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A uniform but time-varying magnetic field B = Kt - C (valid for 0 <= t <= C/K), where K and C are positive constants and t is time, is applied perpendicular to the plane of a circular loop of radius a and resistance R. Find the total charge that flows through a cross-section of the loop.

1. C*pi*a² / R
2. C*pi*a² / (2R)
3. C*pi*a / R
4. 2*C*pi*a / R

Correct answer: C*pi*a² / R

Solution

Total charge through a loop Q = delta_flux / R = (phi_final - phi_initial) / R. At t = 0: B = K*0 - C = -C. At t = C/K: B = K*(C/K) - C = 0. Change in B = 0 - (-C) = C. Flux change = C * pi*a². So Q = C*pi*a² / R.

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