Chapter 9
Electromagnetic Induction

9.1.1  Observation of Induction - Lenz's Law

Symmetry in nature predicts that, if a magnetic field acting on a loop of conductor carrying a current can produce a torque on the loop (assuming that the magnetic moment of the loop is not aligned with the field), then a torque acting on a conducting loop in a magnetic field should produce a current.

Figure 9.1: Symmetry in producing a torque for a loop with an externally maintained current going through it is balanced in nature by having an externally applied torque on a conducting loop produced a current in the opposite direction.

Note that the symmetry cannot be "perfect" in that the direction of the current that is induced in the loop due to the externally maintained torque must be opposite to the direction of current that would create a torque in that direction. This is necessary due to energy conservation. If the effect happened as in our first case (externally maintained current), then more current would go through the loop and the induced torque would increase. This violates energy conservation as the change in angular position due to the external torque would induce still more change in angular position. Hence we note that if the effect of induced current does take place, the current must be induced in a direction that opposes the external torque that causes it.

Faraday was the first to show that this induced current effect does happen. In studying the effect, we find that the magnitude and direction of the induced current effect is due to a change in the magnetic flux through an area in space. This effect is referred to as electromagnetic induction and the quantitative description of the effect is Faraday's Law. Specifically, Faraday's Law states that

e = - dFB dt

(9.1.1.1)

In other words, Faraday expressed his effect by saying that the magnitude of the EMF induced by any change of magnetic flux is proportional to the magnitude of the change in magnetic flux. The induced EMF is what causes the current to flow. To correspond to the necessity of energy conservation, the direction of the EMF must be such that the induced current would oppose the change in the magnetic flux. This was best expressed by H.F.E. Lenz in Lenz's Law which can be succintly stated as:

The direction of any magnetic induction effect must oppose the cause of the effect.

The easiest way to generate a static picture of this is to observe the four cases below of a bar magnet being pushed toward and away from a conducting loop as shown in figure 9.2. The current created by the induced EMF must oppose the change in each case.

Figure 9.2: A stationary conducting loop with a bar magnet that can be alternately pushed toward or away from the loop with either of two orientations of the poles (i.e. North pole toward the loop or away from it).

Make sure you understand figure 9.2. It shows, in each case, how the induced magnetic field from the induced current (which comes from the induced EMF assuming the loop has some amount of resistance) opposes the change in magnetic flux. Since magnetic flux is a scalar, it might be difficult to think of a "direction" for it, but the sign of the flux depends on the arbitrary convention for the sign for area, A. Once that is set, then the dot product of B and A gives the positive or negative sign for the flux. The negative sign in Faraday's Law then gives the appropriate sign for the direction of the induced EMF. In the first case of figure 9.2, the induced current creates an induced field which points opposite to the external magnetic field from the bar magnet since the flux going down through the loop is increasing with time. For the third case, the induced current creates an induced field which points opposite to the external magnetic field since the flux pointing up through the loop is increasing. Verify that the second and fourth cases shown make sense to you following the same argument. The importance of Faraday's effect is hard to overestimate. Most of the world runs on electricity generated as a result of it.

9.1.2  Faraday's Law Applied

If we consider the case for a conducting bar sliding over rails in the presence of a magnetic field, we have a good case for examining both Lenz's Law and considering a quantitative example of Faraday's Law. Figure  9.3 shows the situation. To see a dynamic view of the effect, go to the following Java applet. It's based on the following image of what happens as a bar slides across two rails in the presence of a magnetic field.

Figure 9.3: A conducting bar slides on frictionless conducting rails in the presence of a magnetic field pointing out of the page.

Problem 1:

Find the magnitude and direction of the current induced in the sliding bar of figure 9.3 assuming that the magnitude of the magnetic field, the resistance R and the speed v of the sliding bar are all known for some instant of time. Also find the magnitude of the minimum external force needed to keep the sliding bar in motion at constant speed (assume no friction is acting).

