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A sperical capacitor consists of two con...

A sperical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports. Show that the capacitance of this spherical capacitor is given by `C = (4pi in_(0) r_(1) r_(2))/(r_(1) - r_(2))`,
Where `r_(1) and r_(2)` are radill of outer and inner spheres respectively.

Text Solution

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As is clear Fig. `+Q` charge spreads uniformly on inner surface of outer sphere of radius `r_(1)`. The induced charge `-Q` spreads uniformly on the outer surface of inner `r_(2)`. The outer surface of outer sphere is earthed. Due to electrostic shielding, `E = 0 for r lt r_(2) and E = 0, for r gt r_(1)`.
In the space between the two spheres, electric intensity E axis as shown. Potential difference between the two spheres,
`V = (Q)/(4pi in_(0) r_(2)) - (Q)/(4pi in_(0) r_(1)) = (Q)/(4pi in_(0)) [(1)/(r_(2)) - (1)/(r_(1))] = (Q)/(4pi in_(0)) ((r_(1) - r_(2)))/(r_(1) r_(2))`
As `C = (Q)/(V) :. C = (Q 4pi in_(0) r_(1) r_(2))/(Q(r_(1) - r_(2))) = (4pi in_(0) r_(1) r_(2))/(r_(1) - r_(2))`
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