Question

Define self-inductance of a coil. Obtain the expression for the energy stored in an inductor L connected across a source of emf.

Answer 1

Hint: You have to find the magnetic field of the solenoid and use it to find the magnetic flux through each turn of the solenoid. Then you equate the two equations of total flux to obtain inductance of the solenoid. Using the inductance, you can integrate partial derivatives of work done to find total work. This work done is stored as potential energy (U) in the magnetic field of the inductor.

Formula used:
B=$\dfrac{{{\mu }_{0}}NI}{l}$
$\phi =\dfrac{{{\mu }_{0}}NI}{l}\times A$
$\int\limits_{0}^{W}{dW}=\int\limits_{0}^{{{I}_{0}}}{LIdt}$

Complete step by step answer:
The property for the coil to oppose the change in current flowing through it is known as the self-inductance.
Let us consider the inductance of a long solenoid to be L.

The magnetic field inside such solenoid is constant at any point which is equal to
B =$\dfrac{{{\mu }_{0}}NI}{l}$, where μ0 = absolute magnetic permeability
N=total number of turns
We should calculate the magnetic flux through each turn of solenoid
$\phi =B\times A$, where A=area of cross section of the solenoid
$\phi =\dfrac{{{\mu }_{0}}NI}{l}\times A$
\[Total\text{ }flux\text{ }=\text{ }flux\times total\text{ }number\text{ }of\text{ }turns\]
$N\phi =N\times (\dfrac{{{\mu }_{0}}NI}{l}\times A)$………………. (1)
Nφ=LI …………….. (2)
Let us equate equation 1 and equation 2
 $LI=N\times (\dfrac{{{\mu }_{0}}NI}{l}\times A)$
$L=\dfrac{{{\mu }_{0}}{{N}^{2}}A}{l}$…………….. (3)
The magnitude of emf is given by
$e=L\dfrac{dI}{dt}$…………. (4)
If we multiply I on both sides
eIdt = LIdI
Also I=$\dfrac{dq}{dt}$
Idt=dq
Work done = Voltage*charge
dW=eIdt
On substituting the values in equation 4 we get,
dW = LIdt
Now let’s integrate both sides
$\int\limits_{0}^{W}{dW}=\int\limits_{0}^{{{I}_{0}}}{LIdt}$
$W=\dfrac{1}{2}LI_{0}^{2}$
So this work done is stored as potential energy (U) in the magnetic field of the inductor.
$U=\dfrac{1}{2}LI_{0}^{2}$

Note:
The term inductor is used to indicate a circuit element attaining the qualities of inductance. A coil of wire is an example of an inductor. A coil symbol is used to indicate an inductor in a circuit diagram. A voltage is induced in a wire carrying a changing current which opposes the current itself. The alternating current runs throughout the coil. This creates a magnetic field in and around the coil that increases or decreases as the current changes.