A spherical capacitor is formed from two concentric spherica

A spherical capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has a radius of ra = 12.3 cm , and the outer sphere has a radius of rb = 15.0 cm . A potential difference of 120 V is applied to the capacitor.

What is the capacitance of the capacitor?

Use 0 = 8.85×10¹² F/m for the permittivity of free space.

What is the magnitude E1 of the electric field E at radius r= 12.8 cm , just outside the inner sphere?

What is the magnitude of E at r= 14.6 cm , just inside the outer sphere?

Solution

A. by gauss\'s law

phi = integral(E.dA) = kq/r^2

V = - integral(E.dl) = kq*(rb - ra)/(rb*ra)

Q = CV

C = Q/V = Q*ra*rb/(kQ(rb - ra))

C = 12.3*15*10^-4/(9*10^9*(15 - 12.3)*10^-2) = 7.59*10^-11 F

B. Vab = kQ*(rb - ra)/(rb*ra)

Q = Vab*rb*ra/(k*(rb - ra))

E = kQ/r1^2

use above value of Q

E = ra*rb*Vab*/(r1^2*(rb-ra)) = 12.3*15*10^-4*120/(0.128^2*(15 - 12.3)*10^-2)

E = 5004.88 N/C

C. E = ra*rb*Vab/(r2^2*(rb-ra)) = 12.3*15*10^-4*120/(0.146^2*(15 - 12.3)*10^-2)

E = 3846.87 N/C