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

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.

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.