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A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. The inner sphere has a radius 15.0 cm and the capacitance is 116 pF.

(a) What is the radius of the outer sphere?

(b) If the potential difference between the two spheres is 220 V, what is the magnitude of the charge on each sphere?

Short Answer

Expert verified

The radius of the outer sphere is 17.5cm.

The magnitude of the charge Q on each sphere is 25.52nC.

Step by step solution

01

Capacitance of spherical body.

We were given a spherical capacitor where the radius of the first sphere is

C=4πϵrarbrb-ra

a) The capacitance C is the magnitude Q of the charge on either sphere divided by the potential difference Vabbetween the spheres where we can get the capacitance as the terms of the radius of each sphere

C=4πϵrarbrb-rarbC=raC=4πϵrarbraC=rb(C-4πϵra)ra=rbCC-4πϵra

02

Outer radius

Now Substitute our values for C, rainto the above equation to get the radius of the outer sphere where

rb=raCC-4πϵra=(0.15m)(116×10-12F)116×10-12F-4π8.854×10-12F/m0.15m=0.175m=17.5cm

The radius of the outer sphere is 17.5cm

03

Charge on the sphere.

b) The two spheres have the same magnitude of the charge Q but in the opposite sign. So, we can use the value of the capacitance and the potential difference Vabto get the charge by applying the equation

Q=CVab

where the charge is directly proportional to the capacitance C. Now substitute our values to get the charge Q

Q=CVab=(116×10-12F)(220.0V)=25.52×10-9C=25.52nC

Hence the magnitude of the charge Q on each sphere is 25.52nC

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