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Maxwell's Equations

Module by: a.i. trivedi. E-mail the author

Summary: An introduction to time varying fields

Lecture 1: Electromagnetic waves and radiating systems

Maxwells’ Equations

Static field laws

are summarized by

∇×E=0∮E.ds=0∇×E=0∮E.ds=0 size 12{ nabla times E=0~ lInt {E "." ital "ds"} `=0} {} Irrotational property of static E field, potential depends on the point and not on path…..

∇.D=ρ∮D.da=∫ρdV∇.D=ρ∮D.da=∫ρdV size 12{ nabla "." D=ρ~ lInt {D "." ital "da"= Int {ρ ital "dV"} } } {}charge is a source or sink for E , gausses’ law

∇×H=J∮H.ds=∫J.da∇×H=J∮H.ds=∫J.da size 12{ nabla times H=J~ lInt {H "." ital "ds"} `= Int {J "." ital "da"} } {} ampere’s circuital law

∇.B=0∮B.da=0∇.B=0∮B.da=0 size 12{ nabla "." B=0~ lInt {B "." ital "da"=0} } {} Continuity of magnetic field lines, no magnetic monopoles

∇.J=0∮J.da=0∇.J=0∮J.da=0 size 12{ nabla "." J=0~ lInt {J "." ital "da"=0} } {} equation of continuity for steady conduction currents

Time varying fields:

If magnetic field is time varying, faradays law invokes the electric field as a result.

∮ E . ds = − dΦ dt = − d dt ∮ S B . da = − ∫ S ∂ B ∂ t . da ∮ E . ds = − dΦ dt = − d dt ∮ S B . da = − ∫ S ∂ B ∂ t . da size 12{ lInt {E "." ital "ds"} `= - { {dΦ} over { ital "dt"} } = - { {d} over { ital "dt"} } lInt cSub { size 8{S} } {B "." ital "da"={}} - Int cSub { size 8{S} } { { { partial B} over { partial t} } "." ital "da"} } {}
(1)
∇ × E = − ∂ B ∂ t ∮ E . ds = − ∫ S ∂ B ∂ t . da ∇ × E = − ∂ B ∂ t ∮ E . ds = − ∫ S ∂ B ∂ t . da size 12{ nabla times E= - { { partial B} over { partial t} } ~ lInt {E "." ital "ds"} `= - Int cSub { size 8{S} } { { { partial B} over { partial t} } "." ital "da"} } {}
(2)

The Equation of continuity for time varying case

– needs to account for flows through capacitors and dielectrics – basis of conservation of charge

∮ J . da = − d dt ∫ ρ dV = − ∫ ∂ ρ ∂ t dV ∮ J . da = − d dt ∫ ρ dV = − ∫ ∂ ρ ∂ t dV size 12{ lInt {J "." ital "da"= - { {d} over { ital "dt"} } Int {ρ ital "dV"= - Int { { { partial ρ} over { partial t} } ital "dV"} } } } {}
(3)
∫ ∇ . JdV = − ∫ ∂ ρ ∂ t dV ∫ ∇ . JdV = − ∫ ∂ ρ ∂ t dV size 12{ Int { nabla "." ital "JdV"= - Int { { { partial ρ} over { partial t} } } } ital "dV"} {}
(4)
∇ . J = − ∂ ρ ∂ t = − ∂ ∂ t ∇ . D ∇ . J = − ∂ ρ ∂ t = − ∂ ∂ t ∇ . D size 12{ nabla "." J= - { { partial ρ} over { partial t} } = - { { partial } over { partial t} } nabla "." D} {}
(5)

After Substituting Gausses’ law

∇ . ∂ D ∂ t + J = 0 ∇ . ∂ D ∂ t + J = 0 size 12{ nabla "." left ( { { partial D} over { partial t} } +J right )=0} {}
(6)
∮ ∂ D ∂ t + J . da = 0 ∮ ∂ D ∂ t + J . da = 0 size 12{ lInt { left ( { { partial D} over { partial t} } +J right ) "." ital "da"=0} } {}
(7)
∇ × H = ∂ D ∂ t + J ∇ × H = ∂ D ∂ t + J size 12{ nabla times H= { { partial D} over { partial t} } +J} {}
(8)
∮ H . ds = ∫ ∂ D ∂ t + J . da ∮ H . ds = ∫ ∂ D ∂ t + J . da size 12{ lInt {H "." ital "ds"} `= Int { left ( { { partial D} over { partial t} } +J right ) "." ital "da"} } {}
(9)

Total current density = conduction + displacement current density

Maxwell’s equations:

Show the symmetry between electric and magnetic fields particularly for charge free nonconducting regions like free space. Led Maxwell to hypothesize that if changing magnetic field could create electric field, then changing electric field should create a magnetic field.

