A conducting loop with face-area A and resistance R is oriented perpendicular to a magnetic field B . The loop is completely removed from the field. Determine the charge that flows through any cross-section of the wire during this process. Note that this quantity depends neither on the shape of the loop nor on the method of withdrawal.

Step-by-step solution

Initial magnetic flux through the loop is _i = BA . Final magnetic flux through the loop is _f = 0 . According to Faraday's law of electromagnetic induction, the induced emf is given by E = - d dt . The induced current in the loop is I = E R = - 1 R d dt . The charge flowing through any cross-section in time dt is dq = I dt = - 1 R d . Integrating both sides, the total charge Q is: Q = dq = - 1 R _ _i ^ _f d = _i - _f R Substituting the values of initial and final flux: Q = BA - 0 R = BA R