Question

If the capacity of a spherical conductor is $1$ picofarad, then its diameter would be
(A) $1.8 \times {10^{ - 3}}m$
(B) $18 \times {10^{ - 3}}m$
(C) $1.8 \times {10^{ - 5}}m$
(D) $18 \times {10^{ - 5}}m$

Answer 1

We have to put the expression for the voltage on the sphere for a given charge stored by the conductor into the basic formula of the capacitance to get the expression of the capacitance for a spherical conductor. On putting the value given in the question in the question in that expression, we will get the final answer.

Complete step by step solution:
We know that the capacity, or the capacitance of a conductor is defined as the charge stored by the conductor per unit voltage applied. That is,
$C = \dfrac{Q}{V}$ ............................(1)
Now, according to the question, we have been given the capacity of a spherical conductor and we are asked about its diameter. So we need to deduce the expression for the capacitance of a spherical conductor in terms of its geometrical parameters.
Consider a spherical conductor of radius $R$ which stores a charge of $Q$ on its surface. We know that the potential of this sphere is given by
$V = \dfrac{Q}{{4\pi {\varepsilon _0}R}}$
Putting this in (1) we get
$C = \dfrac{Q}{{\dfrac{Q}{{4\pi {\varepsilon _0}R}}}}$
$\Rightarrow C = 4\pi {\varepsilon _0}R$
So this is the required expression for the capacitance of a spherical conductor in terms of its radius.
According to the question the capacitance of the given spherical conductor is equal to $1$ picofarad. This means that $C = 1pF$ . We know that $1pF = {10^{ - 12}}F$ , so we have $C = {10^{ - 12}}F$ . So we substitute this in the above expression to get
${10^{ - 12}} = 4\pi {\varepsilon _0}R$
$\Rightarrow R = \dfrac{{{{10}^{ - 12}}}}{{4\pi {\varepsilon _0}}}$
We know that $\dfrac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}N{m^2}/{C^2}$ . Substituting this above, we get
$R = 9 \times {10^9} \times {10^{ - 12}}m$
$\Rightarrow R = 9 \times {10^{ - 3}}m$
Now, we know that the diameter of a sphere is equal to twice its radius. So the diameter of the given spherical conductor is
$D = 2R$
$\Rightarrow D = 2 \times 9{\kern 1pt} \times {10^{ - 3}}m = 18 \times {10^{ - 3}}m$
Thus, the diameter of the given spherical conductor is equal to $18 \times {10^{ - 3}}m$ .
Hence the correct answer will be option B.

Note
Do not forget to calculate the diameter of the given spherical conductor. So do not end the solution by calculating the radius of the spherical conductor. Capacitance of a conductor depends on material of the conductor and size and shape of the conductor.