Question

A parallel plate capacitor with air between the plates has a capacitance of $8pF$ the separation between the plates is now reduced by its half and the space between them is filled with a medium of dielectric constant $5$ Calculate the value of capacitance of the capacitor in the second case.

Answer 1

In order to solve this question, we will first calculate the capacitance without dielectric medium in terms of general parameters and then will change the parameters as distance between plates is half and introduced the dielectric medium and then we will substitute given parameters value to find capacitance in second case.

Formula used:
If A, d, C be the area of plates, distance between plates, capacitance of a parallel plate capacitor then,
$C = \dfrac{{{ \in _o}A}}{d}$
where ${ \in _o}$ is the relative permittivity of free space.
if dielectric constant K is introduced between the plates of a capacitor then,
$C = \dfrac{{K{ \in _o}A}}{d}$

Complete step by step answer:
According to the question, we have given that without dielectric medium the capacitance of parallel plate capacitor is $C = 8pF$ so, putting this value in formula we get,
$C = \dfrac{{{ \in _o}A}}{d}$
$\Rightarrow 8 = \dfrac{{{ \in _o}A}}{d} \to (i)$

Now, in second case let d’ be the new distance between the plates and its being half from original distance so we have, $d' = \dfrac{d}{2}$ and dielectric constant of $K = 5$ is introduced so, new capacitance C’ can be written using formula,
$C = \dfrac{{K{ \in _o}A}}{d}$
$\Rightarrow C' = \dfrac{{5{ \in _o}A}}{{d'}}$
$\Rightarrow C' = \dfrac{{10{ \in _o}A}}{d} \to (ii)$

Now, divide the first and second equation we get,
$\dfrac{8}{{C'}} = \dfrac{{\dfrac{{{ \in _o}A}}{d}}}{{\dfrac{{10{ \in _o}A}}{d}}}$
On solving we get,
$C' = 8 \times 10$
$\therefore C' = 80\,pF$

Hence, the capacitance of the parallel plate capacitor in the second case is $C' = 80\,pF$.

Note: It should be remembered that, while changing the distance between the plates, the area of plate don’t change and the pF in the unit of capacitance stands for Pico-farad and this is related as $1pF = {10^{ - 12}}F$ ,capacitance is measured in the units of called Farad.