Question

A charge $Q$ is distributed over two concentric hollow spheres of radii $a$ and $b(a>b)$, so that the surface charge densities are equal. The potential at the common centre is $\frac{1}{4\pi {\epsilon }_0}$ times

• A. $Q\left(\frac{a+b}{a^2+b^2}\right)$
• B. $2Q\left(\frac{a+b}{a^2+b^2}\right)$
• C. Q
• D. $\frac{Q}{2}\left(\frac{a+b}{a^2+b^2}\right)$
• E. $\frac{Q}{4}\left(\frac{a+b}{a^2+b^2}\right)$

Answer 1

Answer: (A)

As, charge $O$ is distributed over two concentric hollow spheres of radii $a$ and $b(a>b)$,
$Q=4\pi a^2\sigma +4\pi b^2\sigma$
$\sigma =\frac{Q}{4\pi \left(a^2+b^2\right)}$
and potential energy, we have
$U=\frac{K{Q}_1}{a}+\frac{K{Q}_2}{b}$
As, potential in common centre,
$K=\left(\frac{4\pi a^2\sigma }{a}+\frac{4\pi b^2\sigma }{b}\right)$
$=K(4\pi a\sigma +4\pi b\sigma )=K\sigma (a+b)$
$=K\cdot 4\pi \left(\frac{Q}{4\pi \left(a^2+b^2\right)}\right)\cdot (a+b)$
$=\frac{K\cdot Q(a+b)}{a^2+b^2}=KQ\left(\frac{a+b}{a^2+b^2}\right)$
Here, $K=\frac{1}{4\pi {\epsilon }_0}$