Question

If the capacity of a spherical conductor is $1\;{\rm{\mu F}}$, then its diameter is
(A). $18 \times {10^3}\;{\rm{m}}$
(B). $1.8 \times {10^7}\;{\rm{m}}$
(C). $1.8 \times {10^3}\;{\rm{m}}$
(D). $18 \times {10^4}\;{\rm{m}}$

Answer 1

We will use the formula of capacitance of spherical capacitor to find the diameter of the capacitor. The formula of capacitance of spherical capacitor is given by $C = 4\pi {\varepsilon _0}r$.

Complete step by step answer:
Consider the formula $C = 4\pi {\varepsilon _0}r$.
Here, $C$ is the capacitance of the capacitor, ${\varepsilon _0}$ is permittivity of the vacuum, and $r$ is radius of the spherical capacitor.
The capacitance of any capacitor is the ratio of charge $\left( q \right)$ stored in that capacitor to the potential difference $\left( V \right)$ across the conductor.
The general formula of self capacitance of conductor is given by,
$C = \dfrac{q}{V}$
The SI unit of capacitance is Farad $\left( {\rm{F}} \right)$ or Micro Farad$\left( {{\rm{\mu F}}} \right)$ or PicoFarad $\left( {{\rm{pF}}} \right)$.
The dimension of the capacitance is $\left[ {{{\rm{M}}^{ - 1}}{{\rm{L}}^{ - 2}}{{\rm{T}}^4}{{\rm{I}}^2}} \right]$.
The capacitance is only dependent upon the geometry of the dielectric material and permittivity of the dielectric material. Sometime, the permittivity and capacitance is independent of the potential difference for many dielectric materials.
The property of a material to store electrical potential energy under the influence of an electric field which is measured by taking the ratio of the capacitance of a capacitor of dielectric material to its capacitance with vacuum as dielectric is called permittivity. It is also termed as dielectric constant.
The unit of permittivity is ${\rm{F/m}}$.
Substituting $1 \times {10^{ - 6}}\;{\rm{F}}$ for $C$, and $9 \times {10^9}$ for $\dfrac{1}{{4\pi {\varepsilon _0}}}$ in the formula $C = 4\pi {\varepsilon _0}r$, we get,
$1 \times {10^{ - 6}} = 4\pi {\varepsilon _0}r\\\ r = \dfrac{{1 \times {{10}^{ - 6}}}}{{4\pi {\varepsilon _0}}}\\\ = \left( {1 \times {{10}^{ - 6}}} \right)\left( {9 \times {{10}^9}} \right)\\\ = 9 \times {10^3}\;{\rm{m}}$
Calculate the diameter as follows.
$d = 2r\\\ = 2\left( {9 \times {{10}^3}} \right)\\\ = 18 \times {10^3}\;{\rm{m}}$

So, the correct answer is “Option A”.

Note:
In the formula of capacitance of spherical capacitor, the radius $r$ is used but sometimes students use diameter in place of $r$ which will result in obtaining the wrong answer.