Electromagnetic InductionFaraday’s Law

ac Circuits

We found in the last lecture that an electric current can give rise to a magnetic field, and that a magnetic field can exert a force on a moving charge.

I wonder if a magnetic field can somehow give rise to an electric current…

49 slides for this lecture. Hang on tight!

Induced emf

An electric current is induced if there is a closed circuit (e.g., loop of wire) in the changing magnetic field.

It is observed experimentally that changes in magnetic fields induce an emf in a conductor.

A constant magnetic field does not induce an emf—it takes a changing magnetic field.

Passing the coil through the magnet would induce an emf in the coil.

T hey need to calibrate their meter!

Note that “change” does not require observable (to you) motion.

A magnet may move through a loop of wire, or a loop of wire may be moved through a magnetic field (as suggested in the previous slide). These involve observable motion.

A changing current in a loop of wire gives rise to a changing magnetic field (predicted by Ampere’s law) which can induce a current in another nearby loop of wire.

In the latter case, nothing observable (to your eye) is moving, although, of course microscopically, electrons are in motion.

“induced emf is produced by a changing magnetic field.”

To quantify induced emf, we define magnetic flux. In an earlier lecture we briefly touched on electric flux. This is the magnetic analog.

Because we can’t “see” magnetic fields directly, we draw magnetic field lines to help us visualize the magnetic field. Remember that magnetic field lines start at N poles and end at S poles.

A strong magnetic field is represented by many magnetic field lines, close together. A weak magnetic field is represented by few magnetic field lines, far apart.

Faraday’s Law

We could, if we wished, actually “calibrate” by specifying the number of magnetic field lines passing through some surface that corresponded to a given magnetic field strength.

Magnetic flux B is proportional to the number of

magnetic field lines passing through a surface.

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html

Mathematically, magnetic flux B

through a surface of area A is defined by

B = BA = B A cos

OSE: B = BA

where B is the component of field perpendicular to the surface, A is the area of the surface, and is the angle between B and the normal to the surface.

BA

B

B

A

When B is parallel to the surface, =90° and B = 0.

When B is perpendicular to the surface, =0° and B = BA.

B

B=0A

=0B=0

B

A

The unit of magnetic flux is the Tm2, called a weber:

1 Wb = 1 T m2 .

In the following discussion, we switch from talking about surfaces in a magnetic field…

…to talking about loops of wire in a magnetic field.

Now we can quantify the induced emf described qualitatively in the previous section

Experimentally, if the flux through N loops of wire changes by an amount B in a time t, the induced emf is

BΔφ = - N .

Δtε

This is called Faraday’s law of induction. It is one of the fundamental laws of electricity and magnetism, and an important component of the theory that explains electricity and magnetism.

I wonder why the – sign…

Not an OSE—not quite yet.

…an induced emf always gives rise to a current whose magnetic field opposes the change in flux—Lenz’s law.*

Experimentally…

Think of the current resulting from the induced emf as “trying” to maintain the status quo—to prevent change.

If Lenz’s law were not true—if there were a + sign in faraday’s law—then a changing magnetic field would produce a current, which would further increased the magnetic field, further increasing the current, making the magnetic field still bigger…

Among other things, conservation of energy would be violated.

*We’ll practice with this in a bit.

Ways to induce an emf:

change B

change the area of the loop in the field

Ways to induce an emf (continued):

change the orientation of the loop in the field

Conceptual example: Induction Stove

An ac current in a coil in the stove top produces a changing magnetic field at the bottom of a metal pan.

The changing magnetic field gives rise to a current in the bottom of the pan.

Because the pan has resistance, the current heats the pan. If the coil in the stove has low resistance it doesn’t get hot but the pan does.

An insulator won’t heat up on an induction stove.

Remember the controversy about cancer from power lines a few years back? Careful studies showed no harmful effect. Nevertheless, some believe induction stoves are hazardous.

Practice with Lenz’s Law

In which direction is the current induced in the coil for each situation shown?

(counterclockwise) (no current)

(counterclockwise) (clockwise)

Rotating the coil about the vertical diameter by pulling the left side toward the reader and pushing the right side away from the reader in a magnetic field that points from right to left in the plane of the page.

