Various formulations of Maxwell equations

This article summarizes the famous Maxwell equations in different forms, namely, time-dependent formulation, time-harmonic formulation and their corresponding (vect{A}-varphi) formulations.

Time-dependent Maxwell equations

Maxwell equations are usually presented in the following time-dependent form with both partial differential equations and constitutive laws.

Partial differential equations:

[
egin{align}
& ext{Ampère's law: } abla imesvect{H} = vect{J} = vect{J}_c + frac{partial vect{D}}{partial t} label{eq:ampere-time} cr
& ext{Faraday's law: } abla imesvect{E} = -frac{partial vect{B}}{partial t} cr
& ext{Gauss’s law for $vect{D}$: } abla cdot vect{D} = ho cr
& ext{Gauss’s law for $vect{B}$: } abla cdot vect{B} = 0
end{align}
]

Constitutive equations:

[
egin{align}
vect{D} &= varepsilon vect{E} cr
vect{B} &= mu vect{H}
end{align}
]

It should be noted that the four partial differential equations are not independent of each other. And their inter-relationships are clarified below.

(a) Apply divergence to Ampère's law, the continuity of total current can be obtained.

[
egin{equation}
abla cdot ( abla imes vect{H}) = abla cdot vect{J} = abla cdot left( vect{J}_c + frac{partialvect{D}}{partial t} ight) = 0.
label{eq:total-current-continuity}
end{equation}
]

Then, if the charge conservation law is prescribed as

[
egin{equation}
frac{partial ho}{partial t} + abla cdot vect{J}_c = 0,
end{equation}
]

we have (frac{partial ho}{partial t} = frac{partial}{partial t} ( abla cdot vect{D})). If we further enforce the initial condition ( ho vert_{t=0} = abla cdot vect{D}vert_{t=0}), Gauss's law for (vect{D}) is obtained.

On the other hand, if Gauss's law is given first, by substituting ( abla cdot vect{D} = ho) into eqref{eq:total-current-continuity}, the charge conservation law can be derived.

For short, Ampère's law implies the continuity of total current. The continuity of total current and charge conservation law as well as Gauss's law for (vect{D}) which is only prescribed at (t=0) lead to the Gauss's law for (vect{D}) at any time. Meanwhile, the charge conservation law can be derived from the continuity of total current and Gauss's law for (vect{D}) at any time.

(b) Apply divergence to Faraday's law, we have ( abla cdot ( abla imes vect{E}) = -frac{partial}{partial t}( abla cdot vect{B}) = 0). If the initial condition ( abla cdot vect{B} vert_{t=0} = 0) is prescribed, i.e. there is no magnetic charge in the universe from the origin of time, then Gauss's law for (vect{B}) can be obtained.

(c) If an additional impressed current (vect{J}_i) is involved, the charge conservation law becomes

[
egin{equation}
frac{partial ho}{partial t} + abla cdot (vect{J}_c + vect{J}_i) = frac{partial ho}{partial t} + abla cdot (sigmavect{E}) + abla cdot vect{J}_i = 0.
end{equation}
]

In non-conducting domain (Omega_I), the conductivity (sigma) is zero and there should be no time-variation of free charge density. Therefore, we have ( abla cdot vect{J}_i = 0), i.e. the impressed current should be divergence-free.

Time-harmonic Maxwell equations

Partial differential equations:

[
egin{align}
& ext{Ampère's law: } abla imesphsvect{H} = phsvect{J} = phsvect{J}_c + miomegaphsvect{D} = (sigma + miomegavarepsilon) phsvect{E} cr
& ext{Faraday's law: } abla imesphsvect{E} = - miomegaphsvect{B} = - miomegamuphsvect{H} cr
& ext{Gauss’s law for $vect{D}$: } abla cdot phsvect{D} = phs{ ho} cr
& ext{Gauss’s law for $vect{B}$: } abla cdot phsvect{B} = 0
end{align}
]

Constitutive equations:

[
egin{align}
phsvect{D} &= varepsilon phsvect{E} cr
phsvect{B} &= mu phsvect{H}
end{align}
]

(a) Substitute Ohm's law into Ampère's law and consider an additional impressed current (phsvect{J}_i), we have

[
egin{equation}
abla imes phsvect{H} = sigmaphsvect{E} + miomegavarepsilonphsvect{E} + phsvect{J}_i = (sigma + miomegavarepsilon)phsvect{E} + phsvect{J}_i.
label{eq:ampere-harmonic-full}
end{equation}
]

Notice that the complex-valued conductivity (sigma + miomegavarepsilon) appears.

