Chapter 29

Electromagnetic Induction

In this chapter we investigate how changing the magnetic flux in a circuit inducesan emf and a current. We learned in Chapter 25 that an electromotive force (E) isrequired for a current to flow in a circuit. Up until now, we used a battery as our emfsource. However, most of our power comes from electric generating stations wherethe source of power is gravitational potential energy, chemical energy, and nuclearenergy. How are these forms of energy converted into electrical energy? Thisis accomplished by electromagnetic induction. In all power-generating stations,magnets move relative to coils of wire to produce a changing magnetic flux in thecoils and hence an emf.

In this chapter we will study Faraday’s law. This relates the induced emf to chang-ing magnetic flux in any loop. We also discuss Lenz’s law, which helps predictthe direction of the induced emfs and currents. These principles are at the heartof electrical energy conversion devices such as motors, generators, and transform-ers.

Finally, electromagnetic induction tells us that a time-varying magnetic field canact as a source of electric field. We will also see how a time-varying electric fieldcan act as a source of magnetic field. These are the principle concepts embodied inwhat has been called Maxwell’s equations. Just as Newton’s laws describe classicalkinematics and dynamics, Maxwell’s equations provide the fundamental frameworkfor electrodynamics.

1 Induction Experiments

Magnetically induced emf was first of observed in the 1830’s by Michael Faraday(England), and by Joseph Henry (United States). The changing flux of magneticfield passing through a closed loop (or circuit) was observed to produce an in-duced current and the corresponding emf required to cause this current is callan induced emf (E).

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Figure 1: This figure shows the measurements observed when a changing magnetic flux occurs in acoil of wire connected to a galvanometer.

The following induction experiments were performed using the appara-tus shown in Fig. 2 below.

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Figure 2: This figure is used to describe the observations made below. The figure shows a coilplaced inside an electromagnet and connected to a galvanometer. The strength of the magneticfield shown with the N and S pole faces can be adjusted by varying the current into the device.

What was observed?

1. When there is no current in the electromagnet, in other words ~B = ~0, thegalvanometer show no current.

2. When the electromagnet is turned on, there is momentary current through themeter as ~B increases.

3. When ~B levels off at a steady value, the current drops to zero.

4. With the coil in a horizontal plane, we squeeze it so as to decrease the cross-sectional area of the coil. The meter detects current only during the deforma-tion, not before or after. When we increase the area to return the coil to itsoriginal shape, there is current in the opposite direction, but only while thearea of the coil is changing.

5. If we rotate the coil a few degrees about a horizontal axis, the meter detectscurrent during the rotation, in the same direction as when we decreased thearea. When we rotate the coil back, there is a current in the opposite directionduring this rotation.

6. If we decrease the number of turns in the coil by unwinding one or more turns,

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there is a current during the unwinding, in the same direction as when wedecreased the area. If we wind more turns onto the coil, there is a current inthe opposite direction during the winding.

7. When the magnet is turned off, there is a momentary current in the directionopposite to the current when it was turned on.

8. The faster we carry out any of these changes, the greater the current.

9. If all these experiments are repeated with a coil that has the same shape butdifferent material and different resistance, the current in each case is inverselyproportional to the total circuit resistance. This shows that the induced emfsthat are causing the current do not depend on the material of the coil but onlyon its shape and the magnetic field.

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2 Faraday’s Law

The common feature in all induction effects is changing magnetic flux through acircuit. For an infinitesimal-area elements d ~A in a magnetic field ~B, the magneticflux dΦB through the area is:

dΦB = ~B · d ~A = B⊥ dA = B dA cosφ

Figure 3: This figure shows how the magnetic flux is calculated as it passes through a surface.

Figure 4: Calculating the flux of a uniform magnetic field through a flat area. Different orientationsof the area result in different amounts of flux ΦB.

ΦB = ~B · ~A = BA cosφ

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Faraday’s Law of Induction states:

E = − dΦB

dt(Faraday’s Law) (1)

E = −(dB

dtA cosφ + B

dA

dtcosφ − BA sinφ

(dφ

dt

))There are 3 ways to induce an emf E:

1. Change the strength of the magnetic field ~B,

2. Change the area of the loop or circuit, and

3. Change the angle the area makes with respect to the ~B field.

2.1 Direction of Induced emf

Figure 5: The magnetic flux ΦB is increasing in (a) and (d) thus inducing an negative emf. Infigures (b) and (c) the magnetic flux ΦB is decreasing, thereby inducing a positive emf

.