(counterclockwise)

Now that we are experts on the application of Lenz’s law, lets make our induced emf equation official:

BΔφ = - N ?

Δtε

BΔφ = N .

Δtε

This means it is up to you to use Lenz’s law to figure out the direction of the induced current (or the direction of whatever the problem wants).

Remember

Example Pulling a Coil from a Magnetic Field

A square coil of side 5 cm contains 100 loops and is positioned perpendicular to a uniform 0.6 T magnetic field. It is quickly and uniformly pulled from the field (moving to B) to a region where the field drops abruptly to zero. It takes 0.10 s to remove the coil, whose resistance is 100 .

B = 0.6 T

5 cm

(a) Find the change in flux through the coil.

Initial: Bi = BA . Final: Bf = 0 .

B = Bf - Bi = 0 - BA = - (0.6 T) (0.05 m)2 = - 1.5x10-3 Wb .

(b) Find the current and emf induced.

The current must flow counterclockwise to induce a downward magnetic field (which replaces the “removed” magnetic field).

The induced emf is

BΔφ = N

Δtε

-3-1.5 10 Wb

= 100 0.1 s

ε

= 1.5 Vε

The induced current is

I = = = 15 mA

R 100 .

ε 1.5 V

(c) How much energy is dissipated in the coil?

Current flows “only*” during the time flux changes.

E = Pt = I2Rt = (1.5x10-2 A) (100 ) (0.1 s) = 2.3x10-3 J .

*If there no resistance in the loop, the current would flow indefinitely. However, the resistance quickly halts the flow of current once the magnetic flux stops changing.

(d) What was the average force required?

The loop had to be “pulled” out of the magnetic field, so the pulling force did work. It is tempting to try and set up a free body diagram and use Newton’s laws. Instead, energy conservation gets the answer with less work.

The flux change occurs only when the coil is in the process of leaving the region of magnetic field.

No flux change.No emf.No current.No work (why?).

Flux changes. emf induced. Current flows. Work done.

D

F

No flux change. No emf. No current. (No work.)

The energy calculated in part (c) is the energy dissipated in the coils while the current is flowing. The amount calculated in part (c) is also the mechanical energy put into the system by the force.

Ef – Ei = [ Wother]If

0Ef – Ei = F D

F = Ef / D

F = (2.3x10-3 J) / (0.05 m)

F = 0.046 N

See 2 slides back for F and D.

emf Induced in a Moving Conductor

In the previous section we calculated the emf induced in a wire loop, by considering the magnetic flux change.

You can also use the magnetic flux change to calculate the emf induced in a wire segment moving in a magnetic field.

See lec28_long.ppt on the supplementary material page for the calculation.

I will introduce the calculation, then give the result.

Recall that one of the ways to induce an emf is to change the area of the loop in the magnetic field. Let’s see how this works.

v

ℓ

B

vtA

A U-shaped conductor and a moveable conducting rod are placed in a magnetic field, as shown.

The rod moves to the right with a speed v for a time t.

The rod moves a distance vt and the area of the loop inside the magnetic field increases by an amount

A = ℓ v t .The resulting flux change induces an emf in the rod.

The result is

OSE = B vε

where B is the component of the magnetic field perpendicular to ℓ and v, and v is the component of the velocity perpendicular to B and ℓ.

v

ℓ

B

vtA

A Changing Magnetic Flux Produces an Electric Field

Force on a charged particle moving in a magnetic field:

OSE: F = q v B sin

For v B, and in magnitude only,

F = q E = q v B

E = v B.

We conclude that a changing magnetic flux produces an electric field. This is true not just in conductors, but any-where in space where there is a changing magnetic field.

OSE: E = F / qDefinition of electric field:

Application: determine blood flow velocity in a blood vessel.

You’ll have to look at lec28_full.ppt!

21.5 Electric Generators

Let’s begin by looking at a simple animation of a generator. http://www.wvic.com/how-gen-works.htm

Here’s a “freeze-frame.”

Normally, many coils of wire are wrapped around an armature. The picture shows only one.

Brushes pressed against a slip ring make continual contact.

The shaft on which the armature is mounted is turned by some mechanical means.

Let’s look at the current direction in this particular freeze-frame.

B is down. Coil rotates counter-clockwise.