(b) Apply divergence to Ampère's law as in Equation eqref{eq:ampere-harmonic-full}, we have

[
egin{equation}
abla cdot left[ (sigma + miomegavarepsilon) phsvect{E} ight] + abla cdot phsvect{J}_i = 0.
end{equation}
]

Because the impressed current (phsvect{J}_i) is divergence-free, the continuity of total current, i.e. conduction current plus displacement current, is obtained.

[
egin{equation}
abla cdot left[ (sigma + miomegavarepsilon) phsvect{E} ight] = 0.
end{equation}
]

Specifically, in non-conducting domain (Omega_I), the conduction current vanishes and we have the Laplace equation

[
egin{equation}
abla cdot left( varepsilon phsvect{E} ight) = 0.
end{equation}
]

(c) Apply divergence to Faraday's law, there is then Gauss's law for (vect{B}),

[
egin{equation}
abla cdot ( abla imes phsvect{H}) = abla cdot (- miomegaphsvect{B}) = abla cdot phsvect{B} = 0.
end{equation}
]

(d) Summarize the above, we know that in conducting domain (Omega_c), Ampère's law leads to the continuity of total current, while in non-conducting domain (Omega_I), it leads to Gauss's law for (vect{D}). On the other hand, Faraday's law implies Gauss's law for (vect{B}).

Eddy current approximation of time-harmonic Maxwell equations

Neglecting the displacement current ( miomegavarepsilonphsvect{E}), we obtain the eddy current approximation of time-harmonic Maxwell equations as follows.

[
egin{align}
& ext{Ampère's law: } abla imesphsvect{H} = phsvect{J} = sigma phsvect{E} cr
& ext{Faraday's law: } abla imesphsvect{E} = - miomegaphsvect{B} = - miomegamuphsvect{H} cr
& ext{Gauss’s law for $vect{D}$: } abla cdot phsvect{D} = phs{ ho} cr
& ext{Gauss’s law for $vect{B}$: } abla cdot phsvect{B} = 0
end{align}
]

(a) Because Ampère's law does not include the displacement current term, Gauss's law for (vect{D}) cannot be definitely derived from Ampère's law any more.

(b) In conducting domain (Omega_c), (sigma eq 0) and apply divergence to Ampère's law. Hence

[
egin{equation}
abla cdot ( abla imes phsvect{H}) = abla cdot (sigma phsvect{E}) = 0.
end{equation}
]

This is the continuity of conduction current due to the absence of displacement current. If the impressed current (phsvect{J}_i) appears in Ampère's law, because it is divergence-free, the continuity of conduction current still holds.

(c) In non-conducting domain (Omega_I), (sigma = 0) and Ampère's law becomes ( abla imes phsvect{H} = 0) or ( abla imes phsvect{H} = phsvect{J}_i), if the impressed current is present. Then apply divergence to it will only produce (0 = 0), which is degenerate. Therefore, Gauss's law for (vect{D}) cannot be derived from Ampère's law anymore and must be explicitly enforced in (Omega_I), i.e. ( abla cdot phsvect{D} = abla cdot left( varepsilon phsvect{E} ight) = 0).

(d) Summarize the above, the eddy current approximation of time-harmonic Maxwell equations is as follows.

[
egin{align}
& ext{Ampère's law: } abla imesphsvect{H} = phsvect{J} = sigma phsvect{E} cr
& ext{Faraday's law: } abla imesphsvect{E} = - miomegamuphsvect{H} cr
& ext{Gauss’s law for $vect{D}$: } abla cdot phsvect{D} = 0 ; ext{in $Omega_I$}
end{align}
]

(vect{A}-varphi) formulation for time-dependent Maxwell equations

Let (vect{B} = abla imes vect{A}) and (vect{E} = - abla varphi - frac{partial vect{A}}{partial t}).

