# A tightly-wound,

Given:

Number of turns per unit length = n

Surface current of sheet = K

Formula used:

Ampere’s circuital law states that the line integral of the magnetic field for a closed surface is μ0 times the current enclosed by the surface.

,

Where

B = magnetic field,

dl = line element,

μ0 = magnetic permeability of vacuum,

I = current in the circuit.

(a) Since the magnetic field near the center of the solenoid is 0, we infer that the magnetic field due to the solenoid = magnetic field due to the sheet

For the sheet, using Ampere’s circuital law, we get:

B x 2l = μ0 x Kl, where Kl = current enclosed by sheet and we take 2l because of the two surfaces of the sheet.

=> Bsolenoid = μ0K/2

Now, magnetic field due to a solenoid Bsolenoid = μ0ni, where μ0 = magnetic permeability of vacuum, i = current, n = number of turns per unit length

Hence, μ0K/2 = μ0ni => i (current in solenoid) = K/2n (Ans)

(b) Now, if the axes of the solenoid and the plate are perpendicular to each other, their magnetic fields are also perpendicular to each other.

Hence, net magnetic field at center,

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