Solution:

First, note that the area defined by the rails and the region to the left of the sliding bar is given at any instant of time by A = Lx where L is the length across the rails and x is the position of the bar relative to the left edge of the circuit. Therefore, the time rate of change of the magnetic flux through this area is

dFB dt = B dA dt = BL dx dt = BLv

(9.1.2.2)

Therefore, the magnitude of the induced EMF in the bar is

|e| = BLv

(9.1.2.3)

The direction of this EMF has to be down the bar as shown in the figure since we wish to have a current in the bar whose magnetic field points down through the area to the left of the bar (verify that this is true by use of the modified right-hand rule applied to the downward current shown in the figure). This EMF opposes the increase in the flux by attempting to decrease the net field through the area A. The magnitude of the current induced is

I = e R = BLv R

(9.1.2.4)

The external magnetic field exerts a force on this current, F_induced = ILB, directed to the left (again, you should check this with the right-hand rule) so there must be an external force F_ext = ILB in order to maintain constant velocity for the sliding bar. Therefore,

Fext = B2L2v R

(9.1.2.5)

Note also that the direction is entirely consistent whether you consider the area to the left of the bar as increasing with time or the area to the right of the bar as decreasing with time (check to make sure that you agree that the current induced should point down the bar in either case).

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Problem 2:

In figure  9.4, a conducting bar slides frictionlessly upward under the influence of some external force (not shown). Neglect gravity and find the current (direction and magnitude) induced in the bar assuming that the velocity of the bar is constant and known, and that the magnetic field magnitude and resistance of the bars is known.

Figure 9.4: A conducting bar with resistance R sits on frictionless conducting rails and is pulled upward at constant velocity.

Solution:

We need to specify the area contained by the sliding bar and the rails. Let's choose the area contained below the bar. Specify the bar position as y above the vertex of the rails and w as the distance from the rail to the midpoint of the bar along the bar (see figure  9.5).

Figure 9.5: Induced current for the problem shown in figure 9.4.

Trigonometry tells us that

tan q 2 = w y

(9.1.2.6)

thus the area beneath the sliding bar is

A = 2 æ ç è 1 2 wy ö ÷ ø = w y = y2tan q 2 .

(9.1.2.7)

We first note that this area is increasing with time, thus the induced EMF and hence the induced current in the sliding bar must be to the left. The magnetic field due to the induced current then points up through area A and opposes the increased flux as the area increases. We can now find the induced current in the bar

Iinduced = einduced R = 1 R B dA dt = B R 2y dy dt tan q 2 = B R (2(vt)v)tan q 2 Iinduced = 2Bv2t R tan q 2 (9.1.2.8)

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Problem 3:

Suppose a conducting loop with width w, length l, and resistance R is falling, due to gravity, in a non-uniform magnetic field pointed into the page. The magnitude of the magnetic field is characterized by

B(y) = B0y

(9.1.2.9)

as shown in figure  9.6. Find the terminal velocity for the loop.

Figure 9.6: A rectangular conducting loop falls in a non-uniform magnetic field. The induced current produces a force that opposes the motion.

Solution:

We see that the non-uniform magnetic field means that the magnetic flux into the page through the loop increases with time. The flux at any instant with the position of the center of mass of the loop at y is

FB = ó õ y+w/2 y - w/2  (B0y¢)·l dy¢ = B0l. y¢ 2 2 ê ê ê y+w/2 y-w/2  = B0l 2 [(y + w/2)2 - (y - w/2)2] FB = B0l 2 [2wy] (9.1.2.10)

The induced EMF, according to Faraday's Law, has magnitude as follows.

|e| = ê ê ê dFB dt ê ê ê = B0l 2 é ê ë 2w dy dt ù ú û |e| = B0lwvy (9.1.2.11)

with v_y being the instantaneous velocity of the center of the loop. The direction of the EMF should be counter-clockwise so as to oppose the flux increase as the loop falls. If the loop has resistance R, then the current through the loop is counter-clockwise with magnitude

Iinduced = |e| R = B0lw R vy

(9.1.2.12)

The net force on the loop, assuming equilibrium, i.e. the loop is at its terminal velocity, is given by the vector sum of the gravitational force and the magnetic forces acting on the top and bottom segments of the loop. Note that, by symmetry, the force on the two sides of the loop must cancel (see figure 9.7).

Figure 9.7: The forces acting due to gravity and the magnetic forces on the induced current in the loop falling through a non-uniform magnetic field.

At equilibrium, i.e. when the loop falls at constant speed v_y, we have

Fnet = 0 = mg + Ftop - FbotÞ mg = |Fbot| - |Ftop| = IinducedlB0ybot - IinducedlB0ytop = IinducedlB0(ybot - ytop) mg = B02l2w R vy(w) Þ vy = mgR (B0wl)2 (9.1.2.13)

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