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Two parallel rectangular superconducting plates, each of length ℓ and width b, are separated by a distance a (with ℓ >> b >> a). They are joined at both ends to form a single-turn loop of negligible resistance carrying a steady current I. Find the energy stored in the magnetic field between the plates.

1. U = μ0 ℓ a I² / b
2. U = μ0 a b I² / (2 ℓ)
3. U = μ0 ℓ b I² / (2 a)
4. U = μ0 ℓ a I² / (2 b)

Correct answer: U = μ0 ℓ a I² / (2 b)

Solution

The surface current per unit width is K = I/b, giving a uniform field between the plates B = μ0 K = μ0 I/b. Energy density u = B²/(2μ0) = μ0 I²/(2 b²). Volume between plates = ℓ * b * a. So U = u * V = [μ0 I²/(2b²)] * (ℓ b a) = μ0 ℓ a I²/(2 b).

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