Name: Faraday’s Law of Induction and Lenz’s Law
Uploaded: 2019-07-19
Duration: 10 min 7 s
Description: Here is an interesting video on electromagnetic induction explaining Faraday's laws of induction and Lenz's law. Get in to it to understand more!

Question

Obtain an expression for the magnetic energy stored in a solenoid in terms of the magnetic field, area and length of the solenoid.

Accepted Answer

A current carrying solenoid has a magnetic field due to current flowing through it. Thus electrical energy is stored as magnetic energy in it. Consider a coil of self-inductance $L$ with number of turns per unit length to be equal to $n$ and area of cross-section $A$ . If $I$ current flows through this, the magnetic field due to this current:

$B = μ_{0} n I$ .....(i)

Self-inductance:

$L = μ_{0} n^{2} A l$ .....(ii)

The self-induced emf in the coil opposes any change in current in the circuit. So work needs to be done against the back emf $(ε)$ in establishing the current. This work done is stored as magnetic potential energy. For the current $I$ at an instant in a circuit, the rate of work done is:

$\frac{d W}{d t} = |ε| I$

$\frac{d W}{d t} = L I \frac{d I}{d t}$

The total amount of work done in establishing the current $I$ is:

$W = \int d W = \int_{0}^{I} L I d I$

Thus, the energy stored in the coil:

$W = \frac{1}{2} L I^{2}$ .....(iii)

On putting the value from (i) and (ii) in (iii), we get:

$W = \frac{1}{2} μ_{0} n^{2} A l I^{2}$

$= \frac{1}{2} \frac{B^{2} A l}{μ_{0}}$ ......(iv)

(iv) is the required relation.

Obtain an expression for the magnetic energy stored in a solenoid in terms of the magnetic field, area and length of the solenoid.

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