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Home » A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (as shown in the figure). Show that the capacitance of a spherical capacitor is given by where and are the radii of outer and inner spheres respectively.
A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (as shown in the figure). Show that the capacitance of a spherical capacitor is given by C=\frac{4 \pi \epsilon_{0} r_{1} r_{2}}{r_{1}-r_{2}} where r_{1} and r_{2} are the radii of outer and inner spheres respectively.
CBSE | Class 12 | Electrostatic Potential and Capacitance | NCERT | Physics
A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (as shown in the figure). Show that the capacitance of a spherical capacitor is given by C=\frac{4 \pi \epsilon_{0} r_{1} r_{2}}{r_{1}-r_{2}} where r_{1} and r_{2} are the radii of outer and inner spheres respectively.

Solution,

According to question information given are:

The radius of the outer shell =r_{1}

Radius of the inner shell =r_{2}

The charge on the inner surface of the outer shell =+Q

The charge on the outer surface of the inner shell =-\mathrm{Q}

The potential difference between the two shells is given as

V=\frac{Q}{4 \pi \epsilon_{0} \Gamma_{2}}-\frac{Q}{4 \pi \epsilon_{0} r_{1}}

Here,

\varepsilon_{0}= Permittivity of free space

Evaluating we have,

V=\frac{Q}{4 \pi \epsilon_{0}}\left[\frac{1}{r_{2}}-\frac{1}{r_{1}}\right]

V=\frac{Q\left(r_{1}-r_{2}\right)}{4 \pi \epsilon_{0} r_{1} r_{2}}

The capacitance of the given system is

\mathrm{C}= Charge (\mathrm{Q}) / \mathrm{P} otential difference (\mathrm{V})

Therefore, C=\frac{4 \pi \epsilon_{0} r_{1} r_{2}}{\left(r_{1}-r_{2}\right)}

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