Question

Define self-inductance of a coil. Derive the expression for magnetic energy stored in the inductor of inductance \[L\], carrying current \[I\].

Answer 1

Hint:Recall the basics of the self-inductance. Recall the mathematical formula for self-inductance in the coil and emf induced in the coil. Use the formula for work done in order to establish the electric current in the coil. Integrate this equation from zero to I and derive the mathematical expression for magnetic energy stored in the coil.

Complete answer:
The property of the coil by virtue of which there is decay or growth of the electric current induced in the coil. The self-inductance of a coil is defined as the ratio of total flux lined with all numbers of turns of the coil to the total electric current flowing through the coil.

The mathematical expression for self-inductance \[L\] of a coil is
\[L = \dfrac{{N\phi }}{I}\]
Here, \[N\] is the number of turns of the coil, \[\phi \] is the flux linked with the coil and \[I\] is electric current through the coil.

Suppose the electric current flowing through the coil is changed, there is change in the flux lined with the coil. Due to this changing flux linked with the coil, there is induction of the electromotive force (emf). This emf \[e\] induced in the coil is given by
\[e = - L\dfrac{{dI}}{{dt}}\]

This electromotive force induced in the coil opposes the change in electric current in the coil. Hence, it is also named as back electromotive force or back emf. Hence, the work needs to be done in order to induce current in the coil. This work done is stored in the coil known as magnetic energy stored in the coil.

The rate at which the work is done to establish an electric current in the coil is given by
\[\dfrac{{dW}}{{dt}} = \left| e \right|I\]
Substitute \[ - L\dfrac{{dI}}{{dt}}\] for \[e\] in the above equation.
\[\dfrac{{dW}}{{dt}} = \left| { - L\dfrac{{dI}}{{dt}}} \right|I\]
\[ \Rightarrow \dfrac{{dW}}{{dt}} = LI\dfrac{{dI}}{{dt}}\]
\[ \Rightarrow dW = LIdI\]

The work is done to establish an electric current for zero to \[I\] in the coil. Hence, the total work done is obtained by integrating the above equation from 0 to \[I\].
\[ \Rightarrow \int {dW} = \int_0^I {LIdI} \]
\[ \Rightarrow W = L\left[ {\dfrac{{{I^2}}}{2}} \right]_0^I\]
\[ \Rightarrow W = L\left[ {\dfrac{{{I^2}}}{2} - 0} \right]\]
\[ \therefore W = \dfrac{1}{2}L{I^2}\]

This is the required expression for magnetic energy stored.

Note:The students should keep in mind that the rate at which the work is done to establish an electric current in the coil includes the mod of the electromotive force. This gives the positive value of the expression of the magnetic energy stored. Otherwise, the final expression will have negative sign which is not the correct expression for magnetic energy stored.