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© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1
LECTURE NOTES 24
MAXWELL’S EQUATIONS
Thus far, we have the following four Maxwell equations (in differential form):
Divergence and curl
of both and
specified
nature of and
is fully defined
E B
E B
G G
G G
( ) ( ) ( )
( )
( ) ( )
( ) ( )( )
0
0
Gauss' Law
no magnetic monopoles 0 no magnetic charges
Faraday's Law
r Ampere's Law
ToT
ToT
E r r
B r
B r E r t
B J r
∇ × = −
∇ × =
G G G G
i
G G G
i
G G
G G G
G G G G G
However, there is a problem with this set of equations…
Recall that ∇ ( ∇ ( r ) × F ( r )) = 0
G G G G G
i always for an arbitrary vector field, F (^) ( r )
G G
Apply this to Faraday’s Law:
B
E B
t t
∇ ∇ × = ∇ ⎜ − ⎟= − ∇ =
⎝ ∂^ ⎠ ∂
G
G G G G G G
i i i OK
Apply this to Ampere’s Law:
∇ ( ∇ × B ) = ∇ ( μ 0 JToT ) = μ 0 ( ∇ JToT )
G G G G G G G
i i i
For steady total currents: ( ∇ JToT ) = 0
G G
i because 0
ToT
t
However, for time-varying situations the continuity equation (total charge conservation)
ToT J (^) ToT t
G G
i BIG PROBLEM!!!
Let us investigate Ampere’s Law (in integral form) for the case of a parallel-plate capacitor:
( ) 0 0 enclosed
S C S
∇ × B da = B d = μ J da =μ I
G G G G G G G
i i A i
v
∫ C B d^ =^ μ^0 I enclosed
G G
i A
v
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2
Suppose we have an electric circuit consisting of sine wave function generator that
supplies/generates a time dependent voltage (^) ( ) (^0)
i t V t V e
ω = and a parallel-plate capacitor:
Complex impedance of a capacitor:
C
Z
i ω c
= (^) ω = 2 π f i ≡ − 1
i i
i i
Complex form of Ohm’s law: V^ ( ) t^ =^ I^ ( ) t Z (^) C n.b. V (^) ( t (^) ), I (^) ( t (^) )and ZC are complex quantities.
For contour loop shown in above figure: (^0) encl 0 c
B d = μ I =μ I
∫
G G
i A v
Get: (^) ( )
0
2
I
B
μ ρ πρ
= as usual – so no problem with this…
What about the following contour loop:
∫ (^) c B d^ =^ μ^0 Iencl =^0
G G
i A v
Cain’t be true!!!
We’re missing something! Energy flows
across the gap between plates of parallel-plate
capacitor… - virtual photons associated with
electric field of ||-plate capacitor!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 4
For the ||-plate capacitor, the electric field (^) ( )
( )
0
t E t
Where: σ (^) ( ) t (^) = Q t ( ) A where A = area of one plate
Thus: (^) ( )
( ) ( )
0 0
t Q t E t A
Thus:
( ) ( ) ( ) 0 0
E t (^) 1 Q t (^) 1 I t
t ε A t ε A
where (^) ( )
Q t ( ) I t t
Then: (^) ( ) (^0) ToT 0 D S C S S
∇ × B da = B d = μ J da +μ J da
∫ ∫ ∫ ∫
G G G G G G G G G
i i A i i v
Thus: (^0 0 )
encl ToT c s
E
B d I da t
∫ ∫
G
G G G
i A i v
For ||-plate capacitor circuit with contour C taken inside the gap of the ||-plate capacitor:
0
0
enclosed
c B d^^ μ^ IToT
=
∫
G G
i A v (^0 0) s
E
da t
∫
G
G
i
( )
0 0 2
in gap
B
G
0
ε A
I t ( ) * A (^) ( )
0
2
I t
⇐ Same answer!!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5
Thus the four “new” Maxwell equations are:
( ) ( ) 0
ToT
E r t ρ r t
G G G G
i (^) (Gauss’ Law)
∇ B r t ( , (^) )= 0
G G G
i (^) (no magnetic monopoles/magnetic charges)
( )
( , ) ,
B r t E r t t
∇ × = −
G G
G G G
(Faraday’s Law)
( ) ( ) ( )
( )
( )
0 0
0 0 0
ToT D
ToT
B r t J r t J r t
E r t J r t t
∇ × = +
G G G G G G G
G G
G (^) G (Ampere’s new Law)
Where: (^) ( )
( ) 0
D ,
E r t J r t t
G G
G G
(Maxwell’s displacement current)
Force Law: FToT ( r t , ) = FE ( r t , ) + Fm ( r t , ) = qE r t ( , ) + qv r t ( , ) × B r t ( , )
G G G G G G G G G G G G
Continuity equation: ( )
( , ) ,
toT ToT
r t J r t t
G
G G G
i (^) ⇐ can now be derived from Maxwell’s eqns!!
