Answer :

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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