Question

Find the capacitance of a spherical conductor of radius $9.0cm$. Also find the charge required to give it a potential of $1000V$.

Answer 1

Hint: In order to find the value of the capacitance of a spherical conductor we use a direct formula for the capacitance and we take air as the dielectric material. Once the capacitance is calculated we can apply that value in the direct formula for the charge.

Complete step by step answer:
Let’s define all the terms which are given in the question
Radius of the spherical conductor, $R = 9.0cm$=$9 \times {10^{ - 2}}m$
Potential of the conductor, $V = 1000V$
In the question, we are asked to find the value of the capacitance of a spherical conductor of radius $9.0cm$ and the charge required to give that conductor a potential of $1000V$
First we are calculating the value of the capacitance of a spherical conductor
It is given that the conductor is spherical in shape and its radius is also given.
It is known that the capacitance of a spherical conductor is given by,
$C = 4\pi {\varepsilon _0}R$………………. (1)
Where, $C$ is the capacitance of a spherical conductor
${\varepsilon _0}$ is the permittivity of the dielectric material (Here, dielectric material is air)
(The permittivity of air, ${\varepsilon _0} = 8.85 \times {10^{ - 12}}$)
$R$ is the radius of the conductor
Applying the known values in equation (1), we get,
$ \Rightarrow C = 4 \times 3.14 \times 8.85 \times {10^{ - 12}} \times 9 \times {10^{ - 2}}$
$ \Rightarrow C = 1000 \times {10^{ - 14}}F$
$ \Rightarrow C = 10 \times {10^{ - 12}}F$
$ \therefore C = 10pF$
That is, the value of the capacitance of a spherical conductor, $C = 10pF$
Now we need to find the charge required to give that conductor a potential of $1000V$
We know the charge of the charge of a conductor is given by the equation,
The charge of a conductor, $Q = CV$
Applying the known values to this equation, we get,
$ \Rightarrow Q = 10 \times {10^{ - 12}} \times 1000$
$ \therefore Q = 10 \times {10^{ -9}}C$
That is, the charge required to give the spherical conductor of radius $9.0cm$ a potential of $1000V$,
$Q = 10 \times {10^{ -9}}C$.

Note: Capacitance is defined as the ratio of the amount of electric charge stored on a conductor to its difference in electric potential. There are two closely related notions of capacitance: mutual capacitance and self capacitance. SI unit of the capacitance is farad ($F$).