(charge conservation)
Note that if magnetic charges gm existed, then Maxwell’s equations would become more
“symmetrical”:
( ) ( ) 0
E
E r t ρ ToT r t
G G G G
i (^) (Gauss’ Law for electric charges)
( ,^ ) 0 ( , )
m
∇ B r t = μ ρ ToT r t
G G G G
i (^) (Gauss’ Law for magnetic charges)
( ) ( )
( ) 0
m ToT
B r t E r t J r t t
∇ × = − −
G G
G G G G
(Faraday’s Law)
( ) ( ) N
( )
2
0 0 0 1
E ToT
c
E r t B r t J r t t
≡
∇ × = +
G G
G G G G G
(Ampere’s Law)
( ,^ ) (^) ( ( ,^ ) ( ,^ ) ( , ))
E FToT r t = q E r t + v r t × B r t
G G G G G G G G
( ) ( ) 2 ( ) ( )
m ToT m F r t g B r t v r t E r t c
= − ×
G G G G G G G G
Please see/read Phy435 lecture notes #18 for more details about magnetic monopoles….
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7
Total electric charge density: (^) ( , (^) ) ( , (^) ) ( , (^) ) ( , (^) ) ( , )
E E E E
ρ ToT r t = ρ free r t + ρ bound r t = ρ free r t − ∇ Ρ r t
G G G G G G G
i
Total electric current density:
( ) ( ) ( ) ( )
( ) ( ) ( )
bound
E E m E ToT free bound P
E free
J r t J r t J r t J r t
J r t r t r t t
= + ∇ × Μ +
G G G G G G G G
G
G G G G G G
Then Gauss’ Law becomes:
( ) ( ) ( ( ) ( ))
( ) ( )
0 0
0 0
Tot free bound
E free
E r t r t r t r t
r t r t
ρ ρ ρ ε ε
ρ ε ε
G G G G G G
i
G G G
i
∇ D r t ( , (^) ) ≡ ε 0 ∇ E r t ( , (^) ) + ∇ Ρ (^) ( r t , (^) ) = ∇ (^) ( ε 0 E r t ( , (^) ) + Ρ( r t , ))
G G G G G G G G G G G G G G
i i i
∇ D r t ( , (^) ) = ρ free ( r t , )
G G G G
i (^) and ( ) ( ) ( ) 0 D r t , ≡ ε E r t , + Ρ r t ,
G G^ G G G
(New) Ampere’s Law (with Maxwell’s displacement current term)
∇ × B r t ( , ) = μ 0 J (^) ToT ( r t , ) +μ 0 J (^) D ( r t , )
G G G G G G G
( (^ )^ (^ )^ (^ ))
( ) 0 0 0
E m E free bound bound
E r t J r t J r t J r t t
μ μ ε
G G
G G G G G G
( ) ( )
( ) ( ) 0 0 0
E free
r t E r t J r t r t t t
= ⎜ + ∇ × Μ + ⎟+ ⎜ ⎟
G G G G
G G G G G
( (^ )^ (^ ))
( ) ( ) 0 0 0
E free
E r t r t J r t r t t t
= + ∇ × Μ + ⎜ + ⎟
G G G G
G G G G G
( (^ )^ (^ )) (^ )^ (^ )
( )
0 0 0
,
E free
D r t
J r t r t E r t r t t
≡
= + ∇ × Μ + ⎡^ + Ρ ⎤
G (^) G
G G G G G G G G G
( ) (^) ( ( ) ( ))
( ) 0 0
E free
D r t B r t J r t r t t
∇ × = − ∇ × Μ +
G G
G G G G G G G G
Then: (^) ( ) ( ) ( ) ( )
( ) 0 0
E free
D r t H r t B r t r t J r t t
μ μ
∇ × ≡ ⎢ ∇ × − ∇ × Μ ⎥ = +
G G
G G G G G G G G G G G
( ) ( )
( ) 0
E free
D r t H r t J r t t
∇ × = +