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2.2 Magnitude and Direction of an Induced EMF

Figure 6: In this figure the uniform magnetic field is decreased at a rate of 0.200 T/s. What is theinduced emf in this 500-loop wire coil of radius 4.00 cm?

2.3 A Simple Alternator

Figure 7: A schematic diagram of an alternator is shown (a). The conducting loop rotates in amagnetic field, producing an emf shown in (b). Connections from each end of the loop to theexternal circuit are made by means of that end’s slip ring. One cycle of the generated emf is shownin the figure.

E = − dΦ

dt= −BA d

dtcosφ = −BA d

dt(cosωt) = BAω sin(ωt)

This is the starting point for producing AC power.

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A DC Generator

Figure 8: This figure is used to describe the observations made on the following page.

The time-averaged back-emf for a coil having N loops is:

Eav =2NωBA

π

2.4 The Slidewire Generator

Figure 9: This figure is used to describe the observations made on the following page.

In this case the magnetic flux changes due to the changing area dA/dt.

E = − dΦ

dt= −B dA

dt= −B (Lv dt)

dt= −BLv

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The power required to move the bar is equal to the power dissipated by the I2Rloss in the current loop.

Papplied =B2L2v2

R

3 Lenz’s Law

The direction of any magnetic induction effect is such as to oppose thecause of the effect

Example: Lenz’s Law and the Slidewire Generator

3.1 Lenz’s Law and the direction of Induced Current

Figure 10: This figure shows how to visualize the induced emf, and current, when the strength ofthe external B-field is increased. Lenz’s law says that there will be an opposing B-field, ( ~Binduced)is produced in response to the increasing external field.

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Figure 11: This figure shows four possible ways of increasing or decreasing the flux with twomagnetic poles. In all cases, the induced B-field is created in a manner that opposes the change tothe electric field. The induced B-field is produced as a result of the induced emf and its associatedcurrent.

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4 Motional Electromotive Force

Figure 12: This figure shows a conducting rod moving in a uniform magnetic field. The velocityof the rod and the field are mutually perpendicular. In part (a) an electric field is induced in the

bar as it moves through the ~B field, and positive charges migrate upward. In part (b) of the figureshows the direction of the induced current in the circuit when the bar is connected to a conductingloop.

The magnitude of the potential difference Vab − Va−Vb is equal to the electric-fieldmagnitude E multipled by the length L of the rod. Since E = vB, we have:

Vab = EL = vBL

The motional emf generated in part (b) of the figure is:

E = vBL

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4.1 Motional emf: General Form

Figure 13: This figure shows how the motional emf arises for a current loop moving in a staticmagnetic field. The velocity ~v can be different for different elements if the loop is rotating orchanging shape. The magnetic field ~B can also have different values at different points around theloop.

The motional emf produced for a segment of the loop is:

dE = (~v × ~B) · d~

For any closed conducting loop, the total emf is:

E =

∮ (~v × ~B

)· d~ (2)

∮~Enc · d~ 6= 0 (non-conservative ~E field)

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5 Induced Electric Fields

Figure 14: This figure shows the winding on a solenoid carrying current I that is increasing at arate dI/dt. The magnetic flux in the solenoid is increasing at a rate dΦ/dt, and this changing fluxpasses through a wire loop. An emf E = −dΦ/dt is induced in the loop, inducing a current I ′ thatis measured by the galvanometer G.

The magnetic flux created by the solenoid is:

Φ = BA = µo nIA

When the solenoid current changes, so does the flux Φ.

E = − dΦ

dt= µo nA

dI

dt

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There is an induced electric field in the conductor caused by the changing mag-netic flux.

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5.1 Nonelectrostatic Electric Fields

Figure 15: Here are some applications of induced electric fields. (a) This hybrid automobile hasboth a gasoline engine and an electric motor. As the car comes to a halt, the spinning wheelsrun the motor backward so that it acts as a generator. The induced current is used to rechargethe car’s batteries. (b) The rotating crankshaft of a piston-engine airplane spins a magnet (i.e.,a magneto) inducing an emf in an adjacent coil and generating the spark that ignites fuel in theengine cylinders. This process continues even if the airplane’s other electrical systems fail.

