Lecture 26 Carl Bromberg - Prof. of Physics

PHY481: Electromagnetism

Magnetic fields (reprise) & vector potential

Lecture 26 Carl Bromberg - Prof. of Physics 1

Magnetic force and fieldMagnetic force: F = qv ! B

Currents make magnetic fields: Biot-Savart Law

dB =µ

0

4!

Id! " r

r2

B(x) =µ

0

4!

Id! " x # $x( )

x # $x3wire

% d3$x

r

r

x

y

z

Id!

Id! k R

R

dB = Id!!

Lorentz force: F = q(E + v ! B)

Magnetic field into the board

Current element

B = !B i

Applications:Short and long wires carrying currentEnd of a wire carrying currentRing of current on axis

Right hand rule #1

Lecture 26 Carl Bromberg - Prof. of Physics 2

Ampere’s Law ! "B = 0 Magnetic fields are “solenoidal” (always true)

! " B = µ

0J

Ampere’s Law (constant currents only.Maxwell’s equations have another term)

B !d!C

"" = µ0

J ! dA

S

""

Via Stokes’s theorem:

B !d!C

"" = µ0I

encl

B(R) =

µ0I

2!R"

dI = J ! dA

Ampere’s Law and symmetries

Straight wire carrying currentField by the easy wayUse right hand rule #2

Lecture 26 Carl Bromberg - Prof. of Physics 3

More Ampere’s law applications

Infinite current slab

Infinite current sheet

B = !

µ0K

2j z > 0; B = +

µ0K

2j z < 0

Inside Outside

B

top= !µ

0Jz j

B

bottom= µ

0Jz j

B

top= !µ

0Ja j

B

bottom= µ

0Ja j

Lecture 26 Carl Bromberg - Prof. of Physics 4

Field equations for B(x) & Vector potential A(x)

! "B = 0

! " ! # B( ) = 0 = µ

0! "J

! "J = 0

Constant currents only.Always true

Constant currents:

x ! "x( )

x ! "x3= !#

1

x ! "x

B(x) = ! " A(x)

Show

! " B = µ

0J

B(x) =µ

0

4!

J( "x ) # x $ "x( )

x $ "x3% d

3"x

Start withBiot-Savart Law:

! " B = µ

0J

B !d!C

"" = µ0I

enclAmpere’s Law:

E = !"V now also B = " # A

Fields from potentialsGet field from a Vector Potential A

Previously used Identity

J( !x ) " #$

1

x # !x

%&'

()* = $ " ( )

B(x) =µ

0

4!J( "x ) #$ %&

1

x % "x

'

()*

+,d

3"x

Lecture 26 Carl Bromberg - Prof. of Physics 5

Required Identity

J( !x ) " #$

1

x # !x

%&'

()* = $ " ( )

!f (x)[ ]k="f (x)

"xk

J( !x ) " #$f (x)[ ] = %ijk

Jj( !x ) #$f (x)[ ]k

= #%ijk

Jj( !x )

&f (x)

&xk

= %ikj

&

&xk

f (x)Jj( !x )'( )*

= $ " f (x) J( !x )[ ]

Let f (x) =

1

x ! "x

Prove:

!

ijk= "!

ikj

J( !x ) " #$

1

x # !x

%&'

()* = $ "

J( !x )

x # !x

%&'

()*

Lecture 26 Carl Bromberg - Prof. of Physics 6

Field equations for B(x) & Vector potential A(x)

x ! "x( )

x ! "x3= !#

1

x ! "x

!

1

x " #x

$%&

'() * J( #x ) = ! *

J( #x )

x " #x

$%&

'()

B(x) = ! "

µ0

4#

J( $x )d3$x

x % $x&

'

()

*

+,

B(x) = ! " A(x)

A(x) =

µ0

4!

J( "x )d3"x

x # "x$Vector potential:

B(x) =µ

0

4!

