maxwell’s equations in vacuum

Maxwell’s Equations in Vacuum

(1) .E = r /eo Poisson’s Equation

(2) .B = 0 No magnetic monopoles

(3) x E = -∂B/∂t Faraday’s Law

(4) x B = moj + moeo∂E/∂t Maxwell’s Displacement

Electric Field E Vm-1

Magnetic Induction B Tesla

Charge density r Cm-3

Current Density Cm-2s-1

Ohmic Conduction j = s E Electric Conductivity Siemens (Mho)

Constitutive Relations

(1) D = eoE + P Electric Displacement D Cm-2 and Polarization P Cm-2

(2) P = eo c E Electric Susceptibility c

(3) e =1+ c Relative Permittivity (dielectric function) e

(4) D = eo e E

(5) H = B / mo – M Magnetic Field H Am-1 and Magnetization M Am-1

(6) M = cB B / mo Magnetic Susceptibility cB

(7) m =1 / (1- cB) Relative Permeability m

(8) H = B / m mo m ~ 1 (non-magnetic materials), e ~ 1 - 50

Electric Polarisation• Apply Gauss’ Law to right and left ends of polarised dielectric

• EDep = ‘Depolarising field’

• Macroscopic electric field EMac= E + EDep = E - P/eo

s± surface charge density Cm-2

s± = P.n n outward normalE+2dA = s+dA/eo Gauss’ LawE+ = s+/2eo

E- = s-/2eo

EDep = E+ + E- = (s++ s-)/2eo EDep = -P/eo P = s+ = s-

s-

E

P s+

E+E-

Electric PolarisationDefine dimensionless dielectric susceptibility c through

P = eo c EMac

EMac = E – P/eo

eo E = eo EMac + Peo E = eo EMac + eo c EMac = eo (1 + c)EMac = eoeEMac

Define dielectric constant (relative permittivity) e = 1 + c

EMac = E /e E = e EMac

Typical values for e: silicon 11.8, diamond 5.6, vacuum 1Metal: e →∞Insulator: e∞ (electronic part) small, ~5, lattice part up to 20

Electric PolarisationRewrite EMac = E – P/eo as

eoEMac + P = eoE

LHS contains only fields inside matter, RHS fields outside

Displacement field, D

D = eoEMac + P = eoe EMac = eoE

Displacement field defined in terms of EMac (inside matter,relative permittivity e) and E (in vacuum, relative permittivity 1). Define

D = eoe E

where e is the relative permittivity and E is the electric field

Gauss’ Law in Matter• Uniform polarisation → induced surface charges only

• Non-uniform polarisation → induced bulk charges also

Displacements of positive charges Accumulated charges

+ +- -

P- + E

Gauss’ Law in MatterPolarisation charge density

Charge entering xz face at y = 0: Py=0DxDz Cm-2 m2 = CCharge leaving xz face at y = Dy: Py=DyDxDz = (Py=0 + ∂Py/∂y Dy) DxDz

Net charge entering cube via xz faces: (Py=0 - Py=Dy ) DxDz = -∂Py/∂y DxDyDz

Charge entering cube via all faces:

-(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z) DxDyDz = Qpol

rpol = lim (DxDyDz)→0 Qpol /(DxDyDz)

-. P = rpol

Dx

Dz

Dy

z

y

x

Py=DyPy=0

Gauss’ Law in MatterDifferentiate -.P = rpol wrt time.∂P/∂t + ∂rpol/∂t = 0Compare to continuity equation .j + ∂r/∂t = 0

∂P/∂t = jpol

Rate of change of polarisation is the polarisation-current density

Suppose that charges in matter can be divided into ‘bound’ orpolarisation and ‘free’ or conduction charges

rtot = rpol + rfree

Gauss’ Law in MatterInside matter.E = .Emac = rtot/eo = (rpol + rfree)/eo

Total (averaged) electric field is the macroscopic field-.P = rpol

.(eoE + P) = rfree

.D = rfree

Introduction of the displacement field, D, allows us to eliminatepolarisation charges from any calculation. This is a form of Gauss’ Law suitable for application in matter.

Ampère’s Law in Matter

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A steady current implies constant charge density, so Ampère’s law is consistent with the continuity equation for steady currents

Ampère’s law is inconsistent with the continuity equation (conservation of charge) when the charge density is time dependent

Continuity equation

Ampère’s Law in MatterAdd term to LHS such that taking Div makes LHS also identically equal to zero:

The extra term is in the bracket

extended Ampère’s Law

Displacement current (vacuum)

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• Polarisation current density from oscillation of charges in electric dipoles• Magnetisation current density variation in magnitude of magnetic dipoles

PMf jjjj ++

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jM = curl M = a cos(ay) i

Total current

MjM x


∂D/∂t is the displacement current postulated by Maxwell (1862)

In vacuum D = eoE ∂D/∂t = eo ∂E/∂t

In matter D = eo e E ∂D/∂t = eo e ∂E/∂t

Displacement current exists throughout space in a changing electric field

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Maxwell’s Equations

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

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

Divergence Theorem 2-D 3-D• From Green’s Theorem

• In words - Integral of A.n dA over surface contour equals integral of div A over surface area

• In 3-D • Integral of A.n dA over bounding surface S equals integral of div V dV

within volume enclosed by surface S• The area element n dA is conveniently written as dS

dV .dA .V

S AnA

A.n dA .A dV

Differential form of Gauss’ Law• Integral form

• Divergence theorem applied to field V, volume v bounded by surface S

• Divergence theorem applied to electric field E

V

SS

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Differential form of Gauss’ Law(Poisson’s Equation)

Stokes’ Theorem 3-D• In words - Integral of ( x A).n dA over surface S equals integral of A.dr

over bounding contour C• It doesn’t matter which surface (blue or hatched). Direction of dr

determined by right hand rule.

