Lectures 16-17 (Ch. 29)

Electromagnetic Induction1. The law of EM induction2. The Lenz’s rule3. Motional emf: slide wire generator, Faraday’s disc dynamo, Ac and dc current generators

Electromagnetic Induction, 1830-1832

Joseph Henry (1797-1878) Michael Faraday (1791-1867)

Change of magnetic flux through the loop of wire induces current (i.e. emf) in the loop.

Law of EM Induction (Faraday’s law)

)(111][

cos

2 weberWbmT

dABdABAdB

B

AAA

B

dt

d B

dt

dl B

0

Flux can be changed by change of B or A or angle between B and dA.In order to find the direction of the induced current it is convenient to write the faraday’s law in the form

It is convenient to choose in such direction that

0l

Ad

where is the unite vector of a circulation in the loop which direction is connected with by a RHR:

0 AdB

Ad

0l

Ad

Wilhelm Weber (1804 – 1891)

Examples. Find direction of the induced current.

0l

0l

00 l00 l

Induced current is in the direction of

0l

Induced current is in the direction opposite to

0l

dt

dl B

0

Lenz’s law

Heinrich Lenz (1804 –1865)

Magnetic field produced by induced current opposes change of magnetic flux

Examples

Example

Motional emfSlide-wire generator

Blvl

lvdtdAdt

dAB

dt

dl B

0

0

Origin of this emf is in separation of charges in a rod caused by its motion in B.

vBl

ldBv

q

ldF

BvqF

a

b

a

b

m

m

)(

Motional emf exists in the conductor moving in B. It does not require the existence of the closed circuit.

The secondary magnetic force

lBIFm

'

lBIFF mext

'

External force is required to keep constant velocity of the rod

R

vBlBlv

RIBlv

dt

dxF

dt

dWP

R

vBl

RRIP

extmech

el

2

222

)(

)(

m’

Example. Find motional emf in the rod.

IL

d

Example. Find induced current in the loop with resistance R.

I

V

V

ExampleA single rectangular loop of wire with the dimensions inside a region of B=0.5 T and part is outside the

field. The total resistance of the loop is 0.2Ω . The loop is pulled from the field with a constant velocity of 5m/s.

1)What is the magnitude and direction of the induced current?2) In which part of the loop an induced emf is developed?3) Find the force required to pull the loop at a constant velocity.4) Explain why such force is required.

xB v0.1m

0.5m

0.75m

Example

30cm

40cmv=2cm/s

xB=1T

Find emf in each side of the loop and the net emf when the loop is the region:a)all inside the region of Bb)partly outside of this regionc)all outside of this region

Faraday’s disk dynamo

rv

BRrdrBrdBv

RR

0

2

0 2

v+

-

Fm

F’m

AC –current generator (alternator)

R

tABtIABB

R

tBA

RI

tBAdt

d

tBA

tBA

222 sinsin

sin

sin

cos

,cos

Induced current results in torque which slows down a rotation. External torque is required to maintain the rotation with a constant frequency.

DC-current generator

Applications

2007 Nobel Prize in Physics

Peter Grünberg Albert Fert

For the discovery of the giant magnetic resistance

Tiny magnetic field triggers large change in electrical resistance.

Better read-out heads for pocket-size devices: miniaturization of PC, ipods, etc.

Big RB

Small RB

Resistance strongly depends on the direction of the spin in the first ferromagnetic layer. When it is the same as in the next ferromagnetic layer R is small, when it’s opposite to it R is big.

I

Eddy currents

Eddy currents responsible for levitation and Meisner effect in superconductors

Meisner’s effect

Bind

B0

vN

S

Eddy currents limit efficiency of transformers

Induced nonelectrostatic electric field

dt

d B

Origin of emf? No motion, moreover no B outside solenoid, i.e. in the region of a wire loop. Then it should be E which results in induced current.

0

0,

ldE

dt

dldE

ldFEqF

dt

d

q

ldF

B

elel

Bel

Nonconcervative force

Nonelectrostatic field

0dt

dI

0dt

dI

B(t) should induce E by independently on the presence of the loop of the wire!Let’s find E(r).

dt

dldE B

r

R

dt

dinE

dt

dBRrE

Rr

rdt

dinE

KniBdt

dBrrE

Rr

m

2

2

0

2

2

2

)22

,

2

.1

E

R

r

R

Displacement current

1

2

dB/dt produces E. Let’s show that dE/dt produces B!Consider the process of charging the capacitor.Calculate B in front of the plate of capacitor at r>R.

encl

line

IldB 0

Using the plane surface 1 we get cirB 02 Using the bulging surface 2 we get 02 rB

We come to contradiction! What is wrong ?

)(

,

,,,

0

000

00

dc

ddd

Ec

iildB

dt

dE

A

iji

dt

d

dt

dEA

dt

dqi

AEqEdVd

ACCVq

1.Now we get the same answer for both surfaces 1 and 2!2. B≠0 between the plates!

)(dt

dildB E

General form of Amper’s law

Let’s find B between the plates.

r

iB

irB

RrR

riB

R

i

A

ij

R

rirjrB

Rr

c

d

c

cdd

cd

2

2

.22

2

.1

0

0

20

2

2

2

02

0

B

rR

μ= Kmμ0, ε=Kε0,

In free space K=1, Km =1

Maxwell’s equations

James Clerk Maxwell (1831 –1879)

)(dt

dildB Eencl

dt

dldE B

enclq

AdE

0 AdB

Two Gauss’s laws + Faraday’s law +Amper’s law

Maxwell introduced displacement current, wrote these four equations together, predicted the electromagnetic waves propagating in vacuum with velocity of light and shown that light itself is e.m. wave.

1865 Maxwell’s theory 1887 Hertz’s experiment1890 Marconi radio (wireless communication)