∇ × H = D . + J ∮ H . ds = ∫ D . + J . da ∇ × H = D . + J ∮ H . ds = ∫ D . + J . da size 12{ nabla times H= {D} cSup { size 8{ "." } } +J~ lInt {H "." ital "ds"} `= Int { left ( {D} cSup { size 8{ "." } } +J right ) "." ital "da"} } {}
(10)
∇ × E = − B . ∮ E . ds = − ∫ S B . . . da ∇ × E = − B . ∮ E . ds = − ∫ S B . . . da size 12{ nabla times E= - {B} cSup { size 8{ "." } } ~ lInt {E "." ital "ds"} `= - Int cSub { size 8{S} } { {B} cSup { size 8{ "." } } "." "." ital "da"} } {}
(11)
∇ . D = ρ ∮ D . da = ∫ ρ dV ∇ . D = ρ ∮ D . da = ∫ ρ dV size 12{ nabla "." D=ρ~ lInt {D "." ital "da"= Int {ρ ital "dV"} } } {}
(12)
∇ . B = 0 ∮ B . da = 0 ∇ . B = 0 ∮ B . da = 0 size 12{ nabla "." B=0~ lInt {B "." ital "da"=0} } {}
(13)
∇ . J = ρ . ∮ J . da = − ∫ ρ . dV ∇ . J = ρ . ∮ J . da = − ∫ ρ . dV size 12{ nabla "." J= {ρ} cSup { size 8{ "." } } ~ lInt {J "." ital "da"= - Int { {ρ} cSup { size 8{ "." } } ital "dV"} } } {}
(14)

Word Statements:

EMF is an electrical voltage, similarly mmf can be considered as a magnetic voltage.

Derivative of displacement density is an electric current, so derivative of magnetic displacement B can be considered as magnetic current.

The Equations can then be interpreted as:

  1. The magnetic voltage around a closed path is equal to the electric current through the path
  2. The electric voltage around a closed path is equal to the magnetic current through the path.
  3. The total electric displacement through a surface enclosing a volume is equal to the total charge within the volume.
  4. The net magnetic flux emeriging through a closed surface is zero (as there are no magnetic charges or monopoles.)

Stated in this form the duality between the two fields is apparent. Note that now electric field lines need not originate from charges only, ie, all flux may not begin and end on a charge. The second equation shows that E or D may have circulation when a changing magnetic field exists, and lines may be closed. Thus D may exist in volume not enclosing charges.

Boundary conditions :

Figure 1
Figure 1 (graphics1.png)

At junction of two dielectrics:

Using Integral form of Maxwell’s second equation

E y2 Δy − E x2 Δx 2 − E x1 Δx 2 − E y1 Δy + E x3 Δx 2 + E x4 Δx 2 = − B . z ΔxΔy E y2 Δy − E x2 Δx 2 − E x1 Δx 2 − E y1 Δy + E x3 Δx 2 + E x4 Δx 2 = − B . z ΔxΔy size 12{E rSub { size 8{y2} } Δy - E rSub { size 8{x2} } { {Δx} over {2} } - E rSub { size 8{x1} } { {Δx} over {2} } - E rSub { size 8{y1} } Δy+E rSub { size 8{x3} } { {Δx} over {2} } +E rSub { size 8{x4} } { {Δx} over {2} } = - { {B} cSup { size 8{ "." } } rSub { size 8{z} } ΔxΔy} cSup {} } {}
(15)

If the dimension Δx→0Δx→0 size 12{Δx rightarrow 0} {}, and assuming B is finite

E y2 Δy − E y2 Δy = 0 ⇒ E y2 = E y1 E y2 Δy − E y2 Δy = 0 ⇒ E y2 = E y1 size 12{E rSub { size 8{y2} } Δy - E rSub { size 8{y2} } Δy=0 drarrow E rSub { size 8{y2} } =E rSub { size 8{y1} } } {}
(16)

Tangential E fields are continuous.

