Electromagnetic induction So far we have found that kQ = E r - - PDF document

Electromagnetic induction

r r kQ E ˆ

2

= v

2

ˆ 4 r r v q B × = v v π µ

So far we have found that

Electric charges create E-fields Moving electric charges create B-fields

But this is not the whole story!

2

Electromagnetic induction

Electric Fields can also be created by a changing Magnetic Field. Magnetic Fields can also be created by a changing Electric Field. This is described by “Faraday’s Law”. This will help complete the set of equations called Maxwell’s equations of electricity and magnetism. Electricity and Magnetism can be unified in a single theory! The electricity and magnetic forces are manifestations of a single “electromagnetic interaction”

Electromagnetic induction Magnetic Flux

Magnetic Flux

The flux through an element of area is: To calculate the flux through the surface, we need to integrate: For the simple case of a flat surface, we obtain: B φ

Magnetic flux

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Clicker Question At what angle of orientation of the surface is the flux through the surface half of the maximum possible flux? A) φ = 0 degrees B) φ = 30 degrees C) φ = 45 degrees D) φ = 60 degrees E) None of the above

B φ

Faraday’s Law of Induction

8

dt d

B

Φ − = ε

Two new quantities introduced.

• 1. Magnetic Flux = ΦB
• 2. Electro Motive Force (EMF) = ε

A changing (time-dependent) magnetic field “induces” (generates) an EMF ε. The magnitude of the EMF equals minus the rate of change of the flux.

Important observations:

• Recall: EMF is not a force, but a source of voltage

capable of generating power.

• The EMF is called “motional” EMF, to distinguish

from “chemical” EMF (batteries)

• Recall: The magnetic flux through a closed surface is

zero! (Gauss’ Law for B-fields)

• The surface for Faraday’s Law is an open surface!

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Clicker Question We have a square imaginary loop with side of length a. There is a Magnetic Field that is uniform throughout the region as shown. How does the Magnetic Flux magnitude |ΦB| change if we halve the Magnetic Field strength and double the sides of the square loop? A) Flux goes to ½ original value B) Flux goes to ¼ original value C) Flux goes to 2 times original value D) Flux does not change E) None of the above

X B

Changing the magnetic flux

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• 1. Change the strength of the Magnetic Field
• 2. Change the area of the loop
• 3. Change the orientation of the loop (A vector) and the B-field

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A loop of wire is moving rapidly through a uniform magnetic field as

• shown. Is a non-zero EMF induced in the loop?

A) No, there in no EMF B) Yes, there is an EMF Clicker Question

13

A loop of wire is spinning rapidly about a stationary axis in uniform magnetic field as shown. True or False: There is a non-zero EMF induced in the loop. A) False, zero EMF B) True, there is an induced EMF Clicker Question

and the angle φ is changing.

φ cos BA F =

Direction of the EMF around a loop

Lenz’s Law

15

The (-) negative sign in Faraday’s Law is more of a reminder. Lenz’s Law is just another way to express Faraday’s Law. Moreover, it can be deduced from it.

dt d

B

Φ − = ε

The induced EMF tends to induce a current in the direction that opposes the changes in magnetic flux

Lenz’s Law

16

If we move the magnet closer to the loop, what is the direction of the induced EMF and thus the direction of the current in the loop?

∫

⋅ = Φ

surf B

A d B v v

• 1. The Magnetic

Flux is increasing.

• 2. Therefore

> Φ dt d

B

• 3. Induced current must create B-field

that fights the change!

Lenz’s Law

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> Φ dt d

B

Induced current must create B-field that fights the change! An induced B-field down through the loop will create a slight decrease in ΦB. Thus, we have determined the direction of the induced current and EMF. i

B

Lenz’s Law

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“The Magnetic Flux ΦB Up Through the Loop is Increasing. To Fight this Change, an Induced B-field is made that Decreases the Flux Up Through the Loop. This is accomplished by the induced current.”

Lenz’s law (summary)

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A bar magnet is positioned below a loop of wire. The magnet is pulled down, away from the loop. As viewed from above, is the induced current in the loop clockwise, counterclockwise, or zero? A: clockwise B: counter-clockwise

eyeball

C: zero Answer: Counterclockwise. As the magnet is pulled away, the flux is decreasing. To fight the decrease, the induced B-field should add to the

• riginal B-field.

Clicker Question i

B

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A loop of wire is sitting in a uniform, constant magnet field as

• shown. Suddenly, the loop is bent into a smaller area loop. During

the bending of the loop, the induced current in the loop is ... A: zero B: clockwise C: counterclockwise

B(in) B(in)

Answer: Clockwise. The flux into the page is decreasing as the loop area decreases. To fight the decrease, we want the induced B to add to the original B. By the right hand rule (version II) , a clockwise induced current will make an induced B into the page, adding to the original B. Clicker Question

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Is there an induced current in this circuit? If so, what is its direction?

