A circular coil of N turns and radius r is placed in a magnetic field that changes as B = B0 e^(-alpha t). The coil is connected to a resistor R. Find the total charge that flows through the resistor from t = 0 to t = infinity.

A. N r^2 B_0R — The total charge is determined by the total change in magnetic flux through the coil divided by the resistance.

JEE Main · PhysicsHard

$A circular coil of N turns and radius r is placed in a magnetic field that changes as B = B_{0} e^{- α t} . The coil is connected to a resistor R . Find the total charge that flows through the resistor from t = 0 to t = \infty .$

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Correct answer: A — $\frac{N π r ^{2} B _{0}}{R}$

$The total charge Q is given by \int I d t = \int (E / R) d t .$
$Since E = - N (d Φ/ d t), the integral becomes Q = - \frac{N}{R} \int_{Φ_{i}}^{Φ_{f}} d Φ .$
$The initial flux is Φ_{i} = N π r^{2} B_{0} and the final flux is Φ_{f} = 0 .$
$Thus, Q = - \frac{N}{R} (0 - N π r^{2} B_{0}) = N π r^{2} B_{0} / R .$
$Hence the answer is (A).$