Electromagnetic Induction, Magnetism | Physics

Learning objectives

Section Key Terms

emf

induction

magnetic flux

Changing Magnetic Fields

In the preceding section, we learned that a current creates a magnetic field. If nature is symmetrical, then perhaps a magnetic field can create a current. In 1831, some 12 years after the discovery that an electric current generates a magnetic field, English scientist Michael Faraday (1791–1862) and American scientist Joseph Henry (1797–1878) independently demonstrated that magnetic fields can produce currents. The basic process of generating currents with magnetic fields is called induction; this process is also called magnetic induction to distinguish it from charging by induction, which uses the electrostatic Coulomb force.

When Faraday discovered what is now called Faraday’s law of induction, Queen Victoria asked him what possible use was electricity. “Madam,” he replied, “What good is a baby?” Today, currents induced by magnetic fields are essential to our technological society. The electric generator—found in everything from automobiles to bicycles to nuclear power plants—uses magnetism to generate electric current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances.

One experiment Faraday did to demonstrate magnetic induction was to move a bar magnet through a wire coil and measure the resulting electric current through the wire. A schematic of this experiment is shown in Figure_20_02_Ammeter. He found that current is induced only when the magnet moves with respect to the coil. When the magnet is motionless with respect to the coil, no current is induced in the coil, as in Figure_20_02_Ammeter. In addition, moving the magnet in the opposite direction (compare Figure_20_02_Ammeter with Figure_20_02_Ammeter) or reversing the poles of the magnet (compare Figure_20_02_Ammeter with Figure_20_02_Ammeter) results in a current in the opposite direction.

View original figure (external; image not reproduced in this text edition)

Original figure description: Movement of a magnet relative to a coil as measured by an ammeter.

Movement of a magnet relative to a coil produces electric currents as shown. The same currents are produced if the coil is moved relative to the magnet. The greater the speed, the greater the magnitude of the current, and the current is zero when there is no motion. The current produced by moving the magnet upward is in the opposite direction as the current produced by moving the magnet downward.

Induced Electromotive Force

If a current is induced in the coil, Faraday reasoned that there must be what he called an electromotive force pushing the charges through the coil. This interpretation turned out to be incorrect; instead, the external source doing the work of moving the magnet adds energy to the charges in the coil. The energy added per unit charge has units of volts, so the electromotive force is actually a potential. Unfortunately, the name electromotive force stuck and with it the potential for confusing it with a real force. For this reason, we avoid the term electromotive force and just use the abbreviation emf, which has the mathematical symbol $ε .$ The emf may be defined as the rate at which energy is drawn from a source per unit current flowing through a circuit. Thus, emf is the energy per unit charge added by a source, which contrasts with voltage, which is the energy per unit charge released as the charges flow through a circuit.

To understand why an emf is generated in a coil due to a moving magnet, consider Figure_20_02_Two_mags, which shows a bar magnet moving downward with respect to a wire loop. Initially, seven magnetic field lines are going through the loop (see left-hand image). Because the magnet is moving away from the coil, only five magnetic field lines are going through the loop after a short time $Δ t$ (see right-hand image). Thus, when a change occurs in the number of magnetic field lines going through the area defined by the wire loop, an emf is induced in the wire loop. Experiments such as this show that the induced emf is proportional to the rate of change of the magnetic field. Mathematically, we express this as

ε \propto \frac{Δ B}{Δ t},

where $Δ B$ is the change in the magnitude in the magnetic field during time $Δ t$ and A is the area of the loop.

View original figure (external; image not reproduced in this text edition)

Original figure description: Part (a) shows seven magnetic field lines going through a wire loop. Part (b) shows that after a period of time, only five magnetic field lines are going through the loop.

The bar magnet moves downward with respect to the wire loop, so that the number of magnetic field lines going through the loop decreases with time. This causes an emf to be induced in the loop, creating an electric current.

