Question 426: Electrostatics & Capacitance

Question

A charge Q is distributed over two concentric
conducting thin spherical shells radii r and
R (R > r). If the surface charge densities on the
two shells are equal, the electric potential at
the common centre is :
![](https://imagex.cdn.examgoal.net/jgy2xukexxq7an/29b875b1-38c7-47e7-ba72-e78d4bac1573/9efceff0-f402-11ea-8293-3b6cc591631c/file-jgy2xukexxq7ao-231w.png)

Accepted Answer

Let the charges on inner and outer spheres are Q
1
and Q
2
.
$ 	herefore $ Q
1
+ Q
2
= Q  ...(1)
Since charge density '$\sigma $' is same for both spheres, so
$\sigma $ = ${{{Q_1}} \over {4\pi {r^2}}} = {{{Q_2}} \over {4\pi {R^2}}}$
$ \Rightarrow $ ${{{Q_1}} \over {{Q_2}}} = {{{r^2}} \over {{R^2}}}$ ....(2)
From (1) and (2),
${{{Q_2}{r^2}} \over {{R^2}}} + {Q_2} = Q$
$ \Rightarrow $ Q
2
= ${{Q{R^2}} \over {\left( {{r^2} + {R^2}} ight)}}$
$ 	herefore $ Q
1
= ${{{r^2}} \over {{R^2}}}.{{Q{R^2}} \over {\left( {{r^2} + {R^2}} ight)}}$ = ${{Q{r^2}} \over {\left( {{r^2} + {R^2}} ight)}}$
Potential at centre 'O' = ${{k{Q_1}} \over r} + {{k{Q_2}} \over R}$
= $k\left[ {{{Q{r^2}} \over {r\left( {{r^2} + {R^2}} ight)}} + {{Q{R^2}} \over {R\left( {{r^2} + {R^2}} ight)}}} ight]$
= ${{{kQ\left( {r + R} ight)} \over {\left( {{r^2} + {R^2}} ight)}}}$
= ${1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} ight)} \over {\left( {{R^2} + {r^2}} ight)}}$Q

Answer

${1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} \right)} \over {\left( {{R^2} + {r^2}} \right)}}$Q

Answer

${1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} \right)} \over {2\left( {{R^2} + {r^2}} \right)}}Q$

Answer

${1 \over {4\pi {\varepsilon _0}}}{{\left( {R + 2r} \right)Q} \over {2\left( {{R^2} + {r^2}} \right)}}$

Answer

${1 \over {4\pi {\varepsilon _0}}}{{\left( {2R + r} \right)} \over {\left( {{R^2} + {r^2}} \right)}}Q$

Electrostatics & Capacitance

A charge Q is distributed over two concentric conducting thin spherical shells radii r and R (R > r). If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

Choose the correct answer: