The capacitance of the Parallel-Plate Capacitor with Dielectric Slab in Series

A capacitor stores electric energy when it is connected to its charging circuit.

And when it is disconnected from its charging circuit, it can dissipate that stored energy, so it can be used like a temporary battery.

The capacitors in a circuit can be arranged in series as well as in parallel combination.

In capacitors, a dielectric material is used in between the plates of the capacitors.

We can use one more dielectric materials in a capacitor along with their different arrangements.

And by saying "the different arrangement" means they can be arranged in parallel or in a series fashion.

And these combinations affects the capacitance of the capacitor.

Let's study the Capacitance of the Parallel-Plate Capacitor with Dielectric Slab in Series

Suppose we have a capacitor in which the three dielectric slabs are in series.

Let the dielectric constants of slabs 1, 2, and 3 are $K_{1}$, $K_{2}$, and $K_{3}$ respectively.

And the width of the dielectric slab 1 is $t_{1}$, width of slab 2 is $t_{2}$ and the width of slab 3 is $t_{3}$.

The area $A$ of each slab is same therefore they are connected in series.

If we introduce the thin metal plates between the slabs then they are separated from each other as these are in series.

Now, these become the capacitors $C_{1}$, $C_{2}$, $C_{3}$ which are connected in a series arrangement.

The equivalent capacitance of these slabs is $C$. Where $C_{1}$ is the capacitance of slab 1, $C_{2}$ is the capacitance of slab 2 and $C_{3}$ is the capacitance of slab 3.

The formula of the capacitance of the capacitor is.

So, the capacitance of the capacitor $C_{1}$ is.

This is the capacitance $C_{2}$ for the second slab.

And the capacitance $C_{3}$ for the slab 3 is,

Now the net capacitance of the Parallel-Plate Capacitor with Dielectric Slabs in Series is

Hence the net effective capacitance of the slabs in series will be given by,

Revision

This is the series arrangement of the capacitors.

The formula of the capacitance of the capacitor is.

The equivalent capacitance of when the slabs are connected in the series.

This is the capacitance of the capacitors connected in the series.