G G
G G G G G
and (^) ( ) ( ) ( ) 0
H r t , B r t , r t ,
G G G G G G
Faraday’s Law: (^) ( )
( , ) ,
B r t E r t t
∇ × = −
G G
G G G
and ∇ B r t ( , (^) )= 0
G G G
i (no magnetic monopoles)
are unaffected by separation of electric charge and electric current into free and bound parts.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 8
The four Maxwell’s Equations for free charges and free currents only:
( ) ( )
( )
( )
( )
, , (Gauss' Law)
, 0 (no magnetic charges)
free charges & currents only , , (Farad
D r t (^) free r t
B r t
B r t E r t t
∇ × = −
G G G G
i G G (^) G i G (^) G G G (^) G
( ) ( )
( )
ay's Law)
, , (Ampere's Law)
E free
D r t H r t J r t t
∇ × = +
G G
G G G G G
The four Maxwell equations for matter (i.e. dielectric and magnetic materials) are:
Gauss’ Law: ( ) ( ) ( ) ( )
0 0
E E E
E r t ρ ToT r t ρ free r t ρ bound r t
G G G G G G
i (^) ( , (^) ) ( , )
E
ρ bound r t ≡ −∇ Ρ r t
G G G G
i
Auxilliary Relation: D r t ( , (^) ) ≡ ε 0 E r t ( , (^) ) + Ρ( r t , )
G G G G G G
( , ) 0 ( , ) ( , ) ( , )
E
∇ D r t = ε∇ E r t + ∇ Ρ r t =ρ free r t
G G G G G G G G G G
i i i
No magnetic monopoles: ∇^ B r t ( ,^ )=^0
G G G
i
Faraday’s Law: ( )
( , ) ,
B r t E r t t
∇ × = −
G G
G G G
Ampere’s Law: ∇ ×^ B r t ( ,^^ ) =^ μ 0 ( J^ ToT ( r t ,^^ ) + J^ D ( r t , ))
G G G G G G G
0 (^ (^ ,^ )^ (^ ,^ )^ (^ ,^ )^ (^ , )) bound
E m
= μ J free r t + J bound r t + J P r t + J D r t
G G G G G G G G
( ) ( )
( ) ( ) 0 0
E free
r t E r t J r t r t t t
= ⎜ + ∇ × Μ + + ⎟
G G G G
G G G G G
With (^) ( )
( ) 0
D ,
E r t J r t t
G G
G G
and (^) ( , (^) ) ( ,)
m J (^) bound r t ≡ ∇ × Μ r t
G G G G G
and (^) ( )
( , ) , Pbound
r t J r t t
G G
G G
Auxilliary Relation: ( ) ( ) ( )
o
H r t B r t r t
G G G G G G
( ) ( )
( ) 0
, (^) free ,
D r t H r t J r t t
∇ × = ⎜ + ⎟
G G
G G G G G
For linear dielectric and/or magnetic media:
Ρ (^) ( r t , (^) ) = ε χ o eE r t ( , )
G G G G
Μ (^) ( r t , (^) ) = χ mH (^) ( r t , )
G G G G
ε = ε (^) o ( 1 + χ e ) Ke ≡ ε ε o = (^) ( 1 + χ e ) μ = μ (^) o ( 1 + χ m ) Km ≡ μ μ (^) o = (^) ( 1 +χ m )
D r t ( , (^) ) = ε E r t ( , )
G G G G
H (^) ( r t , (^) ) = B r t ( , ) μ
G G G G
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 10
- Apply the integral form of Gauss’ Law at a dielectric interface/boundary using infinitesimally