(a) Substitute them into Ampère's law in Equation eqref{eq:ampere-time}, hence

[
abla imes left( frac{1}{mu} abla imes vect{A} ight) = vect{J}_c + frac{partial}{partial t} left[ varepsilon (- ablavarphi - frac{partial vect{A}}{partial t}) ight].
]

Reorder the terms, Ampère's law becomes

[
egin{equation}
abla imes left( frac{1}{mu} abla imes vect{A} ight) + varepsilon abla frac{partial varphi}{partial t} + varepsilon frac{partial^2 vect{A}}{partial t^2} = vect{J}_c.
label{eq:ampere-law-aphi}
end{equation}
]

(b) For Faraday's law, due to the adopted (vect{A}-varphi) potentials,

[
abla imes vect{E} = abla imes left( - ablavarphi - frac{partial vect{A}}{partial t} ight) = - abla imes( ablavarphi) - frac{partial}{partial t} left( abla imesvect{A} ight) = -frac{partial vect{B}}{partial t}.
]

Therefore, Faraday's law is automatically satisfied through the selection of (vect{A}-varphi).

(c) Gauss's law for (vect{D}):

[
egin{align}
abla cdot left[ varepsilon left( - ablavarphi - frac{partial vect{A}}{partial t} ight) ight] &= ho cr
- abla cdot left( varepsilon abla varphi ight) - frac{partial}{partial t} ablacdotleft( varepsilonvect{A} ight) &= ho label{eq:gauss-law-for-d-aphi}.
end{align}
]

(d) Gauss's law for (vect{B}): ( ablacdotvect{B}=0) is automatically satisfied by the selection of (vect{A}) such that (vect{B}= abla imesvect{A}).

(e) It should be noted that the above set of effective Maxwell equations, namely, Ampère's law and Gauss's law for (vect{D}), does not uniquely solve (vect{A}-varphi). This can be further explained below.

If we assume ((vect{A},varphi)) is a solution to the equation system, then for any scalar field (psi), we can define

[
vect{A}'=vect{A}+ ablapsi, ;varphi'=varphi-frac{partialpsi}{partial t}.
]

Substitute ((vect{A}',varphi')) into Ampère's law as in Equation eqref{eq:ampere-law-aphi},

[
egin{aligned}
abla imesleft(frac{1}{mu} abla imesvect{A}' ight) + varepsilon ablafrac{partial varphi'}{partial t} + varepsilonfrac{partial^2 vect{A}'}{partial t^2} &= vect{J}_c cr
abla imesleft[frac{1}{mu} abla imesleft(vect{A}+ ablapsi ight) ight] + varepsilon ablaleft[frac{partial}{partial t}left(varphi-frac{partial psi}{partial t} ight) ight] + varepsilonfrac{partial^2}{partial t^2}(vect{A}+ ablapsi) &= vect{J}_c cr
abla imesleft(frac{1}{mu} abla imesvect{A} ight) + varepsilon ablafrac{partial varphi}{partial t} - varepsilon ablafrac{partial^2 psi}{partial t^2} + varepsilonfrac{partial^2 vect{A}}{partial t^2} + varepsilonfrac{partial^2}{partial t^2}( ablapsi) &= vect{J}_c cr
abla imesleft(frac{1}{mu} abla imesvect{A} ight) + varepsilon ablafrac{partial varphi}{partial t} + varepsilonfrac{partial^2vect{A}}{partial t^2} &= vect{J}_c
end{aligned}.
]

It can be seen that Ampère's law filled with ((vect{A},varphi)) is recovered.

Substitute ((vect{A}',varphi')) into Gauss's law for (vect{D}) as in Equation eqref{eq:gauss-law-for-d-aphi},

[
egin{aligned}
- ablacdot(varepsilon ablavarphi') - frac{partial}{partial t} ablacdot(varepsilonvect{A}') &= ho cr
- ablacdotleft[varepsilon ablaleft( varphi - frac{partialpsi}{partial t} ight) ight] - frac{partial}{partial t} ablacdotleft[varepsilon(vect{A}+ ablapsi) ight] &= ho cr
- ablacdot(varepsilon ablavarphi) + ablacdotleft[varepsilon ablaleft(frac{partialpsi}{partial t} ight) ight] - frac{partial}{partial t} ablacdot(varepsilonvect{A}) - frac{partial}{partial t} ablacdot(varepsilon ablapsi) &= ho cr
- ablacdot(varepsilon ablavarphi) - frac{partial}{partial t} ablacdot(varepsilonvect{A}) &= ho
end{aligned}.
]

Hence, the original Gauss's law for (vect{D}) filled with ((vect{A},varphi)) is recovered. Summarize the above, ((vect{A}',varphi')) is also a solution of the Maxwell equations.