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6 Eddy Currents

In previous examples we observed the induction effects and the induced currentsthey produced. However, many pieces of electrical equipment contain masses ofmetal moving in magnetic fields or located in changing magnetic fields (e.g., trans-formers). In these situations we have induced currents that circulate throughoutthe volume of the material. Because their flow patterns resemble swirling eddies ina river, we call them eddy currents.

Figure 16: These figures (a and b) shows the eddy currents induced in a rotating metal disk as it

rotates through a magnetic field. In figure (b) you can see the braking force ~F that occurs from the

I~L× ~B opposing the rotation of the disk.

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Figure 17: Eddy currents can be used in (a) metal detectors at airport security checkpoints. When

the transmitting coil generates an alternating magnetic field ~Bo, this induces eddy currents in aconducting object carried through the detector. The eddy currents in turn produce an alternatingmagnetic field ~B′, which induces a current in the detector’s receiver coil. The same principle is usedfor portable metal detectors shown in fig. (b).

7 Displacement Current and Maxwell’s Equations

In the previous sections we have seen how varying magnetic fields give rise to aninduced electric field. One of the more remarkable symmetries of nature is that avarying electric field gives rise to a magnetic field. This symmetric phenomenonexplains the existence of radio waves, gamma rays, and visible light, as well as allother forms of electromagnetic waves.

In order to illustrate this symmetric relationship between varying electric fields andmagnetic fields, we return to Ampere’s law:∮

~B · d~ = µo Iencl

However, when we apply this law to the charging of a capacitor (see the figure be-low), we see that this equation is incomplete. No conduction current iC is observedbetween the capacitor plates; however, there is current iC coming into the posi-tiely charged plate and current iC leaving the negatively charged plate. While the

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right-hand side of Ampere’s law is equal to µoiC for the planar surface, it’s equalto zero if we use the bulging surface. However, it cannot be both, so, something ismissing.

However, something else is happening at the bulging surface. While the charge isbuilding up on the capacitor plates, the ~E-field is changing between the plates. Infact, the flux of the ~E-field ΦE is increasing. Let’s try to find a relationship betweenthe conducting current iC and the changing electric field dE/dt. The instantaneouscharge on the capacitor q is related to the instantaneous voltage v by the followingequation: q = vC where C is the capacitance ε/Ad.

Figure 18: This figure shows a current charging a parallel-plate capacitor. The conduction currentthrough the plane surface is iC , but there is no conduction current through the surface bulging outbetween the plates. The two surfaces have a common boundary, so this difference in iencl leads toan apparent contradiction in applying Ampere’s law.

q = Cv =εA

d(Ed) = εEA = εΦE (3)

By taking a time derivative of both sides of this equation, we can relate the cur-rent coming in to the capacitor to the changing E-field building up between theplates

iC =dq

dt= ε

dΦE

dt

where

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iD = εdΦE

dt

is called the displacement current.

We include this fictitious current, along with the real conduction current iC , torewrite Ampere’s law as: ∮

~B · d~ = µo (iC + iD)encl (4)

Figure 19: This figure shows a capacitor being charged by a current iC with a displacement currentequal to iC between the plates, with displacement current density jD = ε dE/dt. This can beregarded as the source of magnetic field between the plates.

The displacement current, along with the magnetic field it produces, is morethan an artifice; it is “real” and a fundamental property of nature.

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8 Maxwells’ Equations of Electromagnetism

We can now summarize all the relationships between electric and magnetic fieldsinto a package of four equations, called Maxwell’s equations. Two of Maxwell’sequations involve an integral of ~E or ~B over a closed surface.

The third and fourth equations involve a line integral of ~E or ~B around a closedpath. Faraday’s law states that a changing magnetic flux acts as a source of electricfield.

1. ∮~E · d ~A =

Qencl

εo

2. ∮~B · d ~A = 0

3. ∮~E · d~ = − dΦB

dt(Faraday’s Law)

4. ∮~B · d~ = µo

(iC + εo

dΦE

dt

)The electric field in Maxwell’s equations includes both conservative and non-conservative components of the ~E field:

~E = ~Ec + ~Enc

N.B. These four equations are highly symmetric in empty space. The third andfourth equations can be rewritten in empty space (iC = 0 and Qencl = 0) asfollow: ∮

~E · d~ = − d

dt

∫~B · d ~A (5)

∮~B · d~ = µoεo

d

dt

∫~E · d ~A (6)

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