Answer Happy

Consider a long solenoid of radius a, length 1 and winding number N with tight winding. The magnetic field at the center can be taken to be homogeneous inside with magni- S tude B(t) = μoI(t), where I (t) the current in the solenoid. N = a) Assume the solenoid is turned of at t = 0 and the current goes down as I (t) Compute the induced electric field Ēind(t). In which direction does it point compared to Ioe-t/T. the current in the wire? b) Derive the characteristic time 7 in terms of the resistance R of the wire and the inductance L = ла² µN²/1 (note that the time-dependence drops out). Why is the turn-off process visible for a solenoid, but not for normal circuits? Hints: a) Use a Maxwell equation (together with an integral theorem) and the symmetry of the fields. b) The potential difference between the ends of the wire due to the current is 1 = IR. After the turn-off, there is an additional potential difference Pind due to the induced electric field Eind = -√ind. To obtain ind, view the solenoid as N consecutive current loops. Since there is no potential applied to the solenoid, these must add up to zero, 0 + ind - 0.