( x a) .ndA

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dA

local value of x A

local value of A

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Integral form of law: enclosed current is integral dS of current density j

Apply Stokes’ theorem

Integration surface is arbitrary

Must be true point wise

Differential form of Ampère’s Law

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Maxwell’s Equations in VacuumTake curl of Faraday’s Law

(3) x ( x E) = -∂ ( x B)/∂t

= -∂ (moeo∂E/∂t)/∂t

= -moeo∂2E/∂t2

x ( x E) = (.E) - 2E vector identity

- 2E = -moeo∂2E/∂t2 (.E = 0)

2E -moeo∂2E/∂t2 = 0 Vector wave equation

Maxwell’s Equations in VacuumPlane wave solution to wave equation

E(r, t) = Re {Eo ei(wt - k.r)} Eo constant vector

2E =(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2)E = -k2E

.E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z = -ik.E = -ik.Eo ei(wt - k.r)

If Eo || k then .E ≠ 0 and x E = 0

If Eo ┴ k then .E = 0 and x E ≠ 0

For light Eo ┴ k and E(r, t) is a transverse wave

2k i.e.

r 2r k

r k .

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Consecutive wave fronts

Plane waves travel parallel to wave vector k

Plane waves have wavelength 2 /k


Eo

Maxwell’s Equations in VacuumPlane wave solution to wave equation

E(r, t) = Eo ei(wt - k.r) Eo constant vector

moeo∂2E/∂t2 = - moeow2E moeow2 = k2

w =±k/(moeo)1/2 = ±ck w/k = c = (moeo)-1/2 phase velocity

w = ±ck Linear dispersion relationship

w(k)

k

Maxwell’s Equations in Vacuum Magnetic component of the electromagnetic wave in vacuum

From Maxwell-Ampère and Faraday laws

x ( x B) = moeo ∂( x E)/∂t

= moeo ∂(-∂B/∂t)/∂t

= -moeo∂2B/∂t2

x ( x B) = (.B) - 2B

- 2B = -moeo∂2B/∂t2 (.B = 0)

2B -moeo∂2B/∂t2 = 0 Same vector wave equation as for E


If E(r, t) = Eo ex ei(wt - k.r) and k || ez and E || ex (ex, ey, ez unit vectors)

x E = -ik Eo ey ei(wt - k.r) = -∂B/∂t From Faraday’s Law

∂B/∂t = ik Eo ey ei(wt - k.r)

B = (k/w) Eo ey ei(wt - k.r) = (1/c) Eo ey e

i(wt - k.r)

For this wave E || ex, B || ey, k || ez, cBo = Eo

More generally

-∂B/∂t = -iw B = x E

x E = -i k x E

-iw B = -i k x E

B = ek x E / c

ex

ey

ez

Maxwell’s Equations in Matter

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

EM wave in a dielectric with frequency at which dielectric is transparentMaxwell’s equations become

x E = -∂B/∂t

x H = ∂D/∂t H = B /mo D = eoe E

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 Matter2E - moeoe ∂2E/∂t2 = 0 E(r, t) = Eo ex Re{ei(wt - k.r)}

2E = -k2E moeoe ∂2E/∂t2 = - moeoe w2E

(-k2 +moeoe 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(wt - k.r) } = Eo ex Re{ei(wt - kz)} k || ez

= Eo ex Re{ei(wt - (n - ik)wz/c)} = Eo ex Re{ei(wt - nwz/c)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

EM wave in a metal with frequency at which metal is absorbing/reflecting

Maxwell’s equations become

x E = -∂B/∂t

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

x B = mo jfree + 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 Matter2E - mos ∂E/∂t - moeoe ∂2E/∂t2 = 0 E(r, t) = Eo ex Re{ei(wt - k.r)} k || ez

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

(-k2 -mos iw +moeoe w2 )E = 0 s >> eoe w for a good conductor

E(r, t) = Eo ex Re{ ei(wt - √(wsmo/2)z)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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Bound and Free Charges

Bound chargesAll valence electrons in insulators (materials with a ‘band gap’)Bound valence electrons in metals or semiconductors (band gap absent/small )

Free chargesConduction electrons in metals or semiconductors

Mion k melectron k MionSi ionBound electron pair

Resonance frequency wo ~ (k/M)1/2 or ~ (k/m)1/2 Ions: heavy, resonance in infra-red ~1013HzBound electrons: light, resonance in visible ~1015HzFree electrons: no restoring force, no resonance


Bound chargesResonance model for uncoupled electron pairs

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Energy in Electromagnetic WavesRate of doing work on a moving charge

W = d/dt(F.dr) F = qE + qv x B

= d/dt{(q E + q v x B) . dr} = d/dt(q E . dr) = qv . E =

j(r) = q v d(r - r’)

W = fields do work on currents in integration volume

Eliminate j using modified Ampère’s law

x H = jfree + ∂D/∂t

W =

Energy in Electromagnetic WavesVector identity .(A x B) = B . () - A . () W = becomes

W =

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Local energy density

Poynting vector

Energy in Electromagnetic WavesEnergy density in plane electromagnetic waves in vacuum

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maxwell’s equations in vacuum

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