Faraday s Law

8/10/2019 Faraday s Law

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What is Electromagnetics?


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Fundamental Laws of

Electromagnetics


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The Story of E and B

Stationary charges cause electric fields

(Coulombs Law, Gauss Law).

Moving charges or currents cause magnetic fields

(Biot-Savart Law). Therefore, electric fields

produce magnetic fields.

Question: Can changing magnetic fields cause

electric fields?


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Maxwell equations :

rotH JD

t

rotE B

t

div D

div B 0

D EB H

JE

Time-independent case : ( Ampres / Biot-Savart ) law

( Gauss ) law

( Gauss ) law for magnetism

( Faradays ) law of induction

( Initial conditions )

( Assumptions )

( Time-dependent equations )

Electromagnetic wave :

( Electric ) field

( Magnetic ) field

propagation speed : v

1

in a vacuum,

v 1

00 c


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Todays agenda:

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

Faradays Law.You must be able to use Faradays Law to calculate the emf induced in a circuit.

Lenzs Law.You must be able to use Lenzs Law to determine the direction induced current, andtherefore induced emf.

Motional emf; Generators (part 1).You must understand how generators work, and use Faradays Law to calculate numerical

values of parameters associated with generators.


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Induced emf and Faradays 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 anelectric current


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An electric current is induced if there is a closed circuit (e.g.,

loop of wire) in the changingmagnetic flux.

It is observed experimentally that changesin magnetic fluxinduce an emf in a conductor.

A constant magnetic flux does not induce an emfit takes achanging magnetic flux.

B

B

I


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Note that change may or may not not require observable (toyou) motion.

A magnet may move through a loop of wire

N S

move magnet toward coil

v region ofmagnetic field

change area of loop

inside magnetic field

N S

rotate coil in

magnetic field

this part of the loop isclosest to your eyes

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 involveobservable motion.

I


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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.

changing I

changing B

induced I

A changing current in a loop of wire gives rise to achanging magnetic field (predicted by Amperes

law)

A changing current in a loop of wire gives rise to achanging magnetic field (predicted by Amperes

law) which can induce a current in another nearbyloop of wire.


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Todays agenda:

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






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

Experimentally, if the flux through N loops of wire changes bydBin a time dt, the induced emf is

Bd = - N .dt

Faradays law of induction is one of the fundamental laws of

electricity and magnetism.

I wonder why thesign

Faradays Law ofMagnetic Induction


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This is sometimes shown as another version of Faradays Law:

Bd = - N ,dt

B

dE ds = -

dt

is the magnetic flux.

Faradays Law ofMagnetic Induction

B B dA

In a future lecture, well work with . E ds

Well use this versionin a later lecture.

In the equation


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N SI

v

+ -

Example: move a magnet towards a coil of wire.

N=5 turns A=0.002 m2

dB = 0.4 T/s

dt

B dA B dd = - N = - Ndt dt

d BA = - N (what assumption did I make here?)

dt

dB = - N Adt

2 T = - 5 0.002 m 0.4 = -0.004 V

s


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Ways to induce an emf:

change B

change the area of the loop in the field

Possible homework hint: if B varies but loop B.B Bd B dA B(t) dA

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


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Ways to induce an emf (continued):

change the orientation of the loop in the field

=90 =45 =0


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Example: a uniform (but time-varying) magnetic field passes through acircular coil whose normal is parallel to the magnetic field. The coils area is10-2m2and it has a resistance of 1 m. B varies with time as shown in thegraph. Plot the current in the coil as a function of time.

.01 T

=1 sFor 0 < t < 3:

B d BAd dB= - = - = - Adt dt dt

A dB= IR I = = -R R dt

.01dB B .01 A dB .01= = I = - = - = - .0333 A

dt t 3 R dt .001 3

For 3 < t < 5: dB

= 0 I = 0

dt


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Example: a uniform (but time-varying) magnetic field passes through acircular coil whose normal is parallel to the magnetic field. The coils area is10-2m2and it has a resistance of 1 m. B varies with time as shown in thegraph. Plot the current in the coil as a function of time.

.01 T

=1 s

For 5 < t < 11:

dB B -.01= =

dt t 6.01A dB -.01

I = - = -R dt .001 6

= + .0167 A

I(t)

-.0333 A


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Todays agenda:

Induced emf.

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






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Lenzs lawAn induced emf always gives rise to a currentwhose magnetic field opposes the change in flux.*

Experimentally

*Think of the current resulting from the induced emf as trying to maintain the status quo

to prevent change.

N SI

v

+ -

If Lenzs law were not trueif there were a + sign inFaradays lawthen a changing magnetic field would producea current, which would further increase the magnetic field,further increasing the current, making the magnetic field stillbigger


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More on Lenzs Law:

An induced current has a direction such that the magnetic field

due to the current opposes the change in the

magnetic flux that induces the current

Question: What is the direction of the current induced in the ring

given Bincreasing or decreasing?