J( "x ) # x $ "x( )

x $ "x3% d

3"xBiot-Savart Law:

E = !"V

B = " # A

Fields from potentials

Vector potential A plays a crucial role in the quantization of the EM field

B(x) =

µ0

4!J( "x ) #$ %&

1

x % "x

'

()*

+,d

3"x

Lecture 26 Carl Bromberg - Prof. of Physics 7

Vector potential calculations

A(x) =

µ0

4!

J( "x )d3"x

x # "x$

A(x) =

µ0

4!

K( "x )d2"x

x # "x$

A(x) =

µ0

4!

Id!

x " #x$

Volume currents

Surface currents

Line currents

If currents are bounded (do not extend to infinity) get A by

If currents extend to infinityfind B, then get A via Stokes’s theorem:

A ! d!

C"" = # $ A( )" ! da = B" ! da

I = Id! is a vector!

For example, a long straight wire !

Vector potential sumsvectors that point in thedirection of the current!

“Real” currents form loops and thus are bounded

Lecture 26 Carl Bromberg - Prof. of Physics 8

Infinite current sheet

A(x) =

µ0

4!

J( "x )d3"x

x # "x$

A ! d!C"" = B" ! da

zB

B

y

x K

L

H

For bounded currents Currents of infinite extent

Vector potential A:

B ! d!"" = µ

0I

encl

Ampere’s law around the loop shown

B =

µ0K

2

! j z > 0

+ j z < 0{

Magnetic field B is easy:

Choose A•dl loop withnormal in one B direction

But infinite extent currents form loops

Counter clockwise around the loop

Lecture 26 Carl Bromberg - Prof. of Physics 9

Aside

Magnetic field B is only in y direction

B2= !

231

"A1

"x3

# !231

"A3

"x1

Involves z dependence of Ax and, x dependence of Az .

zB

B

y

x K

L

H

A(x, z) = f (z) i + g(x) k

B = ! " A

Assume f (0)= 0 and g(0) = 0

Lecture 26 Carl Bromberg - Prof. of Physics 10

Infinite current sheet

A ! d!

C"" = B" ! da

B! " da = #BHL

A ! d!C"" = f (H ) dx

0

L

" + g(L) dz

H

0

" = Lf (H ) # Hg(L)

A x, z( ) =

B

2+ z i ! x k( )

Assume

A(x, z) = f (z) i + g(x) k

zB

B

y

x K

L

H

f (z) = ! Bz 2;g(x) = Bx 2

z > 0

Lf (H ) ! Hg(L) = !BHLFind functions f and g such that:

For currents ofinfinite extent use

Vector potential A:Choose A•dl loop withnormal in one B direction

B = ! " A

Yes! f (z) = + Bz 2;g(x) = !Bx 2

A(x, z) =

B

2!z i + x k( )

z < 0

! "A = 0Also in Coulomb gauge:

Lecture 26 Carl Bromberg - Prof. of Physics 11

Visualizing the vector potential

A = µ0K !z i + x k( ) 4 z > 0

A = µ0K +z i ! x k( ) 4 z < 0

zA

y

x

K

Infinite sheet of current

Az is discontinuous across thecurrent sheet.

Why does A point opposite to Kand curve this way?

zB

B

y

x K

L

HChanges sign above andbelow the sheet

B =

µ0K

2

! j z > 0

+ j z < 0{Magnetic field

Vector potential B 2 = µ

0K 4

Lecture 26 Carl Bromberg - Prof. of Physics 12

Solenoid field and vector potentialMagnetic field

Cut-away drawing showing the axial magnetic field B(uniform), and growing circular vector potential A.

B = !B j = !µ

0nI j

n = turns unit length

I B

A

A = B !z i + x k( ) 2

Ampere’s law gives

Note: B and A directions and compare with a current sheet.

Two half solenoids Current sheet

Vector potential

Infinite solenoid

Full solenoid

End views.