Inside an ideal solenoid with n turns per unit length and current I, what is the magnetic field?

• A. μ0 I
• B. μ0 n I
• C. μ0 n I / 2
• D. μ0 I n^2

Answer: (B)

In an ideal long solenoid, the magnetic field inside is uniform and points along the axis, while outside the field is essentially zero. Use Ampere’s law: ∮ B·dl = μ0 I_enclosed. Take a rectangular loop that runs along the length L inside the solenoid; the field is constant, so ∮ B·dl = B L. The current enclosed is the total current from all turns within length L, which is nL turns times the current I, giving I_enclosed = n L I. Setting B L = μ0 n L I and canceling L gives B = μ0 n I. So the field depends on both the current and the turns per unit length.