(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 ?

Explanation

(a) Consider a solenoid (or an inductor) of cross-sectional area A, length l, volume V = Al and self inductance L. Suppose current passing through it at time t is I and rate of increase of current is `(dI)/(dt)`. At this instant, if magnitude of self induced emf across the inductor is `epsilon` then we have,
`epsilon=L(dI)/(dt)`
Now, if electrical power in the inductor is P then
`P=epsilonI`
`therefore P=(L (dI)/(dt))I`
`therefore Pdt = LI dI`
`therefore dW=LI dI`
Above is the work to be done against self induced emf to increase the current by amount dl. This work done gets stored in the magnetic field inside the solenoid in the form of magnetostatic potential energy `dU_B`. Thus, `dU_B=LI dI`
If current through the solenoid is increased from 0 to I in time interval from 0 to t then if total magnetostatic potential energy stored in the solenoid is `U_B` then,
`int_0^U_B dU_B=L int_0^I I dI`
`therefore {U_B}_0^U_B = L{I^2/2}_0^I`
`therefore U_B=1/2 LI^2`...(1)
Now , magnetic field inside the solenoid, is
`B=mu_0 nI=mu_0 (N/l)I`
`therefore I=(Bl)/(mu_0N)`...(2)
Self inductance of a solenoid is given by
`L=(mu_0N^2A)/l` ...(3)
(where N=total no. of turns in a solenoid)
From equations (1),(2)and (3)
`U_B=1/2 ((mu_0N^2A)/l)((B^2l^2)/(mu_0^2N^2))`
`=1/2B/mu_0(Al)`
`therefore U_B=1/2 B^2/mu_0(V)`....(4) (where V=Al=volume of solenoid)
Above equation gives required expression of total magnetostatic potential energy (or magneic energy ) stored in the magnetic field , inside a current carrying solenoid having volume V.
(b)Magnetostatic potential energy stored in a solenoid (or inductor ) per unit volume is called magnetostatic potential energy density , shown by symbol `rho_B`. Thus,
`rho_B=U_B/V`...(5)
`therefore rho_B=B^2/(2mu_0)`
Above equation is analogous to equation or electrostatic potential energy density `rho_1` (or `U_E`) given as,
`rho_E=1/2 in_0 E^2`...(6)