two expressions for the energy
The internal field of a solenoid is B = μ0 nI = μ0 (N/ℓ)I, and the volume
is V = πr²ℓ. From Problem 7.11, the self-inductance of a long solenoid is
L = μ0πr²N²/ℓ. So the two expressions give the same energy if
1/2 LI² = 1/(2μ0) B²V
⟸⟹ 1/2 · μ0πr²N²/ℓ · I² = 1/(2μ0) · (μ0 NI/ℓ)² · πr²ℓ, (12.365)
which is indeed true.
720 Solutions to the problems