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The capacitance of a spherical condenser is \[1\mu F\] . If the spacing between the two spheres is \[\text{1 mm}\] , then the radius of the outer sphere is
(A) \[30cm\]
(B) \[6m\]
(C) \[5cm\]
(D) \[3m\]

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Last updated date: 05th Mar 2024
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Answer
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Hint: A spherical capacitor consists of a solid or hollow spherical conductor of a certain radius surrounded by another concentric spherical conductor of a larger radius. The capacitance for spherical conductors can be obtained by finding the voltage difference between the conductors for a given charge.

Formula Used:
 \[C=\dfrac{4\pi {{\varepsilon }_{0}}{{R}_{1}}{{R}_{2}}}{{{R}_{2}}-{{R}_{1}}}\]

Complete step by step answer:
We have been provided with the capacitance of the spherical capacitor and the spacing between the two conductors, which is the difference between their radii.
Capacitance of the spherical capacitor \[(C)=1\mu F={{10}^{-3}}F\] since \[1\mu F={{10}^{-6}}F\]
Also, spacing between the conductors \[({{R}_{2}}-{{R}_{1}})=1mm={{10}^{-3}}m\]
Now since the spacing is very small, we can consider the two capacitors to have almost equal radii, that is \[{{R}_{1}}\approx {{R}_{2}}\]
Also, we know the value of the constant in the capacitor formula, that is, \[\dfrac{1}{4\pi {{\varepsilon }_{0}}}=9\times {{10}^{9}}\]
Substituting all the values listed above in our formula, we get
\[\begin{align}
  & C=\dfrac{4\pi {{\varepsilon }_{0}}{{R}_{1}}{{R}_{2}}}{{{R}_{2}}-{{R}_{1}}} \\
 & \Rightarrow {{10}^{-6}}F=\dfrac{{{R}_{1}}{{R}_{2}}}{9\times {{10}^{9}}\times {{10}^{-3}}} \\
 & \Rightarrow {{R}_{1}}{{R}_{2}}=9{{m}^{2}} \\
 & \Rightarrow {{R}_{2}}^{2}=9{{m}^{2}}(\because {{R}_{1}}\approx {{R}_{2}}) \\
 & \Rightarrow {{R}_{2}}=3m \\
\end{align}\]
Hence the outer and the inner radius of the spherical conductors forming the capacitor are approximately equal to \[3m\] (a difference of one millimetre in their radius)
But we are only concerned with the outer radius and hence option (D) is the correct answer.

Additional Information: Every spherical conductor having a certain amount of charge acts as a capacitor, even an isolated sphere is considered as a capacitor whose second plate is at infinity. The applications for an isolated spherical capacitor or a pair of spherical capacitors illustrate that a charged sphere has some stored energy as a result of being charged.

Note: Although we have assumed the radii of the two spheres to be equal, we didn’t take their difference to be zero because the spacing between them, although very negligible, is still a gap and means that their radii are only approximately equal. We had to make the assumption because we didn’t have any other piece of information to help us solve the question.