A spherical capacitor is formed from an inner conducting sphere of radius a = 10

Question

A spherical capacitor is formed from an inner conducting sphere of radius a = 10cm, a dielectric shell with inner radius b = 15cm and outer radius c = 20cm, and an outer conducting shell with inner radius d = 25cm. The dielectric shell has dielectric constant k = 3. For the computation of the capacitance, assume an arbitrary charge of Q on the inner conductor and - Q on the outer conductor. Compute the potential difference across region I, deltaV_I, in terms of Q. Compute the potential difference across region II, deltaV_II, in terms of Q. Compute the potential difference across region III, deltaV_III in terms of Q. Compute the capacitance, both symbolically and numerically.

Explanation / Answer

Potential difference in a spherical capacitor

V = Q / 4 pai epsilon zero ( 1/r1 - 1/ r2 )

a ) V1 = 9*10^9 Q (1/10 - 1/15 ) = 3*10^8 Q V

b ) V 2 = 9*10^9 /3 Q (1/15 -1/ 20 ) = 5*10^7 Q V

c ) V 3 = 9*10^9 Q ( 1/20 -1/25 ) = 9*10^7 Q V

d ) equivalent capacitance of the system is same as three capacitors C1 , C2 , C3 in series

1/ C = 1/C1 + 1/ C2 + 1/C3

C = Q / V so 1/C = V/Q

1 /C1 = 3*10^8 Q/Q = 30*10^7

1/C = 30*10^7 + 5*10^7 +9*10^7

= 44*10^7

C = 2.27 *10^-9 F

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