Note that magnetic field lines that lie in the plane of the wire loop do not actually pass through the loop, as shown by the left-most loop in Figure_20_02_Loops. In this figure, the arrow coming out of the loop is a vector whose magnitude is the area of the loop and whose direction is perpendicular to the plane of the loop. In Figure_20_02_Loops, as the loop is rotated from $θ = 90°$ to $θ = 0°,$ the contribution of the magnetic field lines to the emf increases. Thus, what is important in generating an emf in the wire loop is the component of the magnetic field that is perpendicular to the plane of the loop, which is $B \cos θ .$

This is analogous to a sail in the wind. Think of the conducting loop as the sail and the magnetic field as the wind. To maximize the force of the wind on the sail, the sail is oriented so that its surface vector points in the same direction as the winds, as in the right-most loop in Figure_20_02_Loops. When the sail is aligned so that its surface vector is perpendicular to the wind, as in the left-most loop in Figure_20_02_Loops, then the wind exerts no force on the sail.

Thus, taking into account the angle of the magnetic field with respect to the area, the proportionality $ε \propto Δ B / Δ t$ becomes

ε \propto \frac{Δ B \cos θ}{Δ t} .

View original figure (external; image not reproduced in this text edition)

Original figure description: As a wire loop is rotated through a magnetic field, it generates an emf. The dots show where the magnetic field lines intersect the plane defined by the loop.

The magnetic field lies in the plane of the left-most loop, so it cannot generate an emf in this case. When the loop is rotated so that the angle of the magnetic field with the vector perpendicular to the area of the loop increases to

90°

(see right-most loop), the magnetic field contributes maximally to the emf in the loop. The dots show where the magnetic field lines intersect the plane defined by the loop.

Another way to reduce the number of magnetic field lines that go through the conducting loop in Figure_20_02_Loops is not to move the magnet but to make the loop smaller. Experiments show that changing the area of a conducting loop in a stable magnetic field induces an emf in the loop. Thus, the emf produced in a conducting loop is proportional to the rate of change of the product of the perpendicular magnetic field and the loop area

ε \propto \frac{Δ [(B \cos θ) A]}{Δ t},

where $B \cos θ$ is the perpendicular magnetic field and A is the area of the loop. The product $B A \cos θ$ is very important. It is proportional to the number of magnetic field lines that pass perpendicularly through a surface of area A. Going back to our sail analogy, it would be proportional to the force of the wind on the sail. It is called the magnetic flux and is represented by $Φ$ .

Φ = B A \cos θ

The unit of magnetic flux is the weber (Wb), which is magnetic field per unit area, or T/m². The weber is also a volt second (Vs).

The induced emf is in fact proportional to the rate of change of the magnetic flux through a conducting loop.

ε \propto \frac{Δ Φ}{Δ t}

Finally, for a coil made from N loops, the emf is N times stronger than for a single loop. Thus, the emf induced by a changing magnetic field in a coil of N loops is

ε \propto N \frac{Δ B \cos θ}{Δ t} A.

The last question to answer before we can change the proportionality into an equation is “In what direction does the current flow?” The Russian scientist Heinrich Lenz (1804–1865) explained that the current flows in the direction that creates a magnetic field that tries to keep the flux constant in the loop. For example, consider again Figure_20_02_Two_mags. The motion of the bar magnet causes the number of upward-pointing magnetic field lines that go through the loop to decrease. Therefore, an emf is generated in the loop that drives a current in the direction that creates more upward-pointing magnetic field lines. By using the right-hand rule, we see that this current must flow in the direction shown in the figure. To express the fact that the induced emf acts to counter the change in the magnetic flux through a wire loop, a minus sign is introduced into the proportionality $ε \propto Δ Φ / Δ t .$ , which gives Faraday’s law of induction.

