The graph below shows the magnetic field Bt as a function of

The graph below shows the magnetic field B(t) as a function of time within a conducting loop, with the time divided into equal intervals. Assume that the loop lies in the plane of the page and that positive B means the field points out of the page. (a) Rank the intervals (a. b, c, d, e, f. g, h) according to the magnitude of the induced emf. Explain your reasoning. (b) Identify the direction of the induced current (clockwise, counterclockwise, or zero) in each interval. Explain your reasoning.

Solution

In presence of magnetic field passing through a closed loop, we know that if we change the magnetic flux through the loop, there is an emf induced in the loop.

Part a.) The emf induced in a loop is given as: EMF = d/dt = d(B.A) / dt = A(dB/dt) [Since A is constant for the given situation and makes a fixed angle of 90 degrees with the magnetic field]

Hence we can say that for a B vs T graph the region with maximum rate of change of B is the one with maximum emf induced, that is, the region with the maximum slope signifies maximum EMF.

Therefore region b will have maximum emf, followed by d, e, f and h and then a, c and g

It needs to be noted that the slopes for d = e = f = h and also, a = c = g, hence they will represent same EMFs

Part b.) Lenz\'s Law tells us is that the direction of the induced current in the loop is such that it \"opposes the change in the flux\". Also we know for any current we can curl our fingers around it to find the field it generates. Hence, for the magnetic field coming out of the page, the emf induced would be along clockwise when the field is increased as in that case, the field generated by the current would oppose the magnetic field as in oppose its increasing magnitude.

The current would anti-clockwise when the field is decreased as in that case, the field by the current would be directed out of the page and would try to add up to the field already there, hence trying to oppose the decrease in the magnitude. Also the current induced would be zero for no change in magnetic field. Therefore we have:

Region a: Zero

Region b: Clockwise

Region c: Zero

Region d: Anti-Clockwise

Region e: Anti-Clockwise

Region f: Anti-Clockwise

Region g: Zero

Region h: Anti-Clockwise