A long solenoid with 2930 turns/m carries a current which is dynamically varying

Question

A long solenoid with 2930 turns/m carries a current which is dynamically varying in time according to t = 10 A + (1000 A/s) t as the switch is closed at t = 0. A circular wire loop is located inside the solenoid as shown in the figure. The circular loop has a radius of r = 10 cm and the angle between the solenoid axis and the normal of the loop is theta = 30 degree. a) What is the induced emf in the wire loop? What will be the direction of induced current (clockwise/counterclockwise) in the observer's point of view? b) The circular loop in the solenoid has a capacitor and a resistor in the loop as shown in the figure below. The capacitor was uncharged initially. For R = 100 ohm, and C = 50 mu F, calculate the charge on the capacitor at 5 ms after the switch is closed.

Explanation / Answer

Here ,

a) for the solenoid

magnetic field inside the solenoid = u0 * n * I

Now, for the induced emf in the loop

induced emf = pi * r^2 * cos(theta) * u0 *n * dI/dt

induced emf = pi * 0.10^2 * cos(30 degree) * 4pi *10^-7 * 2930 * d/dt(10 + 1000 * t)

induced emf = pi * 0.10^2 * cos(30 degree) * 4pi *10^-7 * 2930 * 1000

induced emf = 0.1002 V

the induced emf is 0.1002 V

b)

at t = 5 ms = 0.005 s

charge on the capacitor = C * V (1 - e^(-t/(R * C))

charge on the capacitor = 50 * 0.1002 * (1 - e^(-0.005/(50 *10^-6 *100)))

charge on the capacitor = 3.17 uC

the charge on the capacitor at 5 ms is 3.17 uC

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