(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) From Eq. (6.19), the magnetic energy is
`U_B = 1/2 LI^2`
`= 1/2 L ((B)/(mu_0 n))^2 " "`(since B = `mu_0 nI` , for a solenoid)
`=1/2 (mu_0n^2 Al) ((B)/(mu_0 n))^2 " "`(from Eq. (6.17)]
`=1/(2mu_0) B^2 Al`
(b) The magnetic energy per unit volume is,
`u_B = (U_B)/(V) " "` (where V is volume that contains flux)
`= (U_B)/(Al)`
`=(B^2)/(2mu_0)" "` (6.20)
We have already obtained the relation for the electrostatic energy stored per unit volume in a parallel plate capacitor (refer to Chapter 2, Eq. 2.77),
`u_E = 1/2 epsi_0 E^2 " "` (2.77)
In both the cases energy is proportional to the square of the field strength. Equations (6.20) and (2.77) have been derived for special cases: a solenoid and a parallel plate capacitor, respectively. But they are general and valid for any region of space in which a magnetic field or/and an electric field exist.