Faraday's Law

dA

B B

∫ •≡ AdBΦB

rr

dtdΦε B−=

Applications of Magnetic Induction• AC Generator

– Water turns wheelrotates magnetchanges fluxinduces emfdrives current

• “Dynamic” Microphones(E.g., some telephones)– Sound

oscillating pressure wavesoscillating [diaphragm + coil]oscillating magnetic fluxoscillating induced emfoscillating current in wire

Question: Do dynamic microphones need a battery?

More Applications of Magnetic Induction

• Tape / Hard Drive / ZIP Readout– Tiny coil responds to change in flux as the magnetic domains

(encoding 0’s or 1’s) go by.2007 Nobel Prize!!!!!!!!Giant Magnetoresistance

• Credit Card Reader– Must swipe card

generates changing flux– Faster swipe bigger signal

More Applications of Magnetic Induction• Magnetic Levitation (Maglev) Trains

– Induced surface (“eddy”) currents produce field in opposite direction

Repels magnetLevitates train

– Maglev trains today can travel up to 310 mphTwice the speed of Amtrak’s fastest conventional train!

– May eventually use superconducting loops to produce B-fieldNo power dissipation in resistance of wires!

NS

rails“eddy” current

Faraday’s Law of Induction

dtdAdB

dtd BΦ−=∫ ⋅−=

rrε

Recall the definition of magnetic flux is

Faraday’s Law is the induced EMF in a closed loop equal thenegative of the time derivative of magnetic flux change in the loop,

∫ ⋅=Φ AdBBrr

Constant B field,no induced EMFin loop

changing B field,causes induced EMFin loop

Getting the sign EMF in Faraday’s Law of Induction

Define the loop and an area vector, A, who magnitude is theArea and whose direction normal to the surface.

The choice of vector A direction defines the direction of EMFwith a right hand rule. Your thumb in A direction and then yourfingers point to positive EMF direction.

A

Lenz’s Law – easier way!

Heinrich Friedrich Emil Lenz(1804-1865)

The direction of any magnetic inductioneffect is such as to oppose the cause ofthe effect.

Example if an external magnetic field on a loopis decreasing, the induced current creates a field parallel to the that tends to increase thenet field.

⇒ Convenient method to determine I direction

Example if an external magnetic field on a loopis increasing, the induced current creates a fieldopposite that reduces the net field.

Lecture 16, Act 2• A conducting rectangular loop moves with

constant velocity v in the +x direction through a region of constant magnetic field Bin the -z direction as shown.

• What is the direction of the induced current in the loop?

(a) ccw (b) cw (c) no induced current• A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown.

• What is the direction of the induced current in the loop?

2A

X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X X

v

x

y

(a) ccw (b) cw (c) no induced current

2B v

I

x

y

Lecture 16, Act 2

(a) ccw (b) cw (c) no induced current

1A

• There is a non-zero flux ΦB passing through the loop since B is perpendicular to the area of the loop.

• Since the velocity of the loop and the magnetic field are CONSTANT, however, this flux DOES NOT CHANGE IN TIME.

• Therefore, there is NO emf induced in the loop; NO current will flow!!

• A conducting rectangular loop moves with constant velocity v in the +x directionthrough a region of constant magnetic field Bin the -z direction as shown.– What is the direction of the induced

current in the loop?2A

X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X X

v

x

y

Lecture 16, Act 2

(a) ccw (b) cw (c) no induced current

• A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown.

• What is the direction of the induced current in the loop?2B

• The flux through this loop DOES change in time since the loop is moving from a region of higher magnetic field to a region of lower field.

• Therefore, by Lenz’ Law, an emf will be induced which will oppose the change in flux.

• Current is induced in the clockwise direction to restore the flux.

v

I

x

y

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 122cm^2 is rotated in a time interval of 0.04s from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The earth's magnetic field at the lab location is 6x10(-5)T.

What is the total magnetic flux through the coil before it is rotated? What is the total magnetic flux through the coil after it is rotated? What is the average emf induced in the coil?

