Question

Give the expression of energy stored in an inductance coil carrying current.

Answer 1

Hint: To find the expression of energy stored in an inductance coil or inductor carrying current, recall the formula for emf generated in an inductor. Then use this value of emf to find the expression for the rate of work done and using this value find the expression for work done.

Complete answer:
We are asked to find the energy stored in an inductance coil carrying current.Suppose a current is applied to an inductor or inductance coil of inductance $$L$$ such that current through the inductor grows from zero value to a maximum value $$I$$. Let current through the inductor at any instant of time $$t$$ be $$i$$.A emf induced in the inductor which opposes the flow of current and is given by the formula,
$$\epsilon =-L{\displaystyle \frac{di}{dt}}$$
where $$L$$ is the inductance and $${\displaystyle \frac{di}{dt}}$$ is the rate of change of current.
In order to pass current through the current, work must be done by the voltage source against this emf. The formula for rate of work done is given by,
$${\displaystyle \frac{dW}{dt}}=-\epsilon i$$
$$\Rightarrow dW=-\epsilon idt$$
Putting the value of $$\epsilon$$ we get,
$$dW=-\left(-L{\displaystyle \frac{di}{dt}}\right)idt$$
$$\Rightarrow dW=iLdi$$
Now, integrating on L.H.S from $$0$$ to $$W$$ and on R.H.S from $$0$$ to $$I$$, we get
$$\underset{0}{\overset{W}{\int }}dW=\underset{0}{\overset{I}{\int }}iLdi$$
$$\Rightarrow W=L\underset{0}{\overset{I}{\int }}idi$$
$$\Rightarrow W=L{\left[{\displaystyle \frac{i^2}{2}}\right]}_0^I$$
$$\Rightarrow W=L\left[{\displaystyle \frac{I^2}{2}}\right]$$
$$\therefore W={\displaystyle \frac{1}{2}}L{I}^2$$
This energy is stored in the magnetic field generated in the inductor due to the flow of current.

Therefore, the expression for energy stored in an inductance coil carrying current is $$W={\displaystyle \frac{1}{2}}L{I}^2$$.

Note: Remember, one function of an inductor is to store electrical energy. There is one more component called capacitor. A capacitor stores energy in the electric field whereas an inductor stores energy in the magnetic field, students sometimes get confused between these two components.