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http://www.nearingzero.net (nz182.jpg). Today’s agenda: Induced emf. You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf. Faraday’s Law. You must be able to use Faraday’s Law to calculate the emf induced in a circuit.

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Today’s agenda:

Induced emf.

You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf.

You must be able to use Faraday’s Law to calculate the emf induced in a circuit.

Lenz’s Law.

You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf.

Generators.

You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators.

Back emf.

You must be able to use Lenz’s law to explain back emf.

Induced emf and Faraday’s Law

Magnetic Induction

We have found that an electric current can give rise to a magnetic field…

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

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

B

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

I

B

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

N

S

v

move magnet toward coil

Note that “change” may or may not 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. These involve observable motion.

 A magnet may move through a loop of wire

B

I

I

region of magnetic field

this part of the loop is closest to your eyes

change area of loop inside magnetic field

N

S

rotate coil in magnetic field

changing I

induced I

changing B

 A changing current in a loop of wire gives rise to a changing magnetic field (predicted by Ampere’s law)

 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.

 A changing current in a loop of wire

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

Induced emf is produced by a changing magnetic flux.

Today’s agenda:

Induced emf.

You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf.

You must be able to use Faraday’s Law to calculate the emf induced in a circuit.

Lenz’s Law.

You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf.

Generators.

You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators.

Back emf.

You must be able to use Lenz’s law to explain back emf.

We can quantify the induced emf described qualitatively in the last few slides by using magnetic flux.

Experimentally, if the flux through N loops of wire changes by dB in a time dt, the induced emf is

Faraday’s Law of Magnetic Induction

Faraday’s law of induction is one of the fundamental laws of electricity and magnetism.

I wonder why the – sign…

Your text, page 301, shows how to determine the direction of the induced emf. Argh! Lenz’s Law, coming soon, is much easier.

Faraday’s Law of Magnetic Induction

In the equation

is the magnetic flux.

This is sometimes shown as another expression of Faraday’s Law:

We’ll use this version in a “later” lecture.

N

S

I

v

+

-

Example: move a magnet towards a coil of wire.

N=5 turns A=0.002 m2

Possible homework hint: if B varies but loop  B.

Ways to induce an emf:

 change B

 change the area of the loop in the field

Possible homework hint: for a circular loop, C=2R, so A=r2=(C/2)2=C2/4, so you can express d(BA)/dt in terms of dC/dt.

 changing current changes B through conducting loop

a

I

B

b

Possible Homework Hint.

The magnetic field is not uniform through the loop, so you can’t use BA to calculate the flux. Take an infinitesimally thin strip. Then the flux is d = BdAstrip. Integrate from a to b to get the flux through the strip.

=90

=45

=0

Ways to induce an emf (continued):

 change the orientation of the loop in the field

Today’s agenda:

Induced emf.

You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf.

You must be able to use Faraday’s Law to calculate the emf induced in a circuit.

Lenz’s Law.

You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf.

Generators.

You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators.

Back emf.

You must be able to use Lenz’s law to explain back emf.

N

S

Experimentally…

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

I

v

+

-

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 increase the magnetic field, further increasing the current, making the magnetic field still bigger…

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

Possible Demos: induced emf.

Practice with Lenz’s Law.

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

(counterclockwise)

(no current)

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)

You can use Faraday’s Law to calculate the magnitude of the emf (or whatever the problem wants). Then use Lenz’s Law to figure out the direction of the induced current (or the direction of whatever the problem wants).

The direction of the induced emf is in the direction of the current that flows in response to the flux change. We usually ask you to calculate the magnitude of the induced emf ( || ) and separately specify its direction.

Magnetic flux is not a vector. Like electrical current, it is a scalar. Just as we talk about current direction (even though it is not a vector), we often talk about flux direction (even though it is not a vector). Keep this in mind if your recitation instructor talks about the direction of magnetic flux.

Today’s agenda:

Induced emf.

You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf.

You must be able to use Faraday’s Law to calculate the emf induced in a circuit.

Lenz’s Law.

You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf.

Generators.

You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators.

Back emf.

You must be able to use Lenz’s law to explain back emf.

Motional emf: an overview

An emf is induced in a conductor moving in a magnetic field.

Your text introduces four ways of producing motional emf.

1. Flux change through a conducting loop produces an emf:rotating loop.

B

A

side view

derive these

2. Flux change through a conducting loop produces an emf:

2. Flux change through a conducting loop produces an emf:expanding loop.

v

B

B

dA

x=vdt

derive these

3. Conductor moving in a magnetic field experiences an emf:

3. Conductor moving in a magnetic field experiences an emf: magnetic force on charged particles.

v

B

B

+

derive this

You could also solve this using Faraday’s Law by constructing a “virtual” circuit using “virtual” conductors.

4. Flux change through a conducting loop produces an emf:moving loop.

derive these

B

A

side view

S

N

Generators and Motors: a basic introduction

Take a loop of wire in a magnetic field and rotate it with an angular speed .

