First slide

Pontial due to

Question

If on the concentric hollow spheres of radii r and R(>r) the charge Q is distributed such that their surface densities are same then the potential at their common centre is

Moderate

A

$\large \frac{{Q({R^2} + {r^2})}}{{4\pi {\varepsilon _0}(R + r)}}$
B

$\large \frac{{QR}}{{R + r}}$
C

zero
D

$\large \frac{{Q(R + r)}}{{4\pi {\varepsilon _0}({R^2} + {r^2})}}$

Solution

$\large {q_1} + {q_2} = Q$ and $\large \frac{{{q_1}}}{{4\pi {r^2}}} = \frac{{{q_2}}}{{4\pi {R^2}}}$ (given)
$\large {q_1} = \frac{{Q{r^2}}}{{{R^2} + {r^2}}}$ and $\large {q_2} = \frac{{Q{R^2}}}{{{R^2} + {r^2}}}$
Potential at common centre
$\large \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{Q{r^2}}}{{({R^2} + {r^2})r}} + \frac{{Q{R^2}}}{{({R^2} + {r^2})R}}} \right] = \frac{{Q(R + r)}}{{4\pi {\varepsilon _0}({R^2} + {r^2})}}$

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