Expression for energy and average power stored in a pure capacitor

In this article, we will derive an Expression for energy and average power stored in a pure capacitor, so let’s get started…

What is a capacitor?

Expression for energy and average power stored in a pure capacitor
Capacitors

Capacitor definition: Capacitors are passive electronic components consisting of two or more pieces of conductive material separated by an insulating material. The capacitor is a component that has the ability or “capacity” to store energy in the form of an electrical charge by creating a potential difference (static voltage) across its plates, like a small rechargeable battery.

In its basic form, a capacitor consists of two or more parallel conductive (metal) plates that are not connected or touching but are electrically separated by air or a type of highly insulating material called a dielectric. This insulating material can be wax paper, mica, ceramic, plastic, or some type of liquid gel used in electrolytic capacitors.

Expression for Energy stored in a capacitor

Consider a capacitor of capacitance $C$. Suppose the displacement of charge $q$ from one plate to another sets up a potential difference $V$ between its plates. Then
$$
V=\frac{q}{C}
$$
Suppose now a small additional charge $d q$ be displaced from one plate to another. Then work done is
$$
d W=V d q=\frac{q}{C} d q
$$
$\therefore$ Total work done in displacing a charge $q$ from one plate to another is
$$
W=\int_0^q d W=\int_0^q \frac{q}{C} d q=\frac{1}{2} \frac{q^2}{C}
$$

This energy is stored as the electrostatic energy $U$ in the capacitor.
$$
\therefore \quad U=\frac{1}{2} \frac{q^2}{C}=\frac{1}{2} C V^2 \quad[\because q=C V]
$$

Expression for Average power stored in a capacitor

When an a.c. is applied to a capacitor, the current leads the voltage in phase by $\pi / 2$ radian. So we write the expressions for instantaneous voltage and current as follows :
$$V=V_0 \sin \omega t$$
and $\qquad I=I_0 \sin \left(\omega t+\frac{\pi}{2}\right)=I_0 \cos \omega t$

Work done in the circuit in small time $d t$ will be \begin{aligned}d W&=P d t=V I d t \\&=V_0 I_0 \sin \omega t \cos \omega t d t \\&=\frac{V_0 I_0}{2} \sin 2 \omega t d t\end{aligned}
The average power dissipated per cycle in the capacitor is

\begin{aligned}
P_{a v} &=\frac{W}{T}=\frac{1}{T} \int_0^T d W\\&=\frac{V_0 I_0}{2 T} \int_0^T \sin 2 \omega t d t \\
&=\frac{V_0 I_0}{2 T}\left[-\frac{\cos 2 \omega t}{2 \omega}\right]_0^T \\
&=-\frac{V_0 I_0}{4 T \omega}\left[\cos \frac{4 \pi}{T} t\right]_0^T \\
&=-\frac{V_0 I_0}{4 T \omega}[\cos 4 \pi-\cos 0] \\
&=-\frac{V_0 I_0}{4 T \omega}[1-1]=0
\end{aligned}

Thus the average power dissipated per cycle in a capacitor is zero.

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Frequently Asked Questions – FAQs

What is a capacitor in simple words?

capacitor, a device for storing electrical energy, consisting of two conductors in close proximity and insulated from each other. A simple example of such a storage device is the parallel-plate capacitor.

What is the expression for energy stored in a capacitor?

 Energy $U$ is stored in a capacitor is given as $$
U=\frac{1}{2} \frac{q^2}{C}=\frac{1}{2} C V^2 \quad[\because q=C V]
$$

Why is the average power in a capacitor zero?

In AC Circuit the average power in the capacitor and inductor is zero because both of them store energy in the half cycle of the AC and despite energy in the second half.

What is the unit of average power?

The SI Unit of average power is Watt and is represented as ‘w’ where Watt is Joules per second.

What is the working principle of the capacitors?

The principle of a capacitor is based on an insulated conductor whose capacitance is increased gradually when an uncharged conductor is placed next to it.

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