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Question: A spherical capacitor consists of two concentric spherical conductors, held is position by suitable ...

A spherical capacitor consists of two concentric spherical conductors, held is position by suitable insulating supports as shown is figure. The capacitance C, of this spherical capacitor is

A

4πεor1r2r1−r2\frac{4\pi\varepsilon_{o}r_{1}r_{2}}{r_{1} - r_{2}}

B

4πεo(r2−r1)r1r2\frac{4\pi\varepsilon_{o}(r_{2} - r_{1})}{r_{1}r_{2}}

C

r1r24πεo(r2−r1)\frac{r_{1}r_{2}}{4\pi\varepsilon_{o}(r_{2} - r_{1})}

D

(r1−r24πεor1r2\frac{(r_{1} - r_{2}}{4\pi\varepsilon_{o}r_{1}r_{2}}

Answer

4πεor1r2r1−r2\frac{4\pi\varepsilon_{o}r_{1}r_{2}}{r_{1} - r_{2}}

Explanation

Solution

:

As shown in figure, +q+ qcharge spreads uniformly on inner surface of outer sphere of radius r1r_{1}The induced charge −q- qspreads uniformly on the outer surface of inner sphere of radius r2r_{2}. The outer surface of outer sphere is earthed. Due to electrostatic shielding E=0E = 0 for r<r2r < r_{2} and E=0E = 0for r>r1r > r_{1}

In the space between the two spheres, potential difference between two spheres,

V=q4πε0r2−q4πε0r1=q4πε0[1r2−1r1]V = \frac{q}{4\pi\varepsilon_{0}r_{2}} - \frac{q}{4\pi\varepsilon_{0}r_{1}} = \frac{q}{4\pi\varepsilon_{0}}\left\lbrack \frac{1}{r_{2}} - \frac{1}{r_{1}} \right\rbrack

V=q4πε0(r1−r2r1r2)V = \frac{q}{4\pi\varepsilon_{0}}\left( \frac{r_{1} - r_{2}}{r_{1}r_{2}} \right)

Also C=qVC = \frac{q}{V}

∴C=4πε0r1r2r1−r2\therefore C = \frac{4\pi\varepsilon_{0}r_{1}r_{2}}{r_{1} - r_{2}}