• A. Yes, clockwise
• B. Yes, counterclockwise
• C. No

Clicker Question * There is zero Magnetic Flux at all times through the loop.

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A current-carrying wire is pulled away from a conducting loop in the direction shown. As the wire is moving, is there a clockwise current around the loop, a counterclockwise current or no current?

• A. There is a clockwise current around the loop.
• B. There is a counterclockwise current around the loop.
• C. There is no current around the loop.

Clicker Question

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Example with numbers: V 10 cm 1 c m B (increasing) Suppose there are 1000 loops to the wire coil. Area = (0.1m)2=0.01 m2 dB/dt = +0.1 Tesla/second

dt dB A BA dt d dt d

B

= = Φ = ) ( | |ε Volts 001 . ) 1 . )( 01 . ( | | = = ε Volt dt d N dt N d

B B

1 ) 1 ( ) ( | | = Φ = Φ = ε

Generators

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Generators use Faraday’s Law to convert mechanical energy into electrical energy. This is the opposite of motors which convert electrical into mechanical energy.

The alternator

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• Wire loop in a constant external B-field.
• Turning a crank makes the loop rotate.
• This then induces a current in the wire loop.

X X B B i (induced)

The alternator

27

• Wire loop in a constant external B-field.
• Turning a crank makes the loop rotate.
• This then induces a current in the wire loop.

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Clicker Question X X B B i Instead of turning a crank and inducing a current, what if you hook up the wire loop to a battery and push current through it. What happens to the loop? A) Nothing B) It feels a net force C) It feels a net torque D) The current stops flowing Note that the wire loop is a rectangle that is slightly rotated from the plane of the page.

29

Generator Motor Converts mechanical energy into electrical energy. Crank or otherwise mechanically turns a wire loop in an external B-field. This induces an electric current in the loop. Converts electrical energy into mechanical energy. Battery creates a current in a

• loop. This interacts with an

external B-field to create a mechanical torque.

The “sliding bar generator”

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Sliding metal bar on metal rails

L x=vt v X X X X X X B Bar is pulled by an external agent to the right at constant velocity. There is a uniform B- field surrounding the rails and loop system.

The “sliding bar generator”

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L x=vt v X X X X X X B This creates a wire loop whose area is growing with time.

Lvt Lx A = =

Thus, the Magnetic Flux is increasing with time!

BLv Lvt dt d B BA dt d dt d

B

= = = Φ ) ( ) (

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L x=vt v X X X X X X B Clicker Question What is the direction of the induced EMF and thus current in the wire loop? A) Clockwise B) Counter Clockwise Motional EMF induced

BLv = ε

The “sliding bar generator”

33

There is another way to see why the current in the bar is up. Charges in the bar are moving to the right with the bar itself. v X B There is a B-field in this region. Thus, there is an upward force on the charges.

(up) qvB B v q F = × = v v v

Thus, there is an upward current in the bar! i

The “sliding bar generator”

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v X B i One more consequence…. If there is an upward current in the wire and there is an external B-field, the we get?

(left) ILB B L I FMag = × = v v v

FM But, there is also the external agent pulling the rod to the right. FAgent If the velocity is constant, what is true about the forces? They must be equal and opposite!

A conducting loop is halfway into a magnetic

• field. Suppose the magnetic field begins to

increase rapidly in strength. What happens to the loop?

• A. The loop is pushed upward, toward the top of the page.
• B. The loop is pushed downward, toward the bottom of the page.

C.The loop is pulled to the left, into the magnetic field. D.The loop is pushed to the right, out of the magnetic field.

• E. The tension is the wires increases but the loop does not move.

Clicker Question

Eddy currents

If a metal object (something in which electric charges can move) and an external source of Magnetic Field are in relative motion so that there is changing Magnetic Flux through the metal, then Faraday says there is an EMF and thus an Eddy current. If the metal is moving, then the direction of the current is always such as to cause a force which

• pposes the motion (slows the metal).

Eddy currents: Applications

Eddy Current Brakes on Trains Since there is no direct contact, there is minimal friction. Thus the brakes do not wear down!

Eddy currents: Applications

Metal detector Portable metal detector

Eddy currents: Applications

Induction Cooktop

Works via changing B-field, inducing a current in the pot or wok (which must be metal). This current then leads to power loss as heating of the pot. The cooktop surface which is not metal and does not conduct, does not get hot.

