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1. Turning off a solenoid [4 points (hand-in)] 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) = po(t), where I (t) the current in the solenoid. lllll 1 a) Assume the solenoid is turned of at t = 0 and the current goes down as I(t) = Ioe-t/T. Compute the induced electric field Eind(t). In which direction does it point compared to the current in the wire? N b) Derive the characteristic time 7 in terms of the resistance R of the wire and the inductance L = Tа² μN²/l (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-Vind. 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, do + ind - 0.