Q60 of 92 Page 249

A tightly-wound, long solenoid is kept with its axis parallel to a large metal sheet carrying a surface current. The surface current through a width dl of the sheet is Kdl and the number of turns per unit length of the solenoid is n. The magnetic field near the center of the solenoid is found to be zero.

(a) Find the current in the solenoid.


(b) If the solenoid is rotated to make its axis perpendicular to the metal sheet, what would be the magnitude of the magnetic field near its center?


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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57

A tightly-wound solenoid of radius a and length l has n turns per unit length. It carries an electric current i. Consider a length dx of the solenoid at a distance x from one end. This contains n dx turns and may be approximated as a circular current i n dx.

(a) Write the magnetic field at the center of the solenoid due to this circular current. Integrate this expression under proper limits to find the magnetic field at the center of the solenoid.


(b) Verify that if l >> a, the field tends to B = μ0ni and if a >> l, the field tends to


B=. Interpret these results


 58

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A capacitor of capacitance 100 μF is connected to a battery of 20 volts for a long time and then disconnected from it. It is now connected across a long solenoid having 4000 turns per meter. It is found that the potential difference across the capacitor drops to 90% of its maximum value in 2.0 seconds. Estimate the average magnetic field produced at the center of the solenoid during this period.