The external magnetic field oscillates as B = B_0 k cos (omega t) with B_0 = 5 T

Question

The external magnetic field oscillates as B = B_0 k cos (omega t) with B_0 = 5 Tesla, omega = 25 rad/s. The radius of the circle of wire is 0.5 meters. The resistance is 30 Ohm and the capacitance is 1 m F What is the steady-state solution for the current in the circuit? (As drawn, positive current goes counterclockwise.) (You may wish to put a phase constant in your solution, such as cos(omega t - phi) or sin(omega t - phi). Recall that in the book and class presentation, the voltage source was ahead of the current.")

Explanation / Answer

problem 2:

flux through the circle

=B*area

=5*cos(25*t)*pi*0.5^2=3.927*cos(25*t)

emf induced=-time derivative of flux

=3.927*25*sin(25*t)

=98.175*sin(25*t)

steady state current=(98.175/Z)*sin(25*t+theta)

where Z=sqrt(R^2+(1/(w*C))^2)

=50 ohms

theta=arctan(1/(R*w*C))

=53.13 degrees

current=1.9635*sin(25*t+53.13 degrees) A

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