ε = - N \frac{Δ Φ}{Δ t}

Lenz’s law is very important. To better understand it, consider Figure_20_02_LL, which shows a magnet moving with respect to a wire coil and the direction of the resulting current in the coil. In the top row, the north pole of the magnet approaches the coil, so the magnetic field lines from the magnet point toward the coil. Thus, the magnetic field ${\vec{B}}_{mag} = B_{mag} (\hat{x})$ pointing to the right increases in the coil. According to Lenz’s law, the emf produced in the coil will drive a current in the direction that creates a magnetic field ${\vec{B}}_{coil} = B_{coil} (- \hat{x})$ inside the coil pointing to the left. This will counter the increase in magnetic flux pointing to the right. To see which way the current must flow, point your right thumb in the desired direction of the magnetic field ${\vec{B}}_{coil,}$ and the current will flow in the direction indicated by curling your right fingers. This is shown by the image of the right hand in the top row of Figure_20_02_LL. Thus, the current must flow in the direction shown in Figure 4(a).

In Figure 4(b), the direction in which the magnet moves is reversed. In the coil, the right-pointing magnetic field ${\vec{B}}_{mag}$ due to the moving magnet decreases. Lenz’s law says that, to counter this decrease, the emf will drive a current that creates an additional right-pointing magnetic field ${\vec{B}}_{coil}$ in the coil. Again, point your right thumb in the desired direction of the magnetic field, and the current will flow in the direction indicate by curling your right fingers (Figure 4(b)).

Finally, in Figure 4(c), the magnet is reversed so that the south pole is nearest the coil. Now the magnetic field ${\vec{B}}_{mag}$ points toward the magnet instead of toward the coil. As the magnet approaches the coil, it causes the left-pointing magnetic field in the coil to increase. Lenz’s law tells us that the emf induced in the coil will drive a current in the direction that creates a magnetic field pointing to the right. This will counter the increasing magnetic flux pointing to the left due to the magnet. Using the right-hand rule again, as indicated in the figure, shows that the current must flow in the direction shown in Figure 4(c).

View original figure (external; image not reproduced in this text edition)

Original figure description: A bar magnet with a pickup coil to its right. Parts (a) through (c) show what happens as the two are moved toward each other with respect to electron movement and current flow.

Lenz’s law tells us that the magnetically induced emf will drive a current that resists the change in the magnetic flux through a circuit. This is shown in panels (a)–(c) for various magnet orientations and velocities. The right hands at right show how to apply the right-hand rule to find in which direction the induced current flows around the coil.

Induced Current in a Wire

This video explains how a current can be induced in a straight wire by moving it through a magnetic field. The lecturer uses the cross product, which a type of vector multiplication. Don’t worry if you are not familiar with this, it basically combines the right-hand rule for determining the force on the charges in the wire with the equation $F = q v B \sin θ .$

Induced current in a wire | Physics | Khan Academy

What emf is produced across a straight wire 0.50 m long moving at a velocity of (1.5 m/s) $\hat{x}$ through a uniform magnetic field (0.30 T)ẑ? The wire lies in the ŷ-direction. Also, which end of the wire is at the higher potential—let the lower end of the wire be at y = 0 and the upper end at y = 0.5 m)?

0.15 V and the lower end of the wire will be at higher potential
0.15 V and the upper end of the wire will be at higher potential
0.075 V and the lower end of the wire will be at higher potential
0.075 V and the upper end of the wire will be at higher potential

EMF Induced in Conducing Coil by Moving Magnet

Imagine a magnetic field goes through a coil in the direction indicated in Figure_20_02_Coil. The coil diameter is 2.0 cm. If the magnetic field goes from 0.020 to 0.010 T in 34 s, what is the direction and magnitude of the induced current? Assume the coil has a resistance of 0.1 $Ω.$

View original figure (external; image not reproduced in this text edition)

Original figure description: A coil with a magnetic field running through it.

A coil through which passes a magnetic field B.

Strategy

Use the equation $ε = - N Δ Φ / Δ t$ to find the induced emf in the coil, where $Δ t = 34 s$ . Counting the number of loops in the solenoid, we find it has 16 loops, so $N = 16 .$ Use the equation $Φ = B A \cos θ$ to calculate the magnetic flux

Φ = B A \cos θ = B π {(\frac{d}{2})}^{2},

where d is the diameter of the solenoid and we have used $\cos 0° = 1 .$ Because the area of the solenoid does not vary, the change in the magnetic of the flux through the solenoid is

Δ Φ = Δ B π {(\frac{d}{2})}^{2} .

