Maxwell’s Equations in Matter

Free current density from unbound conduction electrons (metals)Polarisation current density from oscillation of charges as electric dipolesMagnetisation current density from space/time variation of magnetic dipoles

PMf jjjj

tP

jP

tooo E

jB

M = sin(ay) k

k

i

j

jM = curl M = a cos(ay) i

Total current

MjM x

Types of Current j

Maxwell’s Equations in Matter

D/t is displacement current postulated by Maxwell (1862) to exist in the gap of a charging capacitor

In vacuum D = eoE and displacement current exists throughout space

tt

tt

t

1t

ff

f

PMf

DjHPEjM

B

EPMj

EjjjB

EjB

oo

o

oo

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Maxwell’s Equations in Matter in vacuum in matter

.E = r /eo .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 = moj + moeo∂E/∂t x H = jfree + ∂D/∂t Maxwell’s Displacement

D = eo e E = eo(1+ c)E Constitutive relation for D

H = B/(mom) = (1- cB)B/mo Constitutive relation for H

Solve with: model e for insulating, isotropic matter, m = 1,rfree = 0, jfree = 0model e for conducting, isotropic matter, m = 1,rfree = 0, jfree = s(w)E

Maxwell’s Equations in Matter

Solution of Maxwell’s equations in matter for m = 1, rfree = 0, jfree = 0

Maxwell’s equations become

x E = -∂B/∂t

x H = ∂D/∂t H = B /mo D = eo e E

x B = moeo e ∂E/∂t

x ∂B/∂t = moeo e ∂2E/∂t2

x (- x E) = x ∂B/∂t = moeo e ∂2E/∂t2

-(.E) + 2E = moeo e ∂2E/∂t2 . e E = e . E = 0 since rfree = 0

2E - moeo e ∂2E/∂t2 = 0

Maxwell’s Equations in Matter

2E - moeo e ∂2E/∂t2 = 0 E(r, t) = Eo ex Re{ei(k.r - wt)}

2E = -k2E moeo e ∂2E/∂t2 = - moeo e w2E

(-k2 +moeo e w2)E = 0

w2 = k2/(moeoe) moeoe w2 = k2 k = ± w√(moeoe) k = ± √e w/c

Let e = e1 + ie2 be the real and imaginary parts of e and e = (n + ik)2

We need √ e = n + ik

e = (n + ik)2 = n2 - k2 + i 2nk e1 = n2 - k2 e2 = 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 MatterSolution of Maxwell’s equations in matter for m = 1, rfree = 0, jfree = s(w)E

Maxwell’s equations become

x E = -∂B/∂t

x H = jfree + ∂D/∂t H = B /mo D = eo e E

x B = mo jfree + moeo e ∂E/∂t

x ∂B/∂t = mo s ∂E/∂t + moeo e ∂2E/∂t2

x (- x E) = x ∂B/∂t = mo s ∂E/∂t + moeo e ∂2E/∂t2

-(.E) + 2E = mo s ∂E/∂t + moeo e ∂2E/∂t2 . e E = e . E = 0 since rfree = 0

2E - mo s ∂E/∂t - moeo e ∂2E/∂t2 = 0

Maxwell’s Equations in Matter

2E - mo s ∂E/∂t - moeo e ∂2E/∂t2 = 0 E(r, t) = Eo ex Re{ei(k.r - wt)} k || ez

2E = -k2E mo s ∂E/∂t = mo s iw E moeo e ∂2E/∂t2 = - moeo e w2E

(-k2 -mo s iw +moeo e w2 )E = 0 s >> eo e w for a good conductor

E(r, t) = Eo ex Re{ ei(√(wsmo/2)z - wt)e-√(wsmo/2)z}

NB wave travels in +z direction and is attenuated

The skin depth d = √(2/wsmo) 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

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