Question

How do you calculate the energy stored in a coil?

Answer 1

Step 1: Given data

The formula for the rate of work done is given by,

$$\begin{gathered} \Rightarrow \frac{\mathrm{dw}}{\mathrm{dt}}=-\mathrm{\epsilon i} \\ \Rightarrow \mathrm{dw}=-\mathrm{\epsilon idt} \end{gathered}$$

1. Where. $\frac{\mathrm{dW}}{\mathrm{dt}}=$rate of change of work done, $\mathrm{\epsilon }=$permittivity
2. But, $\epsilon =-\mathrm{L}\frac{\mathrm{di}}{\mathrm{dt}}$
3. Where$\mathrm{L}=$inductance and $\frac{di}{dt}=$rate of change of current.

Step 2: Calculate the work done

1. In order to pass current through the current work must be done by the voltage source against this emf.

$$\begin{gathered} \Rightarrow \frac{\mathrm{dw}}{\mathrm{dt}}=-\mathrm{\epsilon i} \\ \Rightarrow \mathrm{dw}=-\mathrm{\epsilon idt} \end{gathered}$$

Put the value of $\epsilon$ we get,

$\begin{array}{rcl}& \Rightarrow & \mathrm{dw}=-\left(-\mathrm{L}\frac{\mathrm{di}}{\mathrm{dt}}\right)\mathrm{idt}\\ & \Rightarrow & \mathrm{dw}=\mathrm{iLdi}\end{array}$

Step 3: Integrate the work done for calculating the energy stored in a coil

Now, integrating from$\mathrm{L}.\mathrm{H}.\mathrm{S}\mathrm{from}0\mathrm{to}\mathrm{W}$and on $\mathrm{R}.\mathrm{H}.\mathrm{S}\mathrm{from}0\mathrm{to}I$, we get

$$\begin{gathered} \Rightarrow {\int }_0^{\mathrm{W}}\mathrm{dW}={\int }_0^{\mathrm{I}}\mathrm{iLdi} \\ \Rightarrow W=\mathrm{L}{\int }_0^{\mathrm{I}}\mathrm{idi} \\ \Rightarrow \mathrm{W}=\mathrm{L}{\left[\frac{{\mathrm{i}}^2}{2}\right]}_0^{\mathrm{I}} \\ \Rightarrow \mathrm{W}=\mathrm{L}\left[\frac{{\mathrm{I}}^2}{2}\right] \\ \Rightarrow \mathrm{W}=\frac{1}{2}{\mathrm{LI}}^2 \end{gathered}$$

1. Where, $W=$work done which is equal to energy, $\mathrm{L}=$inductance, and $\mathrm{I}=$current
2. The energy stored in the magnetic field is generated in the inductor due to the current flow.

Hence, the expression for energy stored in an inductance coil carrying current is $\mathrm{W}=\frac{1}{2}{\mathrm{LI}}^2$.