Once we find the emf, we can use Ohm’s law, $ε = I R,$ to find the current.

Finally, Lenz’s law tells us that the current should produce a magnetic field that acts to oppose the decrease in the applied magnetic field. Thus, the current should produce a magnetic field to the right.

Solution

Combining equations $ε = - N Δ Φ / Δ t$ and $Φ = B A \cos θ$ gives

ε = - N \frac{Δ Φ}{Δ t} = - N \frac{Δ B π d^{2}}{4 Δ t} .

Solving Ohm’s law for the current and using this result gives

\begin{matrix} I & = & \frac{ε}{R} = - N \frac{Δ B π d^{2}}{4 R Δ t} \\ = & −16 \frac{(−0.010 T) π {(0.020 m)}^{2}}{4 (0.10 Ω) (34 s)} \\ = & 15 μ A \end{matrix} .

Lenz’s law tells us that the current must produce a magnetic field to the right. Thus, we point our right thumb to the right and curl our right fingers around the solenoid. The current must flow in the direction in which our fingers are pointing, so it enters at the left end of the solenoid and exits at the right end.

Discussion

Let’s see if the minus sign makes sense in Faraday’s law of induction. Define the direction of the magnetic field to be the positive direction. This means the change in the magnetic field is negative, as we found above. The minus sign in Faraday’s law of induction negates the negative change in the magnetic field, leaving us with a positive current. Therefore, the current must flow in the direction of the magnetic field, which is what we found.

Now try defining the positive direction to be the direction opposite that of the magnetic field, that is positive is to the left in Figure_20_02_Coil. In this case, you will find a negative current. But since the positive direction is to the left, a negative current must flow to the right, which again agrees with what we found by using Lenz’s law.

Magnetic Induction due to Changing Circuit Size

The circuit shown in Figure_20_02_Slider consists of a U-shaped wire with a resistor and with the ends connected by a sliding conducting rod. The magnetic field filling the area enclosed by the circuit is constant at 0.01 T. If the rod is pulled to the right at speed $v = 0.50 m/s,$ what current is induced in the circuit and in what direction does the current flow?

View original figure (external; image not reproduced in this text edition)

Original figure description: A slider circuit consisting of a U-shaped wire with a resistor and the ends connected by a sliding conducting rod.

A slider circuit. The magnetic field is constant and the rod is pulled to the right at speed v. The changing area enclosed by the circuit induces an emf in the circuit.

Solution

Faraday’s law of induction gives

ε = - N \frac{Δ Φ}{Δ t} = - B v ℓ .

Solving Ohm’s law for the current and using the previous result for emf gives

I = \frac{ε}{R} = \frac{- B v ℓ}{R} = \frac{- (0.010 T) (0.50 m/s) (0.10 m)}{20 Ω} = 25 μ A .

As the rod slides to the right, the magnetic flux passing through the circuit increases. Lenz’s law tells us that the current induced will create a magnetic field that will counter this increase. Thus, the magnetic field created by the induced current must be into the page. Curling your right-hand fingers around the loop in the clockwise direction makes your right thumb point into the page, which is the desired direction of the magnetic field. Thus, the current must flow in the clockwise direction around the circuit.

Discussion

Is energy conserved in this circuit? An external agent must pull on the rod with sufficient force to just balance the force on a current-carrying wire in a magnetic field—recall that $F = I ℓ B \sin θ .$ The rate at which this force does work on the rod should be balanced by the rate at which the circuit dissipates power. Using $F = I ℓ B \sin θ,$ the force required to pull the wire at a constant speed v is

F_{pull} = I ℓ B \sin θ = I ℓ B,

where we used the fact that the angle $θ$ between the current and the magnetic field is $90° .$ Inserting our expression above for the current into this equation gives

F_{pull} = I ℓ B = - \frac{B v ℓ}{R} (ℓ B) = - \frac{B^{2} v ℓ^{2}}{R} .

