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Maxwell’s Equations in Matter

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k

M= sin(ay) k

j

i

jM= curl M = a cos(ay) i

Types of Current j

Total current

Free current density from unbound conduction electrons (metals)

Polarisationcurrent density from oscillation of charges as electric dipoles

Magnetisation current density from space/time variation of magnetic dipoles

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

in vacuum in matter

.E = r /eo.D = rfreePoisson’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 = eoe E = eo(1+ c)EConstitutive relation for D

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

Solve with:

model e for insulating, isotropic matter, m= 1,rfree= 0,jfree= 0

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

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 /moD = eoeE

x B = moeoe∂E/∂t

x ∂B/∂t = moeoe∂2E/∂t2

x (- x E) = x ∂B/∂t = moeoe∂2E/∂t2

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

2E - moeoe∂2E/∂t2 = 0

- 2E - moeoe∂2E/∂t2 = 0 E(r, t) = Eoex Re{ei(k.r-wt)}
- 2E = -k2E moeoe∂2E/∂t2 = - moeoe w2E
- (-k2+moeoe w2)E = 0
- w2 = k2/(moeoe)moeoew2 = k2 k = ± w√(moeoe) k = ± √ew/c
- Let e = e1+ie2be the real and imaginary parts of e and e = (n+ik)2
- We need √e = n+ik
e = (n+ik)2 = n2- k2 +i 2nke1= n2- k2 e2= 2nk

- E(r, t) = Eoex Re{ ei(k.r- wt) } = Eoex Re{ei(kz- wt)} k || ez
- = Eoex Re{ei((n + ik)wz/c - wt)}= Eoex Re{ei(nwz/c -wt)e- kwz/c)}
Attenuated wave with phase velocity vp = c/n

Solution 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 /moD = eoeE

x B = mojfree+ moeoe∂E/∂t

x ∂B/∂t = mos∂E/∂t + moeoe∂2E/∂t2

x (- x E) = x ∂B/∂t = mos∂E/∂t+moeoe∂2E/∂t2

-(.E) + 2E = mos∂E/∂t+moeoe∂2E/∂t2 . e E= e . E = 0 since rfree= 0

2E - mos∂E/∂t - moeoe∂2E/∂t2 = 0

2E - mos∂E/∂t - moeoe∂2E/∂t2 = 0 E(r, t) = EoexRe{ei(k.r- wt)} k|| ez

2E = -k2E mos∂E/∂t= mosiwEmoeoe∂2E/∂t2 = - moeoe w2E

(-k2-mosiw+moeoew2 )E = 0s >> eoe w for a good conductor

E(r, t) = Eoex 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