Lectures 14-15 (Ch. 28)Sources of Magnetic Field

1. B of a moving charge2. Bio-Savarat law3. Superposition principle4. Force between two currents5. Flux of B6. Amper’s Law7. Bohr’s magneton8. 4 types of magnetic materials

B of a moving charge

200

4 r

rvqB

0204

1r

r

qE

A

Tm70 104

Compare electric and magnetic forces

BvqFEqF me

,

0204

1r

r

qE

200

4 r

rvqB

11

2

2

002

v

c

vF

F

m

e

2

00

1c

s

mc 8103

Biot & Savart law (B of a segment of a current)

dlr

rIBd

vqnAIqnAdldQ

r

rvdQBd

200

200

4

,

4

2/322

20

2/3220

2/322

0

2/3220

200

)(2)(4

2

)(4

)(4

cos;4

Rx

IR

Rx

RIR

Rx

dlIRdBBB

Rx

IRdldB

dBdBr

dlrIBd

xx

x

x

B of a loop of a current

Electromagnet (magnetic coil or solenoid) N loops of the current)

2/3220

2/322

20

)(2)(2 axax

NIaB

Most of magnets used in industry are electromagnets.

Attraction is due to magnetization of iron items.(see the end of the previous lecture)

For closely spaced loops (each loop at the same distance to observation point) of the same radius:

In the center of the coil (x=0)

a

NIB

20

N times stronger than from 1 loop

B of a stright wire

2/12222/322

2/1220

2/3220

22

222

20

200

)(

2

)(

)(2)(4

sin,

sin

4;

4

axx

a

yx

dy

axx

Ia

yx

dyIxB

yx

xyxr

dyr

IdBdl

r

rIBd

a

a

a

a

Long wire: a>>x

x

IB

20

Oersted’s experiments and Oersted’s RH rule

Hans Christian Ørsted (Oersted) (1777 – 1851)

1820: current produces B

Superposition principle

21 BBB

d

IBx

00

Examples. Find B at point P.

A force between two currents

r

IlIlBIFII 2'''' 0

'

r

II

l

FII

2

'

'0'

Al

rFI

IIm

N

l

Fmr

12

',102,1

0

7

NB: currents in the same direction attract each other.Currents in the opposite directions repeal each other.It’s different from charges of the same or opposite signs!

Example. Magnetic bottle again (see also lecture 12)

Now we understand also the structure of magnetic field between as a result of superposition of two fields produced by two coils. NB: rotation direction of a positive charge is opposite to the current in the coils. Repulsive interaction of opposite currents results in trapping the charged particles in the bottle.

Example. Find the force acting on each piece of a rectangular conductor and the net force acting on

the rectangular conductor.

Flux of B

dABdABAdBAAA cos

0enclq

AdE

Gauss’s law for E:

Gauss’s law for B: 0 AdB

Magnetic lines are closed lines.There are no magnetic carges.

dlBldBlineline cos

ldBCirculation of B

Example: Find a circulation of B produced by a current in the long straight wire for suggested

integration passes and circulation directions.

r

IB

20

0)(2 2

2

1

10 r

r

r

rIldB

IldB

circle

0

Ir

rIldB

circle

00

2

2

Irdr

IdlBldB

lineline

0

2

0

0

2cos

Arbitrary shape of the closed line

0 ldBcircle

Amper’s law

encl

line

IldB 0

Example. Find a circulation.

Amper’s law allows one to find B for symmetric configurations of current.

Conducting cylinder

r

IB

IrB

RrR

IBRr

R

IrB

R

IJrJrB

Rr

2

2

.22

:

2

,2

.1

0

0

0

20

22

0

Coaxial cable

0,.4

)1(2

)(

)(2

.32

.22

.1

22

220

22

2200

0

20

Bcrbc

br

r

IB

bc

IJ

brJIrB

crbr

IB

braR

IrB

ar

Self-shielding

Example. An infinite current sheet. There are n conductors per unite length.Each of them carries a current, I. Find B.

22 0

0

InBInllB

B

Example. Prove that if in the absence of currents B is unidirectional it has to be uniform.

2121 0)( BBBBlldB

B1

B2

Solenoid (N loops)

L

NnInB

InLINBLldBb

a

,

'''

0

00

L

L’

In the long solenoid (L>>a) B at the exit =(1/2) B in the center

Toroidal solenoid (toroid)

0,.32

2

2

.2

0,.1

00

0

0

Bbr

Ina

INBbaif

r

INB

INrB

bra

Bar

Example. Find a) the net force on the loop and b)the flux of B through the front surface of the loop.

I2I

a

bd

L

Bohr’s magneton

m

e

Jshh

Lm

e

Lg

m

eL

rm

mrevrI

r

ev

T

e

dt

dQI

LB

LL

L

2

10626.6,2

,...3,2,,0,2

22

2

34

22

Planck’s constant

Bohr’s magneton

e m

e

SgS SSBS

,2

,

Different symmetries of e destributions

S-state(L=0,μL=0) 1e:S≠0,μs ≠ 02e:Stotal=0, μs=0

P-state:L= μL ≠ 0

4 types of magnetic materials1.Paramagnetics:μi≠0, 00 0

Bwhen

VM i

i

Under the action of B0 alignment of μi struggles with chaotic thermal motion, resulting in Curie’s law

T

BCM 0

Inside the material:

Magnetic susceptibility

Relative magnetic permeability

...,,,,,,,

111

,1010~

,1

43

0

0

0

00

OUPlAlNaEuSmCe

K

B

M

KB

BMBB

m

m

2. Ferromagneticsμi≠0, M ≠0 in domains

,...,,,

11

,1010~)(

),(1)(

63

0

00

00

0

steelCoNiFe

K

B

BM

BBKB

B

m

m

Electromagnets typically contain a ferromagnetic core

nIB 00

In the absence of an iron core:

0

000

m

mm

K

BnInIKBKB

In the presence of an iron core:

Magnetic permeability

Narrow hysteresis loop,Often use superconducting wires

Both paramagnetics and ferromagnetics are attracted to the magnet

NS B

1. Alignment (randomly oriented μ or M become parallel to B, i.e. material becomes magnetized )

2. Attraction(opposite poles attract each other)

S NB

S N

NS B0

1. Induced M is anti- parallel to B0

2. Repulsion(similar poles repel each other)

S NB0

N S

DiamagneticsIn the absence of magnetic field μi=0,M=0In the presence of magnetic field M is induced in the direction opposite to external magnetic field (consequence of Lens’s low, see next lecture)These materials a repelled by magnets, though this repulsion is very week.

,...,,,,

11

0,1010|~|

),(1)(

64

00

0

HgPbAuCCu

K

BBKB

B

m

m

Superconductors (R=0)

Perfect diamagnetics:Km=0, No magnetic field inside ofsuperconductors

000 MBB

1

Meisner’s effect