Induced electric field

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In the case where the loop or metal is stationary and the Magnetic field is changing, then there is no Magnetic force on the charges in the wire (since v=0)

B v q FM v v v × =

Thus, the only way to explain the induced current in the wire is to admit that there is the creation of a new Electric Field!

Motional EMF revisited

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∫

⋅ =

Loop

l d E v v ε

Technically, the EMF around a closed Loop is defined as: But, recall that Voltage is defined as:

∫

⋅ − = ∆ l d E V v v

Was not this integral around a closed loop always zero? The motional EMF is not always zero around a closed loop !

Faraday’s Law reformulated

42

dt d

B

Φ − = ε

∫ ∫

⋅ − = ⋅

surface Loop

A d B dt d l d E v v v v

How do the loop and the surface relate?

Motional EMF revisited

43

∫ ∫

⋅ − = ⋅

surf Loop

A d B dt d l d E v v v v

Bexternal R

[ ]

) ( ) 2 (

2

R B dt d R E

ext

π π v v − =

Motional EMF revisited

44

B Wire Loop If B = constant, Then Magnetic Flux = constant dΦB/dt = 0 EMF = 0 and Voltmeter reads 0. If B =increasing, Then Magnetic Flux = changing |dΦB/dt| > 0 EMF > 0 and Voltmeter reads > 0. What if the wire loop was not there? V

Induced electric field

45

A changing B-field creates a new kind of E-field. B If B is decreasing E-field is CCW E B E If B is increasing E-field is Clockwise

Induced electric field

46

The (non-Coulomb) E-field is very different from the Coulomb E- field created by single electric charges. q B E E

≠ ⋅ = ∫

Loop

l d E v v ε

= ⋅ − = ∆

∫

l d E V v v

Time dependent B-fields

So far we have seen that a time varying flux can be generated by moving a magnet, or by mechanically changing the surface area… But what about changing the magnitude of the field? From Ampere’s Law: Replacing in Faraday’s Law we obtain:

Videos

http://www.youtube.com/watch?v=jBrneA-wQGA Tesla

Ampere’s Law revisited

49

Maxwell realized that Ampere’s Law must be incomplete because he discovered a situation where it gave the wrong answer!

? + = ⋅

∫

enc

i l d B µ v v

Ampere’s Law revisited

50

Current I(t) Current I(t) + + + + +

• 1. As the currents are flowing, one has Magnetic Fields around the wires.
• 2. Charge is also building up on the plates, creating an increasing Electric Field

between them. B B E Consider a charging capacitor. The charge builds up in the plates, and there is a time-dependent current flowing through the wires I(t)

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Current I Current I + + + + +

• B=?

Clicker Question Consider the Biot-Savart Law for Magnetic Fields. Is there a Magnetic Field at the point labeled between the plates? A)Yes, there is a B-Field B)No, there is zero B-field

∫ ∫

× = =

2

ˆ 4 r r L d I B d Btot v v v π µ

52

Current I Current I + + + + +

• B=?

Clicker Question Now consider an Amperian Loop as drawn. According to Ampere’s Law is there a Magnetic Field at the point labeled between the plates? A)Yes, there is a B-Field B)No, there is zero B-field Something must be wrong with Ampere’s Law!

Ampere’s Law revisited

Both surfaces have the same boundary, if Ampere’s law applies, then the result should be the same, independently of the surface we choose.

Current I Current I + + + + +

• Ampere’s Law revisited

Both surfaces have the same boundary, if Ampere’s law applies, then the result should be the same, independently of the surface we choose.

Ampere’s law: solving the paradox

The changing Electric Field inside the capacitor must act like a current enclosed for Ampere’s Law.

∫

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Φ + = ⋅ dt d I l d B

E enc

ε µ v v

Where

∫

⋅ = Φ A d E

E

v v

Maxwell defined a fictitious “displacement current”: Thus, a changing Electric Field creates a Magnetic Field !

dt d I

E D

Φ = ε

The displacement current

E

EA Ed d A CV Q Φ = = = = ) ( ε ε ε

C E D

I dt dQ dt d I = = Φ = ε

The B-field in between the plates is exactly the same as the field created by the current outside. The fictitious “displacement current” gives an idea of continuity of the current, even though there is no charge flowing in the space separating the plates Between the plates: ; = Φ =

C E D

I dt d I ε Outside the capacitor: ; ≠ =

C D

I I

Maxwell’s equations

ε

encl Surf

Q A d E = ⋅

∫

r r

Gauss’ Law for E-fields

= ⋅

∫

Surf

A d B r r

Gauss’ Law for B-fields

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Φ + = ⋅

∫

dt d I l d B

E thru loop

ε µ r r

Ampere’s Law

dt d l d E

B loop

Φ − = ⋅

∫

r r

Faraday’s Law