Question

(a) Determine the capacitance of the capacitor.

(b) What is the potential of the inner sphere?

(c) Compare
the capacitance of this capacitor with that of an isolated
sphere of radius 12 cm. Explain why the latter is much smaller.

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Solution

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Radius of the inner sphere, $r_{2}$= 12 cm = 0.12 m

Radius of the outer sphere, $r_{1}$= 13 cm = 0.13 m

Charge on the inner sphere, $q=2.5×10_{−6}C$

Dielectric constant of a liquid, $ε_{r}=32$

(a)

Capacitance, $C=r_{1}−r_{2}4πε_{o}ε_{r}r_{1}r_{2} $

where,

$ε_{o}$= Permittivity of free space $=8.85×10_{−12}C_{2}N_{−1}m_{2}$

$∴C≈5.5×10_{−9}F$

(b)

Potential of the inner sphere is given by,

$V=q/C$

$=4.5×10_{2}V$

(c)

Radius of an isolated sphere, r= 12 cm

Capacitance of the sphere is given by the relation,

$C_{′}=4πε_{o}r$

$=1.33×10_{−11}F$

The capacitance of the isolated sphere is less in comparison to the concentric spheres. This is because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere.

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