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.
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