A very long, tightly-wound solenoid has a circular cross section of radius r = 2

Question

A very long, tightly-wound solenoid has a circular cross section of radius r = 2.8 cm (only a portion of the very long solenoid is shown in the figure). The magnetic field outside the solenoid is negligible. Throughout the inside of the solenoid the magnetic field B is uniform, out of the page as shown, but varying with time t: B = [0.07 + 0.03t^2] tesla. Surrounding the circular solenoid is a loop of N = 3 turns of wire in the shape of a rectangle 11 cm high by 20 cm wide. The total resistance of the 3-turn loop is 0.1 ohm. At t = 3.6 seconds, what is the direction of the current in the 3-turn loop? there is no current in the loop clockwise counter-clockwise At t = 3.6 seconds, what is the magnitude of the current in the 3-turn loop? (remember that the magnetic field is negligible in the region outside the solenoid.) A

Explanation / Answer

Here ,

N = 3

radius ,r = 2.8 cm = 0.028 m

B = 0.07 + 0.03 t^2

dB/dt = 0.06 * t

R = 0.1 Ohm

a) at t = 3.6 s

as the magnetic field is increasing , the induced magetic field will oppose the applied magentic field

hence , the induced magnetic field is into the page

the direction of current is clockwise

b)

magnitude of induced current = rate of change of flux/time

magnitude of induced current = pi * N * dB/dt * r^2/R

magnitude of induced current = pi * 3 * 0.06 * 3.6 * 0.028^2/0.1

magnitude of induced current = 0.0159 A

the magnitude of induced current in the loops is 0.0159 A

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