Choose 0=0. Then

Generators are an application of motional emf.

B

A

side view

If there are N loops in the coil

The NBA equation!

|| is maximum when  = t = 90° or 270°; i.e., when B is zero. The rate at which the magnetic flux is changing is then maximum. On the other hand,  is zero when the magnetic flux is maximum.

emf, current and power from a generator

None of these are on your starting equation sheet!

Example: the armature of a 60 Hz ac generator rotates in a 0.15 T magnetic field. If the area of the coil is 2x10-2 m2, how many loops must the coil contain if the peak output is to be max = 170 V?

“Legal” for me, because I just derived it (but not for you)!

I will not work this in lecture. Please study it on your own! Know where the 2 comes from.

B

B

Another Kind of Generator: A Slidewire Generator

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

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 constant speed v for a time dt.

dA

vdt

x

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

dA = ℓ vdt .

The loop is perpendicular to the magnetic field, so the magnetic flux through the loop is B = = BA. The emf induced in the conductor can be calculated using Faraday’s law:

v

B

B

dA

vdt

x

B and v are vector magnitudes, so they are always +. Wire length is always +. You use Lenz’s law to get the direction of the current.

B

Direction of current?

The induced emf causes current to flow in the loop.

Magnetic flux inside the loop increases (more area).

v

System “wants” to make the flux stay the same, so the current gives rise to a field inside the loop into the plane of the paper (to counteract the “extra” flux).

I

dA

vdt

x

Clockwise current!

What would happen if the bar were moved to the left?

v

B

A constant pulling force, equal in magnitude and opposite in direction, must be applied to keep the bar moving with a constant velocity.

FM

FP

I

x

Power and current.

If the loop has resistance R, the current is

v

B

And the power is

I

(as expected).

x

Mechanical energy (from the pulling force) has been converted into electrical energy, and the electrical energy is then dissipated by the resistance of the wire.

You might find it useful to look at Dr. Waddill’s lecture on Faraday’s Law, from several semesters back. Click here to view the lecture.

If the above link doesn’t work, try copying and pasting this into your browser address bar:

http://campus.mst.edu/physics/courses/24/Handouts/Lec_18.ppt

Example 3 of motional emf: moving conductor in B field.

Motional emf is the emf induced in a conductor moving in a magnetic field.

“up”

v

B

B

If a conductor (purple bar) moves with speed v in a magnetic field, the electrons in the bar experience a force

+

The force on the electrons is “up,” so the “top” end of the bar acquires a net – charge and the “bottom” end of the bar acquires a net + charge. The charges in the bar are “separated.”

This is a simplified explanation but it gives you the right “feel.”

The separated charges in the bar produce an electric field pointing “up” the bar. The emf across the length of the bar is

“up”

The electric field exerts a “downward” force on the electrons:

v

B

B

An equilibrium condition is reached, where the magnetic and electric forces are equal in magnitude and opposite in direction.

+

Homework Hint!

When the moving conductor is “tilted” relative to its direction of motion (when is not perpen-dicular to the conductor), you must use the “effective length” of the conductor.

v

B

B

ℓeffective

+

ℓeffective= ℓ sin  if you use the if you define  as the angle relative to the horizontal

Caution: you do not have to use this technique for homework problems 9.23 (assigned this semester) or 9.26. Be sure to understand why!

Example 4 of motional emf: flux change through conducting loop. (Entire loop is moving.)

I’ll include some numbers with this example.

Remember, it’s the flux change that produces the emf. Flux has no direction associated with it. However, the presence of flux is due to the presence of a magnetic field, which does have a direction, and allows us to use Lenz’s law to determine the “direction” of current and emf.

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

Initial: Bi = = BA .

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

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.

Current will begin to flow when the coil starts to exit the magnetic field. Because of the resistance of the coil, the current will eventually stop flowing after the coil has left the magnetic field.

final

initial

The current must flow clockwise to induce an “inward” magnetic field (which replaces the “removed” magnetic field).

v

The induced emf is

dA

x

x

“uniformly” pulled

The induced current is

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

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

E = P t = I2R t = (1.5x10-2 A)2 (100 ) (0.1 s) = 2.25x10-3 J

(d) Discuss the forces involved in this example.

The loop has to be “pulled” out of the magnetic field, so there is a pulling force, which does work.

The “pulling” force is opposed by a magnetic force on the current flowing in the wire. If the loop is pulled “uniformly” out of the magnetic field (no acceleration) the pulling and magnetic forces are equal in magnitude and opposite in direction.

*Remember: if there were 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.

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?).

Fapplied

D

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

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

I

(e) Calculate the force necessary to pull the coil from the field.

Remember, a force is needed only when the coil is partly in the field region.

Multiply by N because there are N loops in the coil.

where L is a vector in the direction of I having a magnitude equal to the length of the wire inside the field region.