B due to induced current

B due to induced current


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You can use Faradays Law to calculate the magnitude of theemf (or whatever the problem wants). Then use Lenzs Law tofigure out the direction of the induced current (or the direction

of whatever the problem wants).

Faradays LawBd = - Ndt

The direction of the induced emf is in the direction of thecurrent 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.


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Todays agenda:

Induced emf.

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






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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. Wewill cover the first two in this lecture.

B

d= -

dt

side view

BA

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

rotating loop.

= NBA sin t

NBA

I= sin tR

P= INBA sin t

start with this

derive these


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B

d= -

dtv

B

x=vdtdA

B

MF = I B

= B v

B vI = =R R

PP = F v =I Bv

start with these

derive these

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


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

F= q E+v B

v

B

B

+

= E (Mr. Ed)

Next time we will look at two more examples of motional emf

start with these

derive this

= B v

You could also solve this using Faradays Law by constructing a virtual circuit using virtual conductors.

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


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

B

d= -

dt

start with this

derive these

= B v

B vI = P =I Bv

R


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Generators.You must understand how generators work, and use Faradays Law to calculate numericalvalues of parameters associated with generators.


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Generators and Motors: a basic introduction

Take a loop of wire in a magnetic field

and rotate it with an angular speed .S N

side view

BA

B =B A = BA cos

Choose 0=0. Then 0= t = t .

B = BA cos t

Bd = -dt

Generators are an application of motional emf.


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If there are N loops in the coil

Bd = - Ndt

d BA cos t = - N

dt

= NBA sin t

side view

BA

|| is maximum when = t = 90 or 270; i.e., when Biszero. The rateat which the magnetic flux is changing is thenmaximum. On the other hand, is zero when the magnetic fluxis maximum.

The NBA equation!


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emf, current and power from a generator

= NBA sin t

NBA

I= = sin tR R

P= I = INBA sin t

None of these are on your starting equation sheet!


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= N B A sin t

max = N B A

maxN =B A

-2 2 -1

170 VN =0.15 T 210 m 2 60 s

N = 150 (turns)

Example: the armature of a 60 Hz ac generator rotates in a0.15 T magnetic field. If the area of the coil is 2x10-2m2, howmany loops must the coil contain if the peak output is to be

max= 170 V?Legal for me,

not for you!


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The induced emf causes current to flow in the loop.

v

B

vdtdA

Magnetic flux inside the loopincreases (more area).

System wants to make the fluxstay the same, so the current givesrise to a field inside the loop intothe plane of the paper (tocounteract the extra flux).

Clockwise current!

I

Direction of current?

x


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As the barmoves through the magnetic field, it feels a force

B

I

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

MF = I B

P MF = F = I B

v

FM FP

x


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If the loop has resistance R, the current is

v

B

(as expected).I

Power and current.

B vI = = .

R R

And the power is

PP = F v =I Bv

2P = I IR = I R

Mechanical energy (from the pulling force) has beenconverted into electrical energy, and the electrical energy isthen dissipated by the resistance of the wire.

x


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Todays agenda:

Motional emf (part 2).You must be able to apply Faradays and Lenzs Laws to calculation motional emf, as wellas current and power in circuits powered by motional emf.

Motors and Generators (part 2).We use Faradays Law to calculate numerical values of parameters associated with morekinds of generators. You must also understand conceptually how motors and generatorswork.

Back emf.You must be able to use Lenzs law to explain back emf.


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Motional emf

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

In the last lecture you learned about two examples ofmotional emf.

B

d= -

dt

side view

BA


rotating loop.

= NBA sin t

NBA

I= sin tR

P= INBA sin t

start with this

derive these


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

B

d= -

dtv

B

x=vdt

dA

B

MF = I B

= B v

B vI = =R R

PP = F v =I Bv

start with these

derive these


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

F= q E+v B

v

B

B

+

= E (Mr. Ed)

Today well look at two more examples of motional emf.

start with these

derive this

= B v

You could also solve this using Faradays Law by constructing a virtual circuit using virtual conductors.


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

B

d= -

dt

start with this

derive these

= B v

B vI = P =I Bv

R


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Lets work out these two examples now.

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

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

Remember, its the flux changethat produces the emf. Flux has no direction associated withit. However, the presence of flux is due to the presence of a magnetic field, which doeshavea direction, and allows us to use Lenzs law to determine the direction of current and emf.


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Motional emf is the emf induced in a conductor moving in a

magnetic field.

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

v

B

B

If a conductor (purple bar) moveswith speed v in a magnetic field,

the electrons in the bar experiencea force

MF = qv B= -ev B

The force on the electrons is up, so the top end of thebar acquires a netcharge and the bottom end of the baracquires a net + charge.

up

+


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The separated charges in the bar produce an electric fieldpointing up the bar. The emf across the length of the bar is

v

B

B

The electric field exerts adownward force on the electrons:

EF = qE= -eE

An equilibrium condition is reached,where the magnetic and electricforces are equal in magnitude and

opposite in direction.

up

+

evB= eE= e

= E

= B v


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

Ill include some numbers with this example.


