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# Maxwell’s Equations in Matter PowerPoint PPT Presentation

k. M = sin(ay) k. j. i. j M = curl M = a cos(ay) i. Maxwell’s Equations in Matter. Types of Current j. Total current. Free current density from unbound conduction electrons (metals) Polarisation current density from oscillation of charges as electric dipoles

Maxwell’s Equations in Matter

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k

M= sin(ay) k

j

i

jM= curl M = a cos(ay) i

### Maxwell’s Equations in Matter

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

### 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

### Maxwell’s Equations in Matter

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

### 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 /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

### Maxwell’s Equations in Matter

• 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

### Maxwell’s Equations in Matter

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

### Maxwell’s Equations in Matter

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