Class 12 > Unit # 19:Electromagnetic Induction > Energy stored in an Inductor

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Electromagnetic Induction - Energy Stored in an Inductor

Q. Describe an expression for the energy stored in an inductor.

Conceptual Background

Consider an inductor connected to a DC power source through a switch. When the switch is closed, the current in the inductor gradually rises from zero until it reaches its maximum value, denoted as $I$. This changing current leads to a corresponding change in the magnetic flux within the coil, establishing an induced electromotive force (emf) in accordance with Lenz's law.

This back emf works to oppose the growth of current driven by the battery. Consequently, the battery must perform work against this back emf to build up the current $I$ to its maximum value. This work is stored within the magnetic field of the inductor as potential energy.

Figure: Building up current in an inductor and storing energy in its magnetic field.

Mathematical Derivation

The infinitesimal work $dW$ done by the source in moving a small charge $dq$ against the induced back emf $\mathcal{E}$ is expressed as:

$dW = \mathcal{E} \, dq$

Since electric current is the rate of flow of charge ($I = \frac{dq}{dt} \implies dq = I \, dt$), and the magnitude of the back emf in the inductor is $\mathcal{E} = L \frac{dI}{dt}$, we can substitute these into the work equation:

$dW = \left( L \frac{dI}{dt} \right) (I \, dt) = L I \, dI$

To find the total work ($W$) done in building the current from zero to its final maximum value $I$, we integrate both sides from $0$ to $I$:

$W = \int_{0}^{I} L I \, dI = L \int_{0}^{I} I \, dI = L \left[ \frac{I^2}{2} \right]_{0}^{I}$

$W = \frac{1}{2} L I^2$

By the work-energy relation, this total work done by the battery is stored inside the inductor as magnetic potential energy ($E$):

$E = \frac{1}{2} L I^2$

Terms & Interpretation

• $E$: Energy stored in joules ($\text{J}$).
• $L$: Self-inductance of the inductor in Henries ($\text{H}$).
• $I$: Current flowing through the inductor in amperes ($\text{A}$).

This formula indicates that the energy stored in an inductor is directly proportional to the square of the current passing through it and depends linearly on its self-inductance. As the current increases, the stored magnetic energy increases correspondingly.

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