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A toroid has a circular cross-section of radius b and major (central) radius R (with R >> b). It carries N total turns and current I. Find the total energy stored in the toroid.

1. mu₀ * N² * I² * b² / (2R)
2. mu₀ * N² * I² * b² / (3R)
3. mu₀ * N² * I² * b² / (6R)
4. mu₀ * N² * I² * b² / (4R)

Correct answer: mu₀ * N² * I² * b² / (4R)

Solution

For a toroid of major radius R (circumference 2*pi*R) and minor radius b (cross-section area A = pi*b²), in the limit R >> b the field inside is uniform: B = mu₀*N*I/(2*pi*R). The inductance is L = mu₀*N²*A/(2*pi*R) = mu₀*N²*(pi*b²)/(2*pi*R) = mu₀*N²*b²/(2R). Total energy stored: U = (1/2)*L*I² = (1/2)*[mu₀*N²*b²/(2R)]*I² = mu₀*N²*I²*b²/(4R).

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