Energy Stored in a Capacitor Derivation

Energy Stored in the Capacitor

The energy stored in a capacitor is the same as the work done to build up the charge (0 to Q) on the plates by an external source (battery) of voltage.

Energy Stored in a Cipacitor

Derivation

Consider a capacitor of a capacitance of C and it is given a small amount of charge (dq) at a given voltage (V), the work done is

dw = V dq

The total work to charge the capacitor to a charge zero to Q is

`W\ =\ \int_{0}^{Q} V dq`

`W\ =\ \int_{0}^{Q}\frac{q}{C}\ dq`

`W\ =\ \frac{1}{C}\int_{0}^{Q}q\ dq`

`W\ =\ \frac{1}{C} [\frac{q^2}{2}]_0^Q`

`W\ =\ \frac{1}{2\ C}(Q^2\ -\ 0)`

`W\ =\ \frac{Q^2}{2\ C}`

This work is stored as energy in the capacitor (W = U).

`U =\ \frac{Q^2} {2 C}`

We know that Q = CV then

`U =\ \frac{(C V)^2}{2\ C}`

`U = \frac{C^2\ V^2}{2\ C}`

`U =\frac{1}{2}\ C\ V ^2`

Now, putting `V = \frac{Q}{C}`

`U\ =\frac{1}{2}\ C\ \frac{Q^2}{C^2}`

`U\ =\frac{1}{2}\ \ \frac{Q^2}{C}`

`U\ =\ \frac{Q^2}{2\ C}`

Thus, Energy stored in a capacitor

`U\ =\ \frac{Q^2}{2\ C\} =\ \frac{1}{2}\ C\ V^2=\ \frac{Q^2}{2\ C}`

Where,

U = Energy stored in the capacitor (in joules)

C = Capacitance (in farads)

V = Voltage across the capacitor (in volts)

Q = Charge (in Coulomb)

Explanation of Formula

Capacitance (C)

Capacitance is a measure of the capacitor’s ability to store charge. The higher the capacitance, the more energy a capacitor can store.

Voltage (V)

This is the potential difference across the capacitor’s terminals. The stored energy increases with the square of the voltage.

Practical Example

To understand how much energy a capacitor can store, let’s consider an example. If you have a capacitor with a capacitance of 100 microfarads and voltage of 10 volts across it, the energy stored is –

`U\ =\frac{1}{2}\ C\ V^2\ =\ \frac{1}{2}\ (100\ \times\ 10^{-\ 6})\ 10^2\ =\ \ \frac{1}{2}\ \times\ 10^{-\ 2}\ =\ 0.005` Joules

So, with these values, the capacitor stores 0.005 joules of energy.

Safety Note

Capacitors can store significant energy, expecially at high voltages, and can release it suddenly, posing risks of electric shock or damage to circuits. Always cischarge capacitors safely and take precautions when handling them in high-voltage applications.