Similarly if D.D. size 12{ {D} cSup { size 8{ "." } } } {}and JJ size 12{J} {} are finite, similar arguments lead to

H y2 = H y1 H y2 = H y1 size 12{H rSub { size 8{y2} } =H rSub { size 8{y1} } } {}
(17)

Tangential H fields are continuous.

At a dielectric – Conductor boundary:

Infinite conductivity, and infinite current density ideally, but practical conductors have very high conductivity. The current flow in time varying case confined within a depth of penetration,

Approximated by a sheet current, infinite conductivity, infinite current density JJ size 12{J} {}but such that as Δx→0Δx→0 size 12{Δx rightarrow 0} {}, J.ΔxJ.Δx size 12{J "." Δx} {} remains finite and becomes a surface current density JsJs size 12{J rSub { size 8{s} } } {}.

lim Δx → 0 JΔx = J s lim Δx → 0 JΔx = J s size 12{ {"lim"} cSub { size 8{Δx rightarrow 0} } JΔx=J rSub { size 8{s} } } {}
(18)

Then Maxwell’s equations with assumption of 0 fields in conductor body give

H y2 Δy − H y1 Δy = J sz Δy ⇒ H y1 = H y2 − J sz ⇒ H y1 = − J sz ⇒ J s = n × H H y2 Δy − H y1 Δy = J sz Δy ⇒ H y1 = H y2 − J sz ⇒ H y1 = − J sz ⇒ J s = n × H size 12{H rSub { size 8{y2} } Δy - H rSub { size 8{y1} } Δy=J rSub { size 8{ ital "sz"} } Δy drarrow H rSub { size 8{y1} } =H rSub { size 8{y2} } - J rSub { size 8{ ital "sz"} } drarrow H rSub { size 8{y1} } = - J rSub { size 8{ ital "sz"} } drarrow J rSub { size 8{s} } =n times H} {}
(19)

Normal components - Dielectric-dielectric :

Assuming finite charge density

Figure 2
Figure 2 (graphics2.png)
D n1 da − D n2 da + Ψ edge = ρΔ xda D n1 da − D n2 da + Ψ edge = ρΔ xda size 12{D rSub { size 8{n1} } ital "da" - D rSub { size 8{n2} } ital "da"+Ψ rSub { size 8{ ital "edge"} } =ρΔ ital "xda"} {}
(20)

As Δx→0Δx→0 size 12{Δx rightarrow 0} {}, edge components vanish,

D n1 da − D n2 da = 0 ⇒ D n1 = D n2 D n1 da − D n2 da = 0 ⇒ D n1 = D n2 size 12{D rSub { size 8{n1} } ital "da" - D rSub { size 8{n2} } ital "da"=0 drarrow D rSub { size 8{n1} } =D rSub { size 8{n2} } } {}
(21)

Thus Normal component of D is continuous

Dielectric – conductor:

Perfect Metallic conductor cannot hold and charge within its volume, can have only surface charge. Practical metallic conductors have very high conductivity, approximate the ideal

ρ = ρ s Δx ρ = ρ s Δx size 12{ρ= { {ρ rSub { size 8{s} } } over {Δx} } } {}
(22)
lim Δx → 0 ρΔx = ρ s lim Δx → 0 ρΔx = ρ s size 12{ {"lim"} cSub { size 8{Δx rightarrow 0} } ρΔx=ρ rSub { size 8{s} } } {}
(23)

The above analysis leads to

D n1 − D n2 = ρ s D n1 − D n2 = ρ s size 12{D rSub { size 8{n1} } - D rSub { size 8{n2} } =ρ rSub { size 8{s} } } {}
(24)

But Dn2=0Dn2=0 size 12{D rSub { size 8{n2} } =0} {} hence Dn1=ρsDn1=ρs size 12{D rSub { size 8{n1} } =ρ rSub { size 8{s} } } {}

Incase of magnetic field similar treatment leads to

B n1 = B n2 B n1 = B n2 size 12{B rSub { size 8{n1} } =B rSub { size 8{n2} } } {}
(25)

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