Put your fingers along the direction of movement. Stick out your thumb.

Bend your fingers 90°. Rotate your hand until the fingers point in the direction of B. Your thumb points in the direction of conventional current.

Alternative right-hand rule for current direction.

B is down. Coil rotates counter-clockwise.

Make an xyz axes out of your thumb and first two fingers.

2nd finger then points in direction of conventional current.

Thumb along component of wire velocity to B. 1st finger along B.

Hey! The picture got it right!

I know we need to work on that more. Let’s zoom in on the armature.

v vB

vB

B

I

Forces on the charges in these parts of the wire are perpendicular to the length of the wire, so they don’t contribute to the net current.

For future use, call the length of wire shown in green “h” and the other lengths (where the two red arrows are) “ℓ”.

One more thing…

This wire…

…connects to this ring…

…so the current flows this way.

Later in the cycle, the current still flows clockwise in the loop…

…but now this wire…

…connects to this ring…

…so the current flows this way.

Alternating current! ac!

Again: http://www.wvic.com/how-gen-works.htm

Transformers

A transformer is a device for increasing or decreasing an ac voltage.

Power Substation

Pole-mounted transformer

ac-dc converter

No, no, no…

A transformer is basically two coils of wire wrapped around each other, or wrapped around an iron core.

When an ac voltage is applied to the primary coil, it induces an ac voltage in the secondary coil.

A “step up” transformer increases the output voltage in the secondary coil; a “step down” transformer reduces it.

http://www.howstuffworks.com/power.htm

We skipped a discussion of counter emf. It explains why motors burn out when they can’t turn, and why your house lights might dim when the fridge comes on.

Electromagnetic Waves

Every student of E&M should be “exposed” to electromagnetic waves. Here is your exposure.

Maxwell’s Equations

You saw three prior to this lecture. Where was the fourth?

These equations are versions of the fourth:

BΔφ = N

Δtε = B vε

They tell us that an electric field is produced by a changing magnetic field.

To get the “high-quality” version, you integrate the electric field around a closed loop, and find that the result is proportional to the rate of change of magnetic flux through the loop.

In words, Maxwell’s equations are:

1—a generalized form of Coulomb’s law, relating electric fields to their sources (charges)

2—a law relating magnetic fields to magnetic poles

3—an equation describing how an electric field is produced by a changing magnetic field (Faraday’s Law)

4—an equation describing how a magnetic field is produced by an electric current or changing electric field (Ampere’s Law)

That a changing electric field can produce a magnetic field is not one of the predictions of Ampere’s law; it was hypothesized by Maxwell and verified after his death.

Every student should be thoroughly exposed to Maxwell’s equations, so here they are in all their glory…

Gauss’ law for electricity Gauss’ law for magnetism

Faraday’s law Ampere-Maxwell law

Maxwell’s equations are to E&M as Newton’s laws are to mechanics, except Maxwell’s equations are relativistically correct, and Newton’s laws are not.

Production of Electromagnetic Waves

Electromagnetic waves can be produced by oscillating charges on conductors.

These waves travel through space even long after they are far away from the charges that produced them.

E and B in the radiation fields drop off as 1/r, and the intensity of the waves drops off as 1/r2.

The electric and magnetic fields are perpendicular to each other and to the direction of propagation of the wave. They are also in phase.

From Maxwell’s equations you can show that electromagnetic waves travel with a speed v = 1/(00)1/2 which is equal to 3x108 m/s, the speed of light.

Light was known to behave like a wave long before Maxwell showed that the speed of E&M waves is the same as the speed of light. Eventually, it came to be recognized that light is just an example of an E&M wave.

Light as an E&M Wave: The Electromagnetic Spectrum

The frequencies of visible light lie between about 4x10-7 and 7.5x10-7 m, or 400 to 750 nm. The frequency, wavelength, and speed of a wave are related by v = f, so for E&M waves, and for light, c = f.

Visible light represents only a minute portion of the electromagnetic spectrum.

The sun emits large amounts of IR, visible, and UV radiation. We detect the IR as heat, the visible as light, and the UV through skin damage. X-rays, gamma rays, and radio waves are “just” E&M waves, like light, only of different frequencies.

(sorry, rushed scan)