The power contributed by the agent pulling the rod is $F_{pull} v, or$

P_{pull} = F_{pull} v = - \frac{B^{2} v^{2} ℓ^{2}}{R} .

The power dissipated by the circuit is

P_{dissipated} = I^{2} R = {(\frac{- B v ℓ}{R})}^{2} R = \frac{B^{2} v^{2} ℓ^{2}}{R} .

We thus see that $P_{pull} + P_{dissipated} = 0,$ which means that power is conserved in the system consisting of the circuit and the agent that pulls the rod. Thus, energy is conserved in this system.

Practice Problems

The magnetic flux through a single wire loop changes from 3.5 Wb to 1.5 Wb in 2.0 s. What emf is induced in the loop?

–2.0 V
–1.0 V
+1.0 V
+2.0 V

What is the emf for a 10-turn coil through which the flux changes at 10 Wb/s?

–100 V
–10 V
+10 V
+100 V

Check Your Understanding

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Section Summary

Faraday’s law of induction states that a changing magnetic flux that occurs within an area enclosed by a conducting loop induces an electric current in the loop.
Lenz’ law states that an induced current flows in the direction such that it opposes the change that induced it.

Key Equations

magnetic flux	$Φ = B A \cos θ$
emf	$ε = - N \frac{Δ Φ}{Δ t}$

Concept Items

Yes or no—Is an emf induced in the coil shown when it is stretched? If so, state why and give the direction of the induced current.

View original figure (external; image not reproduced in this text edition)

Original figure description: Two images of a circular wire embedded in a magnetic field that is pointing into the page. The left figure shows the circular wire enclosing four of the magnetic field points with two hands on either end pulling on the wire initially shaped like a circle. The image on the right shows the same wire stretched out into an ellipse.

No, because induced current does not depend upon the area of the coil.
Yes, because area of the coil decreases; the direction of the induced current is counterclockwise.
Yes, because area of the coil decreases; the direction of the induced current is clockwise.
Yes, because the area of the coil does not change; the direction of the induced current is clockwise.

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Explain how magnetic flux can be zero when the magnetic field is not zero.

If angle between magnetic field and area vector is 0°, then its sine is also zero, which means that there is zero flux.
If angle between magnetic field and area vector is 45°, then its sine is also zero, which means that there is zero flux.
If angle between magnetic field and area vector is 60°, then its cosine is also zero, which means that there is zero flux.
If the angle between magnetic field and area vector is 90°, then its cosine is also zero, which means that there is zero flux.

Critical Thinking

To obtain power from the current in the wire of your vacuum cleaner, you place a loop of wire near it to obtain an induced emf. How do you place and orient the loop?

A loop of wire should be placed nearest to the vacuum cleaner wire to maximize the magnetic flux through the loop.
A loop of wire should be placed farthest to the vacuum cleaner wire to maximize the magnetic flux through the loop.
A loop of wire should be placed perpendicular to the vacuum cleaner wire to maximize the magnetic flux through the loop.
A loop of wire should be placed at angle greater than 90° to the vacuum cleaner wire to maximize the magnetic flux through the loop.

A magneto is a device that creates a spark across a gap by creating a large voltage across the gap. To do this, the device spins a magnet very quickly in front of a wire coil, with the ends of the wires forming the gap. Explain how this creates a sufficiently large voltage to produce a spark.

The electric field in the coil increases rapidly due to spinning of magnet which creates an emf in the coil that is proportional to the rate of change of the magnetic flux.
The magnetic field in the coil changes rapidly due to spinning of magnet which creates an emf in the coil that is proportional to the rate of change of the magnetic flux.

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Problems

What is the current in a wire loop of resistance 10 Ω through which the magnetic flux changes from zero to 10 Wb in 1.0 s?

–100 A
–2.0 A
–1.0 A
+1.0 A

An emf is induced by rotating a 1,000 turn, 20.0 cm diameter coil in Earth’s 5.00 × 10^–5 T magnetic field. What average emf is induced, given the plane of the coil is originally perpendicular to Earth’s field and is rotated to be parallel to the field in 10.0 ms?

