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Energy Stored In Inductor

Energy Stored In Inductor :- When an inductor is connected to an AC source, it opposes the increase in current due to self-induction. Suppose at some instant the current in the circuit is i, then the induced electromotive force (emf) in the inductor is :

$\displaystyle \varepsilon =-L\frac{di}{dt}$

Small amount of work done by the alternating source for the increase in current during a small time interval dt,

$\displaystyle dW=-(\varepsilon )\left( i \right)dt$

$\displaystyle dW=-(-L\frac{di}{dt})\left( i \right)dt$

$\displaystyle \Rightarrow dW=Lidi$

The total work done by the AC source to raise the current in the inductor from 0 to I,

$\displaystyle W=\int{dW}=\int{Lidi=L\int\limits_{0}^{I}{idi}}$

$\displaystyle \Rightarrow W=L\left[ \frac{i{}^{2}}{2} \right]_{0}^{I}=L\left[ \frac{{{I}^{2}}}{2}-0 \right]$

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

This work gets stored in the inductor as magnetic potential energy (U_B); therefore, the energy stored in the inductor is,

$\displaystyle {{U}_{B}}=\frac{1}{2}L{{I}^{2}}$ …..(1)

Comparison Of The Inductance Of An Inductor With Inertia

(Energy Stored In Inductor)

The behavior of an inductor in an electric circuit is similar to the behavior of mass in translational motion in mechanics. This can be understood as follows :

Inductor in an electric circuit :

An inductor, due to its property of inductance, opposes any change in the current flowing through it. The induced electromotive force in an inductor is :

$\displaystyle \varepsilon =-L\frac{di}{dt}$ …..(2)

Mass in translational motion :

In mechanics, the mass of an object resists any change in its velocity due to its property of inertia. From Newton’s second law :

$\displaystyle F=m\frac{dv}{dt}$ …..(3)

Similarities between inductance and mass :-

(1). Inductance (L) corresponds to mass (m) because both oppose change — an inductor opposes change in current, and mass opposes change in velocity.

(2). The induced emf (ε) across an inductor is analogous to the external force (F) applied on a mass.

(3). The current (I) in an inductor is analogous to the velocity (v) of a mass.

(4). The rate of change of current ( $\displaystyle \frac{di}{dt}$ ) is analogous to the rate of change of velocity (acceleration) ( $\displaystyle \frac{dv}{dt}$ ).