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B = 0.6 T

5c

m

A square coil of side 5 cm contains 100 loops and is positionedperpendicular to a uniform 0.6 T magnetic field. It is quicklyand uniformly pulled from the field (moving to B) to a regionwhere the field drops abruptly to zero. It takes 0.10 s toremove the coil, whose resistance is 100 .


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Final: Bf= 0 .

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

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

Initial: Bi= = BA .B dA


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

initialfinal

(b) Find the current and emf induced.

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


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The induced emf is

Bd d(BA) dA = - N = - N = - N B

dt dt dt

v

A

d xdA dx = = = v

dt dt dt

x

NB


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m

= 100 0.6 T 0.05 m 0.5 s

= 1.5 V

uniformly pulled

= - N B v

x 5 cm mv = = = 0.5t 0.1 s s

The induced current is

I = = = 15 mA .R 100

1.5 V


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B = 0.6 T

5c

m

A square coil of side 5 cm contains 100 loops and is positionedperpendicular to a uniform 0.6 T magnetic field. It is quicklyand uniformly pulled from the field (moving to B) to a region

where the field drops abruptly to zero. It takes 0.10 s toremove the coil, whose resistance is 100 .

And now, back to our

regularly-scheduled lecture.


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Current flows only* during the time flux changes.

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

*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 loop has to be pulled out of the magnetic field, so thereis a pulling force, which does work.

The pulling force is opposed by a magnetic force on thecurrent flowing in the wire. If the loop is pulled uniformly

out of the magnetic field (no acceleration) the pulling andmagnetic forces are equal in magnitude and opposite indirection.

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

(d) Discuss the forces involved in this example.


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

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


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Flux changes. emf induced. Current flows. Workdone.

D

Fapplied


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

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


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

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

I

F =N IL B

Multiply by N because there


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magF = N IL B

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

I

L1

L2 L3

F1

F2

F3

Sorry about the busy slide!The forces should be shownacting at the centers of thecoil sides in the field.

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

Fpull

L4=0

are N loops in the coil.


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-2 -2 -2mag pullF = NILB= 100 1.5 10 5 10 0.6 =4.5 10 N=FMagnitudes only (direction shown in diagram):

I

L1

L2 L3

F1

F2

F3

Fpull

This calculation assumes the coil is pulled out uniformly; i.e.,

no acceleration, so Fpull= Fmag.


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Todays agenda:




electric motors and web applets


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electric motors and web applets

Generator: source of mechanical energy rotates a current loopin a magnetic field, and mechanical energy is converted intoelectrical energy.

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

Other useful animations here.

A dc motor animation is here.

Details about ac and dc motors at hyperphysics.
http://home.a-city.de/walter.fendt/phe/phe.htmhttp://www.walter-fendt.de/ph14e/electricmotor.htmhttp://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/mothow.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/mothow.htmlhttp://www.walter-fendt.de/ph14e/electricmotor.htmhttp://home.a-city.de/walter.fendt/phe/phe.htm


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Lets look at the current direction in this particular freeze-


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pframe.

B is down.

Put your fingers

along thedirection ofmovement. Stickout your thumb.

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

B is down. Coil

rotates counter-clockwise.

One more thing


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g

This wire

connects to thisring

so the currentflows this way.

Another way to generate electricity with hamsters: give them littlemagnetic collars, and run them through a maze of coiled wires.http://www.xs4all.nl/~jcdverha/scijokes/2_16.html#subindex

Later in the cycle the current still flows clockwise in the


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Later in the cycle, the current still flows clockwise in theloop

but now this*wire

connects to thisring

so the currentflows this way.

Alternating current! ac!

Without commutatordc. *Same wire as before, in different position.


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Todays agenda:





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The effect is like that of friction


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The effect is like that of friction.

The counter emf is like friction that opposes the originalchange of current.

Motors have many coils of wire, and thus generate a large

counter emf when they are running.

Goodkeeps the motor from running away. Badrobsyou of energy.

Example: An airplane travels 1000 km/h in a region where the


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v

Example: An airplane travels 1000 km/h in a region where theearths field is 5x10-5T and is vertical. What is the potentialdifference induced between the wing tips that are 70 m apart?

The electrons pile up on the left hand wing of the plane,


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p p g p ,leaving an excess of + charge on the right hand wing.

v

-

+

The equation for at the bottom of

slide 10 gives the potential difference.(Youd have to derive this on a test.)

= B v

-5 = 510 T 70 m 280 m/s

= 1 V

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

Example: Blood contains charged ions, so blood flow can be


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= B v

v = / (B )

If B is applied to the blood vessel, then B is to v. Theions flow along the blood vessel, but the emf is induced across

the blood vessel, so is the diameter of the blood vessel.

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

v = 0.63 m/s

p g ,measured by applying a magnetic field and measuring theinduced emf. If a blood vessel is 2 mm in diameter and a 0.08T magnetic field causes an induced emf of 0.1 mv, what is theflow velocity of the blood?

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Faraday s Law