Maxwell’s Equations in Matter
Types of Current j
  B  o j   o  o
jP 
E
t
j  jf  jM  jP
Total current
P
t
k
jM   x M
M = sin(ay) k
j
i
jM = curl M = a cos(ay) i
Free current density from unbound conduction electrons (metals)
Polarisation current density from oscillation of charges as electric dipoles
Magnetisation current density from space/time variation of magnetic dipoles
Maxwell’s Equations in Matter
  B  o j   o  o

1
o
E
t
  B  jf  jM  jP   o
 jf    M 
E
t
P
E
 o
t
t
B


D
    M   jf   oE  P     H  jf 
t
t
 o

D/t is displacement current postulated by Maxwell (1862) to exist in the
gap of a charging capacitor
In vacuum D = oE and displacement current exists throughout space
Maxwell’s Equations in Matter
in vacuum
in matter
.E = r /o
.D = rfree
Poisson’s Equation
.B = 0
.B = 0
No magnetic monopoles
 x E = -∂B/∂t
 x E = -∂B/∂t
Faraday’s Law
 x B = oj + oo∂E/∂t  x H = jfree + ∂D/∂t
Maxwell’s Displacement
D = o E = o(1+ c)E
Constitutive relation for D
H = B/(o) = (1- cB)B/o
Constitutive relation for H
Solve with:
model  for insulating, isotropic matter,  = 1,rfree = 0, jfree = 0
model  for conducting, isotropic matter,  = 1,rfree = 0, jfree = s(w)E
Maxwell’s Equations in Matter
Solution of Maxwell’s equations in matter for  = 1, rfree = 0, jfree = 0
Maxwell’s equations become
 x E = -∂B/∂t
 x H = ∂D/∂t
H = B /o
D = o E
 x B = oo ∂E/∂t
 x ∂B/∂t = oo ∂2E/∂t2
 x (- x E) =  x ∂B/∂t = oo ∂2E/∂t2
-(.E) + 2E = oo ∂2E/∂t2
2E - oo ∂2E/∂t2 = 0
.  E =  . E = 0 since rfree = 0
Maxwell’s Equations in Matter
2E - oo ∂2E/∂t2 = 0 E(r, t) = Eo ex Re{ei(k.r - wt)}
2E = -k2E
oo ∂2E/∂t2 = - oo w2E
(-k2 +oo w2)E = 0
w2 = k2/(oo)
oo w2 = k2
k = ± w√(oo)
k = ± √ w/c
Let  = 1 + i2 be the real and imaginary parts of  and  = (n + ik)2
We need √ = n + ik
 = (n + ik)2 = n2 - k2 + i 2nk
1 = n2 - k2 2 = 2nk
E(r, t) = Eo ex Re{ ei(k.r - wt) } = Eo ex Re{ei(kz - wt)}
k || ez
= Eo ex Re{ei((n + ik)wz/c - wt)} = Eo ex Re{ei(nwz/c - wt)e- kwz/c)}
Attenuated wave with phase velocity vp = c/n
Maxwell’s Equations in Matter
Solution of Maxwell’s equations in matter for  = 1, rfree = 0, jfree = s(w)E
Maxwell’s equations become
 x E = -∂B/∂t
 x H = jfree + ∂D/∂t
H = B /o
D = o E
 x B = o jfree + oo ∂E/∂t
 x ∂B/∂t = os ∂E/∂t + oo ∂2E/∂t2
 x (- x E) =  x ∂B/∂t = os ∂E/∂t + oo ∂2E/∂t2
-(.E) + 2E = os ∂E/∂t + oo ∂2E/∂t2
2E - os ∂E/∂t - oo ∂2E/∂t2 = 0
.  E =  . E = 0 since rfree = 0
Maxwell’s Equations in Matter
2E - os ∂E/∂t - oo ∂2E/∂t2 = 0
2E = -k2E
E(r, t) = Eo ex Re{ei(k.r - wt)}
os ∂E/∂t = os iw E
(-k2 -os iw +oo w2 )E = 0
k || ez
oo ∂2E/∂t2 = - oo w2E
s >> o w for a good conductor
1
(1 i ) ws
2
E(r, t) = Eo ex Re{ ei(√(wso/2)z - wt)e-√(wso/2)z}
k 2  iwso k   i ws 
NB wave travels in +z direction and is attenuated
The skin depth d = √(2/wso) is the thickness over which incident radiation is
attenuated. For example, Cu metal DC conductivity is 5.7 x 107 (Wm)-1
At 50 Hz d = 9 mm and at 10 kHz d = 0.7 mm