Y&F Problem 29.2

B

2527

2725

1083.104.0

1031.7)(

090cos1031.7)0122.0)(106(

0cos

TmxsTmx

t

ABTmxmTx

BAdABAdB

if

f

oi

−−

−−

=+=Δ

Φ−Φ−=

==Φ==

==•=Φ ∫ ∫

ε

rr

Incredible shrinking loop: a circular loop of wirewith a magnetic flux is shrinking with time. Inwhich direction is the induced current?

(a) There is none. (b) CW. (c) CCW

X X X X X X

X X X X X X

X X X X X X

X X X X X X

Determine the direction of the induced current in a – b when

1.) Switch is opened after having been closed for a long time.

(a) No current. (b) left to right. (c) right to left.

2.) The resistance R is decreased while switch remains closed.

(a) No current. (b) left to right. (c) right to left.

Slide Wire Generator

The area is LS(t) and dS/dt equals velocity of slider

S

BLvSdtdBLBLS

dtdBA

dtd

dABdtdAdB

dtd

dtd

B

−=−=−=−=

−=•−=Φ−= ∫∫rr

ε

Direction of current and ε as shown from Lenz’s Law.

S

BLv=ε

Slide Wire Generator; work and powerwe found

we now ask what ispower dissipated, Pdand power applied, Paassume total resistanceis R, then I=Ɛ/R andPd =I2R=B2L2v2/R

The power applied, Pa=Fv, is the rate of work to the bar to the right against the force, F, which is pointed to the left. F=ILB= (Ɛ/R)LB = B2L2v/R. Thus Pa= Fv = B2L2v2/R = Pd and we find the power put into the moving the bar equals the heat dissipated in the circuit.

⇒ Power needed to move bar = Power dissipated in circuit

S

Slide Wire Generator; use Lenz’s Law to get I direction

Lenz’s Law saysdirection creates fieldthat opposes changein magnetic flux.

If we pull bar to right, the net magnetic flux in rectangle increasesinto screen, hence the I direction must induce opposite B field which is out of screen and is correct in drawing.

Suppose Lenz’s law were reversed, then I would be reversed and F would go right and the bar would be accelerated to the right, w/o need of external positive work and heat would be dissipated at the same time. This violates Conservation of Energy, so Lenz’s law is correct.

Motional Electromotive Force

In Faraday’s Law, we can induce EMF in the loop when the magnetic flux, ΦB, changes as a function of time. There are twoCases when ΦB is changing,1) Change the magnetic field (non-constant over time) 2) Change or move the loop in a constant magnetic field

dtdAdB

dtd BΦ−=∫ ⋅−=

rrε

The slide wire generator is an example of #2 and the induction ofEMF via moving parts of the loop is called, motional EMF.

Alternator Generator Wire loop area A rotates with respect to constant magnetic field.

If the angular frequency is ω, then

and EMF in the loop is

φcosBAAdBB ∫ =•=Φrr

( )tBAB ωcos=Φ

( )tBAdtd

B ωω sin−=Φ

( )tBAdtd

B ωωε sin=Φ−=

Slide Wire Generator; revisited again

Suppose we move a conducting bar in a constant B field, then a forceFB =q v×B moves + charge up and – charge down. The charge distribution produces an electric field, E, force, FE, and EMF, Ɛ, between a & b. This continues until equilibrium is reached: FE = -FB.

Bvq

BvqqF

qF

E BErr

rrrrr

×−=×

−=−

== vBlldEb

a=∫ ⋅=

rrε

In effect the bar motional EMF is an equivalent to a battery EMF

B

EE

Slide Wire Generator; revisited again

vBlldEb

a

=⋅= ∫rr

ε

If the rod is on the U shaped conductor, the charges don’t build up at the ends but move around the U shaped portion. They produce an electric field in the U shaped circuit. The wire acts as a source of EMF – just like a battery. Called motional electromotive force.

Direct Current Homopolar Generator invented by Faraday

Rotate a metal disk in a constant perpendicular magnetic field. The charges in the disk when movingreceive a radial force. The causescurrent to flow from center to pointb.

Faraday’s originalapparatus had the diskfixed with a meter and thenhe spun a magnet underthe disk.

2

0 21 BRdrBr

Rωωε == ∫