$(a)$ 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.
$(b)$ How does this magnetic energy compare with the electrostatic energy stored in a capacitor?

(N/A) The magnetic energy stored in an inductor is given by $U_{B} = \frac{1}{2} L I^{2}$.
For a solenoid,the magnetic field is $B = \mu_{0} n I$,where $n = N/l$ is the number of turns per unit length.
Thus,$I = \frac{B}{\mu_{0} n}$.
The self-inductance of a solenoid is $L = \mu_{0} n^{2} A l$.
Substituting these into the energy formula:
$U_{B} = \frac{1}{2} (\mu_{0} n^{2} A l) \left(\frac{B}{\mu_{0} n}\right)^{2} = \frac{1}{2} (\mu_{0} n^{2} A l) \left(\frac{B^{2}}{\mu_{0}^{2} n^{2}}\right) = \frac{B^{2} A l}{2 \mu_{0}}$.
$(b)$ The magnetic energy density (energy per unit volume) is $u_{B} = \frac{U_{B}}{V} = \frac{U_{B}}{A l} = \frac{B^{2}}{2 \mu_{0}}$.
The electrostatic energy density stored in a parallel plate capacitor is $u_{E} = \frac{1}{2} \varepsilon_{0} E^{2}$.
In both cases,the energy density is proportional to the square of the field strength ($B^{2}$ or $E^{2}$). These expressions are general and valid for any region of space where electric or magnetic fields exist.