4). Ampere’s Law and Applications

• As far as possible, by analogy with Electrostatics

• B is “magnetic flux density” or “magnetic induction”

• Units: weber per square metre (Wbm-2) or tesla (T)

• Magnetostatics in vacuum, then magnetic mediabased on “magnetic dipole moment”

Biot-Savart Law• The analogue of Coulomb’s Law is

the Biot-Savart Law

• Consider a current loop (I)

• For element dℓ there is an associated element field dB

dB perpendicular to both dℓ’ and r-r’same 1/(4r2) dependence o is “permeability of free space” defined as 4 x 10-7 Wb A-1 m-1

Integrate to get B-S Law

O

r

r’

r-r’

dB(r)

dℓ’

3o )x('d

4)(d

r'r

r'rrB

3

o )x('d

4)(

r'r

r'rrB

B-S Law examples

(1) Infinitely long straight conductor

dℓ and r, r’ in the page dB is out of the page

B forms circlescentred on the conductor

Apply B-S Law to get:

r

I

2B o

r - r’

dB

I

dℓ

B

r’

Or

sin = cos 1/222 zr

r

z

B-S Law examples(2) “on-axis” field of circular loop

Loop perpendicular to page, radius a

dℓ out of page and r, r’ in the page On-axis element dB is in the page, perpendicular to r - r’, at to axis.

Magnitude of element dB

Integrating around loop, only z-components of dB surviveThe on-axis field is “axial”

1/2222o

z2o

za

a

'-

acoscos

'-

d

4dB

'-

d

4dB

rrrrrr

II

r - r’ dB

I

dℓ

zdBz

r’r

a

On-axis field of circular loop

Introduce axial distance z, where |r-r’|2 = a2 + z2

2 limiting cases:

3

2o

2o

2o

zaxison

'-2

aa2cos

'-4

dcos'-4

dBB

rrrr

rr

I

I

I

23

22

2o

axisonza2

aB

I

3

2oaz

axisono0z

axison 2z

aBand

2aB

I

I

r - r’ dB

I

dℓ

zdBz

r’r

a

Magnetic dipole momentThe off-axis field of circular loop ismuch more complex. For z >> a it is identical to that of the electric dipole

m “current times area” vs p “charge times distance”

loop currentby enclosed area

z a oramwhere

sincos 2r4

m

sin cos 2r4

p

22

3o

3o

ˆI I I

ˆ ˆ

ˆˆ

m

rB

rE

rm

B field of large current loop• Electrostatics – began with sheet of electric monopoles• Magnetostatics – begin sheet of magnetic dipoles• Sheet of magnetic dipoles equivalent to current loop• Magnetic moment for one dipole m = I area

for loop M = I A area A

• Magnetic dipoles one current loop• Evaluate B field along axis passing through loop

B field of large current loop• Consider line integral B.dℓ from loop• Contour C is closed by large semi-circle which contributes

zero to line integral

2za

dza

circle)(semi 0za

dza

2.d

3/222

2

o3/222

2o

II

C

B

.dB

oI/2

oI

z→+∞z→-∞

I (enclosed by C)

Ca

Electrostatic potential of dipole sheet

• Now consider line integral E.dℓ from sheet of electric dipoles• m = I I = m/ (density of magnetic moments)

• Replace I by Np (dipole moment density) and o by 1/o

• Contour C is again closed by large semi-circle which contributes zero to line integral

.dE

Np/2o

-Np/2o

0

C

.dcircle)(semi 0.d EE

Electric magnetic

Field reverses no reversal

Differential form of Ampere’s Law

S

SjB .d.d oenclo I

S

j

B

dℓ

Sj.dd I

Obtain enclosed current as integral of current density

Apply Stokes’ theorem

Integration surface is arbitrary

Must be true point wise

SS

SjSBB .dd.d o.

jB o

Ampere’s Law examples(1) Infinitely long, thin conductor

B is azimuthal, constant on circle of radius r

Exercise: find radial profile of B inside and outside conductor of radius R

r2Br 2 B.d o

oenclo

I II

B

B

r 2B

R 2

rB

oRr

2o

Rr

I

I

B

rR

SolenoidDistributed-coiled conductorKey parameter: n loops/metre

If finite length, sum individual loops via B-S Law

If infinite length, apply Ampere’s Law B constant and axial inside, zero outsideRectangular path, axial length L

(use label Bvac to distinguish from core-filled solenoids)

solenoid is to magnetostatics what capacitor is to electrostatics

I II nBnLLB.d ovacovacenclovac B

IL

B

I

Relative permeability

Recall how field in vacuum capacitor is reduced when dielectric medium is inserted; always reduction, whether medium is polar or non-polar:

is the analogous expression

when magnetic medium is inserted in the vacuum solenoid.

Complication: the B field can be reduced or increased, depending on the type of magnetic medium

vacrr

vac BBE

E

Magnetic vector potential

0 x x

-.d

0

E

EE For an electrostatic field

We cannot therefore represent B by e.g. the gradient of a scalarsince

Magnetostatic field, try

B is unchanged by

)

o

o

.( always 0 . also

zero) not (x

EB

jB rhs

later) (see xx x

0 x

x

AB

AB

AB

..

0xx'x

AAA

AA

'