–1.6 × 10^-4 V
+1.6 × 10^-4 V
+1.6 × 10^-1 V
–1.6 × 10^-1 V

Test Prep Multiple Choice

What does emf stand for?

electromotive force
electro motion force
electromagnetic factor
electronic magnetic factor

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True or false—If you drop a bar magnet through a copper tube, it induces an electric current in the tube.

false
true

Test Prep Short Answer

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The plane of a square wire circuit with side 4.0 cm long is at an angle of 45° with respect to a uniform magnetic field of 0.25 T. The wires have a resistance per unit length of 0.2. If the field drops to zero in 2.5 s, what magnitude current is induced in the square circuit?

35 µA
87.5 µA
3.5 mA
35 A

Yes or no—If a bar magnet moves through a wire loop as shown in the figure, is a current induced in the loop? Explain why or why not.

View original figure (external; image not reproduced in this text edition)

Original figure description: This figure shows a round circle representing a wire loop and a bar magnet with an arrow pointing upward as it moves through the wire loop. The south and north poles run parallel with the surface area enclosed by the loop.

No, because the net magnetic field passing through the loop is zero.
No, because the net magnetic field passing through the loop is nonzero.
Yes, because the net magnetic field passing through the loop is zero.
Yes, because the net magnetic field line passing through the loop is nonzero.

What is the magnetic flux through an equilateral triangle with side 60 cm long and whose plane makes a 60° angle with a uniform magnetic field of 0.33 T?

0.045 Wb
0.09 Wb
0.405 Wb
4.5 Wb

Test Prep Extended Response

When you insert a copper ring between the poles of two bar magnets as shown in the figure, do the magnets exert an attractive or repulsive force on the ring? Explain your reasoning.

View original figure (external; image not reproduced in this text edition)

Original figure description: The figure shows south pole of a bar magnet at the top of the figure and a north pole of a bar magnet at the bottom of the figure. A copper ring is approaching and almost inside the location between the two magnetic poles as shown by an arrow. The copper ring runs in and out of the page so that the maximum surface area enclosed by the loop cannot be seen in the figure.

Magnets exert an attractive force, because magnetic field due to induced current is repulsed by the magnetic field of the magnets.
Magnets exert an attractive force, because magnetic field due to induced current is attracted by the magnetic field of the magnets.
Magnets exert a repulsive force, because magnetic field due to induced current is repulsed by the magnetic field of the magnets.
Magnets exert a repulsive force, because magnetic field due to induced current is attracted by the magnetic field of the magnets.

The figure shows a uniform magnetic field passing through a closed wire circuit. The wire circuit rotates at an angular frequency of about the axis shown by the dotted line in the figure.

View original figure (external; image not reproduced in this text edition)

Original figure description: The diagram has two rectangles that are folded at the middle at about 90 degrees. This creates an upper flap of the rectangle that points to the right and a lower flap that points straight down. The variable lowercase L labels the length of the lower flap of the rectangle. The label lowercase L over 2 is on the height of the lower flap of the rectangle. The label lowercase L over 2 is also on the width of the upper flap of the rectangle. Vector B labels four downward-pointing, diagonal arrows that each pass through the fold in the rectangle at a 45 degree angle.

What is an expression for the magnetic flux through the circuit as a function of time?

expression for the magnetic flux through the circuit Φ(t) = BAcos⁡ωt
expression for the magnetic flux through the circuit $Φ (t) = \sqrt{2} B A cos ω t$
expression for the magnetic flux through the circuit $Φ (t) = \sqrt{3} B A cos ω t$
expression for the magnetic flux through the circuit Φ(t) = 2BA cos ωt

Glossary

emf: rate at which energy is drawn from a source per unit current flowing through a circuit
induction: rate at which energy is drawn from a source per unit current flowing through a circuit
magnetic flux: component of the magnetic field perpendicular to the surface area through which it passes and multiplied by the area

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