thin Gaussian pillbox extending slightly into dielectric material on either side of interface:
0 0 0 0 0
(^1) enclosed (^1) enclosed (^1) enclosed 1 1
S ToT^ free^ bound^ S free^ S bound
E da Q Q Q σ da σ da
∫ ∫ ∫
G G
i v v v
Gives: (^2 )
0 0 0
free bound ToT above below
E a E a σ a σ a σ a
G G G G
i i (at interface)
or: (^2 1) ( ) 0 0
ToT free bound above below
E E σ σ σ
⊥ ⊥ − = = + (at interface)
Here, the positive direction is from medium 2 (below) to medium 1 (above)
enclosed S free^ S free
D da = Q = σ da ∫ ∫
G G
i v v
⇒ (^2 1) free above below
D D σ
⊥ ⊥ − = (at interface)
Likewise:
enclosed bound bound S S
Ρ da = Q = − σ da ∫ ∫
G G
i v v ⇒ (^2 1) bound above below
P P σ
⊥ ⊥ − = (at interface)
Since: E ≡ −∇ V
G G
( )
2 1
interface^0
above below
ToT free bound
V V
n n
⎜ −^ ⎟ = −^ = −^ +
⎝ ∂^ ∂ ⎠
(at interface)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11
Since: D = ε E = − ∇ε V
G G G
2 1 2 1 interface
above below
free
V V
n n
⎜ −^ ⎟ = −
⎝ ∂^ ∂ ⎠
(at interface)
- Similarly, for 0 v S
∇ Bd τ ′= B da =
∫ ∫
JG G G
G
i i v (no magnetic monopoles), then at an interface:
above above B a − B a =
G G G G
i i ⇒ 2 1 0 above below
B B
⊥ ⊥ − = or: (^2 ) above below
B B
⊥ ⊥ = (at interface)
Since:
o
H B
G G G
Then: B = μ o ( H + Μ)
G G G
o^ (^ )^0 S S
B da = μ H + Μ da =
∫ ∫
G G G G G
i i v v or: S S
H da = − Μ da ∫ ∫
G G G G
i i v v
Then: (^2 1) ( 2 1 )
above below above below H a − H a = − Μ a − Μ a
G G G G G G G G
i i i i (at interface)
Or: (^2 1 2 )
bound magnetic above below above below
H H σ
⎜ −^ ⎟ = −^ ⎜ Μ^ − Μ^ ⎟= −
(at interface)
Effective bound magnetic charge at interface
- For Faraday’s Law: EMF, (^) ( ) m C S
d d E d B da dt dt
∫ ∫
G G G G
i A i v v at an interface / boundary
between two different media, taking a closed contour C of width l extending slightly
(i.e. infinitesimally) into the material on either side of interface, as shown below:
F ≡ (^) { Ε, D orΡ}
JG JG JG G
or: (^) { B , H or M }
JG JJG JJG
Side View:
above below S
d E E B da dt
∫
G G G G G G
iA iA i v
(in limit area of contour loop → 0, magnetic flux enclosed → 0)
Thus: 2 1 0
above below
E − E =
& & (at interface) or: (^2 ) above below
E = E
& & (at interface)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13
In the limit that the enclosing Amperian loop contour C (of width l) shrinks to zero height
above/below interface, causing area of enclosed loop contour → 0, then:
2 1
above below encl encl
B − B = μ o ITOT +μ o ID
G G^ G G
iA iA (^) ( )
0
encl
μ o I TOT KTOT n
=
= = ×
G G
iA
⇒ 2 1 ˆ^ ( ) ˆ
m o TOT o free bound above below
B − B = μ K × n = μ K + K × n
& &
G G G
(at interface)
Since:
o
H B
G G G
and:
o
B H
G G G
then:
( 2 1 ) ( 2 1 ) ( 2 1 ) ( ) ( )
above below above below above below free bound o
B B H H K n K n
− = − + Μ − Μ = ⎡^ × + × ⎤
G G^ G G^ G G^ G G^ G G^ G G G G
iA iA iA iA iA iA
(at interface)
We also see that: (^2 1) free ˆ
above below
H − H = K × n
& &
G
(at interface)
and: 2 1 ˆ
m bound above below
Μ − Μ = K × n
& &
G
(at interface)
& - components of B
G
are discontinuous at interface by μ o KTOT × n ˆ
G
& - components of H
G
are discontinuous at interface by K (^) free × n ˆ
G
& - components of Μ
G
are discontinuous at interface by ˆ
m Kbound × n
G
If B = ∇ × A
G G G
where A
G
is the magnetic vector potential, then:
2 1 0
TOT ˆ
above below
B B K n
⎜ ⎟ ⎢ −^ =^ ×
⎝ ⎠⎣^ ⎦
& &
G
(at interface) is equivalent to:
2 1
interface
above below
TOT o
A A
K
μ n n
⎛ ⎞ ⎛^ ∂ ∂ ⎞
⎜ ⎟ ⎜^ −^ ⎟ = −
⎝ ⎠ ⎝ ∂^ ∂ ⎠
G G
G
(at interface)
For linear magnetic media: B = μ H
G G
or:
H B
G G
Then: (^2 1) free ˆ
above below
H H K n
− = ×
& &
G
(at interface) is equivalent to:
2 1
2 interface^1 interface
above below
free
above below
A A
K
μ n μ n
⎜ ⎟ ∂^ ⎜ ⎟∂
G G
G
(at interface)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 14
Summary of Maxwell’s Equations In Differential and Integral Forms
(Suppress Explicit (^) ( r t , )
G
Dependence)
Gauss’ Law: (^) ( )
TOT free bound o o
E ρ ρ ρ
JG G
i bound
JG G
i
Use of the auxiliary relation: D^ =^ ε oE + Ρ
G G G
Yields: ∇^ D^ =^ ε o ∇^ E + ∇ Ρ =^ ρ free
JG G JG G JG G
i i i
No magnetic monopoles: ∇^ B =^0
JG G
i
Faraday’s Law:^
B
E
t
∇ × = −
G
JG G
Ampere’s Law: ∇ × B = μ o ( J (^) TOT + JD )
JG G G G
bound
m J (^) TOT = J (^) free + J (^) bound + J Ρ
G G G G
Use of auxiliary relation:
o
H B
G G G
with (^) D o
E
J
t
G
G
Yields: (^) o free
D
H J
t
∇ × = +
G
JG G G
Gauss’ Law:
(^1) encl (^1) encl encl
v S tot^ free^ bound o o
E d τ E da Q Q Q
Ε
∫ ∫ ⎣ ⎦
G G G G
i i v (Electric Flux)
Use of the auxiliary relation: D^ =^ ε oE + Ρ
G G G
Yields:
encl
v Dd^^ τ^ SD da^ Qfree^ D
∫ ∫
JG G G G
i i v (Electric Displacement Flux)
encl
v d^^ τ^ S da^ Qbound Ρ
∫ ∫
JG G G G
i i v (Electric Polarization Flux)