F3

L2

L3

I

L4=0

Sorry about the “busy” slide!

The forces should be shown acting at the centers of the coil sides in the field.

Fpull

F2

L1

F1

There must be a pulling force to the right to overcome the net magnetic force to the left.

Magnitudes only (direction shown in diagram):

F3

L2

L3

I

Fpull

F2

L1

F1

This calculation assumes the coil is pulled out “uniformly;” i.e., no acceleration, so Fpull = Fmag.

Work done by pulling force:

x

I

Fpull

D

The work done by the pulling force is equal to the electrical energy provided to (and dissipated in) the coil.

If you build a generator and it doesn’t seem to be working…

European model shown. US model also available.

…or if you want to test a wall socket to see if it is “live…”

…simply purchase a Vilcus Plug Dactyloadapter from ThinkGeek

( http://www.thinkgeek.com/stuff/41/lebedev.shtml )

and follow the instructions.

Today’s agenda:

Induced emf.

You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf.

You must be able to use Faraday’s Law to calculate the emf induced in a circuit.

Lenz’s Law.

You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf.

Generators.

You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators. Skip to back emf (slide 67). Please study the next 5 slides on your own!

Back emf.

You must be able to use Lenz’s law to explain back emf.

electric motors and web applets

Generator: source of mechanical energy rotates a current loop in a magnetic field, and mechanical energy is converted into electrical energy.

Electric motor: a generator “in reverse.” Current in loop in magnetic field gives rise to torque on loop.

A dc motor animation is here.

Details about ac and dc motors at hyperphysics.

Other useful animations here.

True Fact you didn’t know: all electrical motors operate on smoke. Every motor has the correct amount of smoke sealed inside it at the factory. If this smoke ever gets out, the motor is no longer functional. I can even provide the source of this true information. http://www.xs4all.nl/~jcdverha/scijokes/2_16.html#subindex

detailed look at a generator (if time permits)

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

brush

Here’s a “freeze-frame.”

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

slip ring

Brushes pressed against a slip ring make continual contact.

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

Disclaimer: there are some oversimplifications in this analysis which an expert would consider “errors.” Anyone who is an expert at generators is invited to help me correct these slides!

B is down. Coil rotates counter-clockwise.

B is down.

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

Rotate your hand until your palm points in the direction of B. Your thumb points in the direction of conventional current.

One more thing…

This wire…

…connects to this ring…

…so the current flows this way.

Another way to generate electricity with hamsters: give them little magnetic collars, and run them through a maze of coiled wires.

http://www.xs4all.nl/~jcdverha/scijokes/2_16.html#subindex

…but now this* wire…

…connects to this ring…

…so the current flows this way.

Alternating current! ac!

Without commutator—“dc.”

*Same wire as before, in different position.

Back emf.

You must be able to use Lenz’s law to explain back emf.

back emf (also known as “counter emf”) (if time permits)

A changing magnetic field in wire produces a current. A constant magnetic field does not.

We saw how changing the magnetic field experienced by a coil of wire produces ac current.

But the electrical current produces a magnetic field, which by Lenz’s law, opposes the change in flux which produced the current in the first place.

http://campus.murraystate.edu/tsm/tsm118/Ch7/Ch7_4/Ch7_4.htm

The effect is “like” that of friction.

The counter emf is “like” friction that opposes the original change of current.

Motors have many coils of wire, and thus generate a large counter emf when they are running.

Good—keeps the motor from “running away.” Bad—”robs” you of energy.

If your house lights dim when an appliance starts up, that’s because the appliance is drawing lots of current and not producing a counter emf.

When the appliance reaches operating speed, the counter emf reduces the current flow and the lights “undim.”

Motors have design speeds their engineers expect them to run at. If the motor runs at a lower speed, there is less-than-expected counter emf, and the motor can draw more-than-expected current.

If a motor is jammed or overloaded and slows or stops, it can draw enough current to melt the windings and burn out. Or even burn up.

Two brief examples (for you to review outside of lecture):

Induced emf on an airplane wing.

Blood flow measurement.

Please study these examples on your own!

v

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

The electrons “pile up” on the left hand wing of the plane, leaving an excess of + charge on the right hand wing.

The equation for  at the bottom of slide 10 gives the potential difference. (You’d have to derive this on a test.)

-

v

+

No danger to passengers! (But I would want my airplane designers to be aware of this.)

Example: Blood contains charged ions, so blood flow can be measured by applying a magnetic field and measuring the induced emf. If a blood vessel is 2 mm in diameter and a 0.08 T magnetic field causes an induced emf of 0.1 mv, what is the flow velocity of the blood?

 = Bℓ v

v =  / (Bℓ)

If B is applied  to the blood vessel, then B is  to v. The ions flow along the blood vessel, but the emf is induced across the blood vessel, so ℓ is the diameter of the blood vessel.

v = (0.1x10-3 V) / ( (0.08 T)(0.2x10-3 m) )

v = 0.63 m/s