Chapter 21

Electromagnetic Induction and Faraday’s Law

Units of Chapter 21

• Induced EMF

• Faraday’s Law of Induction; Lenz’s Law

• EMF Induced in a Moving Conductor

• Changing Magnetic Flux Produces an Electric Field

• Electric Generators

• Back EMF and Counter Torque; Eddy Currents

• Transformers and Transmission of Power

Units of Chapter 21

• Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI

• Inductance

• Energy Stored in a Magnetic Field

• LR Circuit

• AC Circuits and Reactance

• LRC Series AC Circuit

• Resonance in AC Circuits

21.1 Induced EMF

Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:

21.1 Induced EMF

He found no evidence when the current was steady, but did see a current induced when the switch was turned on or off.

21.1 Induced EMF

Therefore, a changing magnetic field induces an emf.

Faraday’s experiment used a magnetic field that was changing because the current producing it was changing; the previous graphic shows a magnetic field that is changing because the magnet is moving.

21.2 Faraday’s Law of Induction; Lenz’s Law

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

Magnetic flux: (21-1)

Unit of magnetic flux: weber, Wb.

1 Wb = 1 T·m2

This drawing shows the variables in the flux equation:

21.2 Faraday’s Law of Induction; Lenz’s Law

The magnetic flux is analogous to the electric flux – it is proportional to the total number of lines passing through the loop.

21.2 Faraday’s Law of Induction; Lenz’s Law

Example 21-2A square loop of wire 10.0 cm on a side is in a 1.25 T magnetic field B. What are the maximum and minimum values of flux that can pass through the loop?

€

Max value : θ = 0o

ΦB = BAcosθ = (1.25 T)(0.100 m)(0.100 m)cos0 = 0.0125 Wb

Min value : θ = 90o

ΦB = BAcosθ = 0 Wb

Faraday’s law of induction:

[1 loop] (21-2a)

[N loops] (21-2b)

21.2 Faraday’s Law of Induction; Lenz’s Law

The minus sign gives the direction of the induced emf:

A current produced by an induced emf moves in a direction so that the magnetic field it produces tends to restore the changed field.

21.2 Faraday’s Law of Induction; Lenz’s Law

Magnetic flux will change if the area of the loop changes:

21.2 Faraday’s Law of Induction; Lenz’s Law

Magnetic flux will change if the angle between the loop and the field changes:

21.2 Faraday’s Law of Induction; Lenz’s Law

Problem Solving: Lenz’s Law

1. Determine whether the magnetic flux is increasing, decreasing, or unchanged.

2. The magnetic field due to the induced current points in the opposite direction to the original field if the flux is increasing; in the same direction if it is decreasing; and is zero if the flux is not changing.

3. Use the right-hand rule to determine the direction of the current.

4. Remember that the external field and the field due to the induced current are different.

21.2 Faraday’s Law of Induction; Lenz’s Law

Example 21-5A square coil of wire with a side of length 5.00 cm contains 100 loops and is positioned perpendicular to a uniform 0.600 T magnetic field. It is quickly pulled from the field at constant speed to a region where B drops abruptly to zero. At t=0, the right edge of the coil is at the edge of the field. It takes 0.100 s for the whole coil to reach the field-free region. The coil’s total resistance is 100 Ω. Find (a) the rate of change in flux through the coil, and (b) the emf and current induced. (c) How much energy is dissipated in the coil? (d) What was the average force required?

€

(a) A = length2 = (5.00x10-2 m)2 = 2.50x10-3 m2

ΦB = BA = (0.600 T)(2.50x10-3 m2) =1.50x10-3 Wb

ΔΦB

Δt=

0 - (1.50x10-3 Wb)

0.100 s= -1.50x10-2 Wb/s

(b) ε = -NΔΦB

Δt= -(100)(-1.50x10-2 Wb/s) =1.50 V

I =ε

R=

1.50 V

100 Ω=1.50x10-2 A =15.0 mA

Lenz's law : current must be clockwise to produce more r B into

the page and oppose the decreasing flux into the page.

(c) E = Pt = I2Rt = (1.50x10-2 A) 2(100 Ω)(0.100 s) = 2.25x10-3 J

(d) F =W

d=

2.25x10-3 J

5.00x10-2 m= 0.0450 N

Exercise BWhat is the direction of the induced current in the circular loop due to the current shown in each part?

21.3 EMF Induced in a Moving Conductor

This image shows another way the magnetic flux can change:

21.3 EMF Induced in a Moving Conductor

The induced current is in a direction that tends to slow the moving bar – it will take an external force to keep it moving.

21.3 EMF Induced in a Moving Conductor

The induced emf has magnitude

(21-3)

Measurement of blood velocity from induced emf:

Example 21-6An airplane travels 1000 km/h in a region where the earth’s magnetic field is 5.0x10-5 T and is nearly vertical. What is the potential difference induced between the wing tips that are 70 m apart?

€

v =1000 km/h = 280 m/s

ε = Blv = (5.0x10-5 T)(70 m)(280 m/s) =1.0 V

Example 21-7The rate of blood flow in our body’s vessels can be measured using the apparatus shown below since blood contains charged ions. Suppose that the blood vessel is 2.0 mm in diameter, the magnetic field is 0.080 T, and the measured emf is 0.10 mV. What is the flow of velocity of the blood?

€

ε = Blv → v =ε

Bl

v =(1.0x10-4 V)

(0.080 T)(2.0x10-3 m)= 0.63 m/s

21.4 Changing Magnetic Flux Produces an Electric Field

A changing magnetic flux induces an electric field; this is a generalization of Faraday’s law. The electric field will exist regardless of whether there are any conductors around.

21.5 Electric Generators

A generator is the opposite of a motor – it transforms mechanical energy into electrical energy. This is an ac generator:

The axle is rotated by an external force such as falling water or steam, inducing an emf in the rotating coil. The brushes are in constant electrical contact with the slip rings.

21.5 Electric Generators

A dc generator is similar, except that it has a split-ring commutator instead of slip rings.