No magnetic monopoles: 0 v S
∇ Bd τ ′= B da =
∫ ∫
JG G G G
i i v
Faraday’s Law: EMF
encl m S C S
d d E da E d B da dt dt
= ∇ × = = − = −
∫ ∫ ∫
JG G G G G G G
i i A i v
Ampere’s Law:
( ) ( )
( ) ( ) bound
o TOT D S C S encl encl encl encl encl encl o TOT D o free bound D
B da B d J J da
I I I I I I
∫ ∫ ∫
JG G G G G G
G G
i i i A i v
Use of auxiliary relation(s):
o
H B
G G G
and D = ε oE + Ρ
G G^ G
Yields: (^) ( )
encl free S C S
d H da H d I D da dt
∇ × = = + ⎡^ ⎤
∫ ∫ (^) ⎣ ∫ ⎦
JG G G G G
G G
i i A i v
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 16
BC’s Specific to Linear Homogeneous Isotropic Dielectric and/or Magnetic Media:
Ρ = ε χ o e E
G G
Μ = χ m H
G G
D = ε E = ε oE + Ρ
G G G^ G
ε = ε (^) o ( 1 + χ e ) μ = μ (^) o ( 1 + χ m ) B = μ H
G G
or
H B
G G
e^1 e o
K
= = + (^) m ( (^1) m ) o
K
o
H B
G G G
The boundary conditions at the interface between linear dielectric or magnetic media become:
Gauss’ Law: 2 1 ( )
TOT free bound above below (^) o o
E E σ σ σ
⇒ ( )
2 1
interface
above below
TOT free bound o o
V V
n n
⎜ −^ ⎟ = −^ = −^ +
⎝ ∂^ ∂ ⎠
2 1 free above below
D D σ
⇒ (^2 2 1 1) free above below
ε E ε E σ
2 1 bound above below
⎜ Ρ^ − Ρ^ ⎟= −
o e 2 (^) 2 e 1 1 bound above below
ε χ E χ E σ
⎜ −^ ⎟= −
2 1 2 above 1 below interface interface
free
V V
n n
No magnetic monopoles: 2 1 0 above below
B B
bound magnetic above below above below
H H M M σ
above below
μ H μ H
2 1 2 1
above below
B B
⊥ ⊥
Faraday’s Law: 2 1 0
above below
E E
⎜ −^ ⎟=
& & ⇒ (^2 1 2 ) above below above below
D D
⎜ −^ ⎟ =^ ⎜ Ρ^ − Ρ ⎟
& & & &
2 1 2 1
above below
D D
& & ⇒ 2 2 1 1 2 2 1 1 above below
o e e above below
ε E ε E ε χ E χ E
⎜ −^ ⎟ = −^ ⎜ − ⎟
& & & &
Ampere’s Law:
2 1 ˆ^ (^ ) ˆ
m o TOT o free bound above below
B − B = μ K × n = μ K + K × n
& &
G G G
2 1
interface
above below
TOT o
A A
K
μ n n
⎛ ⎞ ⎛^ ⎞
⎜ ⎟ ⎜^ −^ ⎟ = −
JG JG
G
2 2 1 1 above below
μ H μ H
& & using B^ =^ ∇ × A
G JG G
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17
2 1
2 1 2 1
free above below
above below above below
H H K n
B B
− = ×
& &
& &
G
and 2 1
2 1
2 1
2 1
2 1 2 1
above below
above below
m bound above below
m m above below
m m
above below above below
M M K n
H H
B B
⎜ −^ ⎟=^ ×
& &
& &
& &
G
2 1 above below 2 interface^1 interface
free
A A
K
μ n μ n
⎜ ⎟ −^ ⎜ ⎟ = −
⎝ ⎠ ∂^ ⎝ ⎠∂
G G
G
using B^ = ∇ × A
G JG G
