Parallel plate capacitor with dielectric slab

Capacitance formula derivation of parallel plate capacitor with dielectric slab inserted of thickness t.

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Well, In this article, we are going to derive the expression for capacitance of parallel plate capacitor when a dielectric slab of thickness t is inserted between the plates. Wants to know all about the derivation, stay tuned with us till end…

DERIVATION FOR THE CAPACITANCE

Let’s take a parallel plate capacitor having plates surface area is A. A dielectric slab of thickness t is inserted between the plates. And the dielectric slab have the dielectric constant K and the permittivity of it’s medium is ε. See figure below :

Parallel plate capacitor with dielectric slab
Image source : sciencehq.com

After putting dielectric slab of thickness t the remaining distance will be (d – t). Let the electric field inside the dielectric slab is E and field inside the plates and outside the slab is E_0.

DERIVATION PROCESS

The electric field inside the slab is given as-

    \[E=\frac{\sigma}{\epsilon}=\frac{\sigma}{K\epsilon_0}=\frac{q}{\epsilon_0KA}\]

And the electric field outside the slab is given as –

    \[E_0=\frac{\sigma}{\epsilon_0}=\frac{q}{\epsilon_0A}\]

The total potential difference between the plates is given as the sum of the potential difference due to the slab and due to without slab. Which is clearly written below :

    \begin{align*} V&=E_0\left(d-t\right)+Et\\ &=\frac{q\left(d-t\right)}{\epsilon_0A}+\frac{qt}{\epsilon_0KA}\\ &=\frac{Kq\left(d-t\right)+qt}{\epsilon_0KA}\\ &=\frac{Kqd-kqt+qt}{\epsilon_0KA}\\ &=\frac{q\left[kd-kt+t\right]}{\epsilon_0KA} \end{align*}

Capacitance of any capacitor is given as charge per unit potential difference. Let’s put the value of potential difference in the formula of capacitance.

    \begin{align*} C&=\frac{q}{V}\\ &=\frac{q}{\left(\frac{q\left[kd-kt+t\right]}{\epsilon_0KA}\right)}\\ C&=\frac{\epsilon_0KA}{Kd-kt+t}\\ &=\frac{\epsilon_0KA}{Kd+t\left(1-K\right)} \end{align*}

This is the required value of capacitance of parallel plate capacitor when a dielectric slab of thickness t is inserted between the plates.

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One comment

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