Q.Obtain the expression for the magnetic energy stored in a solenoid in terms of magnetic field B, area A and length l of the solenoid and hence obtain formula for magnetic energy per unit volume.

Magnetic Energy Density

A magnetic field stores energy. Building up a current in an inductor requires work against the induced back-emf, and that work is stored in the field around it. The magnetic energy density is the energy stored per unit volume of the field.

Energy stored in an inductor

When the current in an inductor of self-inductance $L$ grows, the induced back-emf is $\varepsilon = -L\,dI/dt$. The work done by the source to push charge $dq = I\,dt$ against this emf is

Integrating from $0$ to the final current $I$,

This energy is stored in the magnetic field of the inductor.

From inductor to field: the energy density

Take a long solenoid with $n$ turns per unit length, cross-sectional area $A$ and length $l$. Its self-inductance is $L = \mu_0 n^2 A l$, and the field inside is $B = \mu_0 n I$, so $I = B/(\mu_0 n)$. The stored energy becomes

Since $Al$ is the volume occupied by the field, the energy per unit volume is

Although derived for a solenoid, this result is general: wherever a magnetic field $B$ exists, it carries an energy density $B^2/2\mu_0$, measured in $\text{J/m}^3$.

Comparison with the electric field

The electric field stores energy at density $u_E = \tfrac{1}{2}\varepsilon_0 E^2$. The magnetic analogue $u_B = B^2/2\mu_0$ has the same structure, and together they give the energy carried by electromagnetic waves. …