon }_o}$

Potential inside sphere of radius $r$ is:

$V_r=\frac{q_2}{4\pi {\epsilon }_{o}r}=\frac{\sigma r}{{\epsilon }_o}$

Net potential at center is the sum of the two potentials.

$V=V_R+V_r=\frac{\sigma }{{\epsilon }_o}(R+r)$