Question

DERIVE AN EXPRESSION FOR ENERGY STORED IN A COIL CARRYING CURRENT.

Answer 1

Hi,

Let an ac current is applied to a inductor of inductance L such that current in the inductor grows from zero to maximum value I₀. Let I be the current at any instant t.The emf(back emf) induced in the inductor is given by
$E=L\frac{dI}{dt}$
This will oppose any change in the current in the circuit. For the current I at an instant t, the rate of doing work is
$\frac{dW}{dt}=EI$
If we iqnore the resistive losses, then

$$\begin{gathered} \frac{dW}{dt}=EI=L\frac{dI}{dt}\times I \\ \Rightarrow dW=LIdI \end{gathered}$$

Total amount of work done in establishing the current I is
$$\begin{gathered} W=\int dW \\ ={\int }_0^{I}LIdI \\ =\frac{1}{2}L{I}^2 \end{gathered}$$
Total energy stored in a coil carrying current is
$U=W=\frac{1}{2}L{I}^2$