wed. fri. 6.2 field of a magnetized object hw10 mon., 6.3...

Wed. Fri.

6.1 Magnetization 6.2 Field of a Magnetized Object

HW10

Mon., 6.3, 6.4 Auxiliary Field & Linear Media HW11

Wed. 12 noon Exam 3 (Ch 7, 10, 4, 6)

Linear Dielectrics Chunk of induced dipoles

Ep

Everyone’s field but its own

Point along field Linearly proportional

d

pdP

Polarization = Dipole density

so E

d

d

d

EdP

Define “electric susceptibility” to be the proportionality constant (and provide convenient factor of eo.)

P e0eE

e

d

d

o

e

1

PED o

e

Always linear dielectric

EE eoo

ee Eeo

e 1

ED

e

e e0 1 e Permittivity (of not-so-free space)

Dielectric Constant

er oe

ee 1

ED ro

ee

Or, in terms of polarization

PDe

11

or PD

e

e

1

for individual induced dipole

For chunk of induced dipoles

Linear Dielectrics

o

Example: Alternate / iterative perspective on field in dielectric. Consider again a simple capacitor with dielectric. We’ll find the electric field in terms of what it would have been without the dielectric. We’ll do this iteratively and build a series solutions.

sfree

-sfree

o

-sb.o

0. Say we start with no dielectric. Initially there’s the field simply due to the free charge; Eo.

oEE

and the associated surface charges contribute a field of their own, where so in the opposite direction.

E1 -P0 e0 -eE0

oeoo EP

e

zEo

ob ˆ.1

e

s

nPobˆ

0.

s

oe E

-

1

1. This field induces a little counter polarization,

P1 e0eE1 -e0e

2E0

E2 -P1 e0 -e 2E0

oe E

2-

2. See a pattern?

...

1

2

o

n

n

e EE

-0

As long as e< 1, this converges to

Einside 1

1 e

E0

1

er

E0

Same result as we got previously

We insert the dielectric and that field induces a polarization,

Which is means a surface charge and resulting field contribution of its own

Linear Dielectrics

o

Exercise: Try it for your self. A sphere made of linear dielectric

material is placed in an otherwise uniform electric field

o. Find the electric field inside the sphere in terms of the material’s dielectric constant, er.

sfree

-sfree oEE

You can take it as a given that a sphere of uniform polarization contributes field E -P 3e0

Example: A coaxial cable consists of a copper wire of radius a surrounded by a concentric copper tube of inner radius c. The space between is partially filled (from b to c) with material of dielectric constant er as shown below. Find the capacitance per length of the cable.

a b

c

V

QC

L

For the sake of reasoning this out, say there’s charge Q uniformly distributed along the surface of the central wire.

er

enclfQadD .

-

c

a

ldEV

QsLD 2

Gaussian cylinder of some radius a<s<c.

sL

QD

2

csbssL

Q

bsassL

Q

E

ro

o

ˆ2

ˆ2

ee

e

-

c

a

ldEV

--

c

b ro

b

a o

dssL

Qds

sL

QV

eee 22

bc

ab

orL

QV lnln

21e

e-

csbD

bsaDE

ro

o

1

1

ee

e

--

c

b

b

a

ldEldE

bc

ab

o

rL

C

lnln

21e

e

Exercise: There are two metal spherical shells with radii R and 3R. There is material with a dielectric constant er = 3/2 between radii R and 2R. What is the capacitance? R

2R

3R

er

Recall Multi-pole Expansion of Vector Potential

1

r1

r

r

r

n0

n

Pn cos

cosnPObservation location

nth Legendre polynomial

P0 1

P1 u u

2

13 2

2

-

uuP

r

vdqrA o

4)(

Continuous current distribution

014

cos1

)(n

n

n

ndrJPr

rrA o

drJP

r

r

rrA

n

n

n

o

0

4cos

1)(

Re-ordering sums

r

r

r

-

...2

1cos3cos

3

22

2 r

drJr

r

drJr

r

drJ

Monopole Dipole term term

Observation location

r

vdqrA o

4)(

Continuous current distribution

r

r

r

...cos

24r

drJro

Dipole term Dipole’s integral

drJr

cos drJrr

ˆ lIdrr

ˆ

Steady current

ldrrI

ˆ

to mathland and back

- adrI

ˆ - adrI

ˆ arI- ˆ raI ˆ

Recall Multi-pole Expansion of Vector Potential

r

vdqrA o

4)(

...

ˆ24

r

raIo

Magnetic Dipole Moment

aIm

...

ˆ)(

24r

rmrA o

Dipole term for a loop Observation location

r

Magnetic Dipole Moment

aIm

...

ˆ)(

24r

rmrA o

r

m

I

zRIm ˆ2

...

ˆˆ)(

2

2

4r

rzRIrA o

...ˆsin

2

2

4

r

RIo

Same direction as current AB

3

ˆˆ3

4)(

r

rr m-mrB o

dip

If m at origin and pointing up

ˆsinˆcos24

)(3

rr

mrB o

dip

(yes, same form as E for p)

Magnetic Dipoles

...

ˆ)(

24r

rmrA o

aIm

ˆsinˆcos24

)(3

rr

mrB o

dip

From a distance much greater than the current distribution’s size, the dipole term dominates

Recall Torque for real current loop

extB

RightF

LeftF

a

I

extBaIN

sinwILBN ext

extBmN

Like for electric dipoles

extEpN

Recall Force for real current loop

extB

RightF

leftF

a

I

xILBILBFFF LeftRightleftRightnetˆ-


extEpF

extBmF

If m is constant, the same as

extBmF

In terms of dipole moment In terms of dipole moment

Change in energy

EpU

-


-- LLRR ldFldFU

- rightright ldF

2

-

dBIb a2

)(2

- dBIb a2

sin)(2

f

i

BIba

cos)(-

f

i

mBU

cos-

BmU

-

In terms of dipole moment

Magnetic Dipoles

...

ˆ)(

24r

rmrA o

aIm

ˆsinˆcos24

)(3

rr

mrB o

dip


extBmN

extBmF


extBmF

BmU

-

Exercise: you have two magnetic dipoles; find the torque m1 applies on m2.

m1

m2

r

y x

z

Dipole B

Dipole A

Magnetic Dipoles

...

ˆ)(

24r

rmrA o

aIm

ˆsinˆcos24

)(3

rr

mrB o

dip


extBmN

extBmF


extBmF

BmU

-

Exercise: find the force on dipoles A, B, C in and near a slab of uniform current

Dipole C

a

-a

2a

zJJ ˆ

infront ˆ

inside ˆ

yJa

yJxB

o

o

B

Dipole B

Dipole A

Magnetic Dipoles

...

ˆ)(

24r

rmrA o

aIm

ˆsinˆcos24

)(3

rr

mrB o

dip


extBmN

extBmF


extBmF

BmU

-

Exercise: Force between two dipoles (read, “bar magnets”). What’s the force A exerts on B?

r = z

Considering ‘real’ dipoles (with real radii), roughly sketch the field and resulting forces on the current loops

Effect of Magnetic Field on Dipoles

Para-magnetic: Rotate the loop (torque)

Dia-magnetic: Stretch the loop; changing field(s) – Faraday-effect: impede the current

Wed. Fri.

6.1 Magnetization 6.2 Field of a Magnetized Object

HW10

Mon., 6.3, 6.4 Auxiliary Field & Linear Media HW11

Wed. 12 noon Exam 3 (Ch 7, 10, 4, 6)

wed. fri. 6.2 field of a magnetized object hw10 mon., 6.3...

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