A very long solenoid of length L has n layers. There are N turns in each layer. Diameter of the solenoid is D and it carries current I. The magnetic field at the centre of the solenoid is .

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(3) Independent of D. Given: a long solenoid of length L with n layers of N turns each, so nN turns in all, carrying a current I. Idea: apply Ampere's law to a rectangle with one side of length ℓ running along the axis inside and the opposite side outside, where the field is negligible. Only the inside side contributes, so Bℓ=μ₀N'I with N' the number of turns threaded by the rectangle. That gives B=μ₀n₀I, where n₀=(N')/ℓ is the number of turns per unit length. Here n₀=(nN)/L, so B=(μ₀nNI)/L. The diameter never enters: the interior field is uniform and is set by the winding density and the current alone. So the field is independent of D.

A very long solenoid of length L has n layers. There are N turns in each layer. Diameter of the solenoid is D and it carries current I. The magnetic field at the centre of the solenoid is ______.

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