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?

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.

**Read Also**

- Sharpness of Resonance: Q-Factor in LCR circuit, class 12
- Power in an AC circuit: definition, and formula derivation
- Choke coil – principle, working, and construction, class 12
- Resonance condition in a series LCR circuit
- Wattless current class 12
- Expression for energy and average power stored in an inductor

## 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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