To uniquely determine the solution ((vect{A},varphi)), Coulomb gauge should be appended to the system, which defines the divergence of (vect{A}), i.e. ( ablacdot(varepsilonvect{A})=0). With Coulomb gauge adopted, Equation eqref{eq:gauss-law-for-d-aphi} becomes

[
egin{equation}
- ablacdot(varepsilon ablavarphi) = ho,
end{equation}
]

which is the classical Poisson equation for electric scalar potential.

(f) Finally, (vect{A}-varphi) formulation for the time-dependent Maxwell equations is given below.

[
egin{align}
& ext{Ampère's law: } abla imesleft(frac{1}{mu} abla imesvect{A} ight) + varepsilon ablafrac{partialvarphi}{partial t} + varepsilonfrac{partial^2vect{A}}{partial t^2} &= vect{J}_c label{eq:ampere-law-aphi-final} cr
& ext{Gauss's law for $vect{D}$: } - ablacdot(varepsilon ablavarphi) = ho label{eq:gauss-law-for-d-aphi-final} cr
& ext{Coulomb gauge: } ablacdot(varepsilonvect{A}) = 0. label{eq:coulomb-gauge-aphi-final}
end{align}
]

(vect{A}-varphi) formulation for time-harmonic Maxwell equations

The system of equations can be directly derived from Equation eqref{eq:ampere-law-aphi-final}, eqref{eq:gauss-law-for-d-aphi-final} and eqref{eq:coulomb-gauge-aphi-final}.

[
egin{align}
& ext{Ampère's law: } abla imesleft(frac{1}{mu} abla imesphsvect{A} ight) + miomegavarepsilon ablaphs{varphi} - varepsilonomega^2phsvect{A} &= phsvect{J}_c label{eq:ampere-law-aphi-harmo} cr
& ext{Gauss's law for $vect{D}$: } - ablacdot(varepsilon ablaphs{varphi}) = phs{ ho} label{eq:gauss-law-for-d-aphi-harmo} cr
& ext{Coulomb gauge: } ablacdot(varepsilonphsvect{A}) = 0. label{eq:coulomb-gauge-aphi-harmo}
end{align}
]

(a) Similar to the discussion in section 1(a), here we also have the fact that Ampère's law and Gauss's law lead to the charge conservation law with only the difference that Coulomb gauge needs to be substituted into the system. Apply divergence to Ampère's law, we have

[
ablacdotleft[ abla imesleft(frac{1}{mu} abla imesphsvect{A} ight) ight] + miomega ablacdot(varepsilon ablaphs{varphi}) - omega^2 ablacdot(varepsilonphsvect{A}) = ablacdotphsvect{J}_c,
]
where the first and third terms on the left hand side are zero. Then charge conservation law is obtained:

[
egin{equation}
ext{Charge conservation law: } miomegaphs{ ho} + ablacdotphsvect{J}_c = 0.
label{eq:charge-conservation-harmo}
end{equation}
]

This suggests us that the charge density can be actually represented by the conduction current density (phsvect{J}_c) as long as (omega>0), whereas (phsvect{J}_c) itself can be further represented by electric field (phsvect{E}) due to Ohm's law as below.

[
egin{equation}
phsvect{J}_c = sigmaphsvect{E} = sigma(- ablaphs{varphi}- miomegaphsvect{A}) = -sigma ablaphs{varphi} - misigmaomegaphsvect{A}.
label{eq:current-density-aphi}
end{equation}
]

Hence, (phs{ ho}) can be eliminated from the system with its final representation in (phsvect{A}-phs{varphi}),

[
egin{equation}
phs{ ho} = frac{ mi}{omega} ablacdot(-sigma ablaphs{varphi}- misigmaomegaphsvect{A}).
label{eq:charge-density-aphi}
end{equation}
]

Substitute Equation eqref{eq:current-density-aphi} into Equation eqref{eq:ampere-law-aphi-harmo} and Equation eqref{eq:charge-density-aphi} into Equation eqref{eq:gauss-law-for-d-aphi-harmo}, we have

[
egin{align}
& ext{Ampère's law: } abla imesleft(frac{1}{mu} abla imesphsvect{A} ight) + (sigma+ miomegavarepsilon) ablaphs{varphi} + miomega(sigma+ miomegavarepsilon)phsvect{A} = 0 label{eq:ampere-law-aphi-harmo-final} cr
& ext{Gauss's law for $vect{D}$: } miomega ablacdot(varepsilon ablaphs{varphi}) = - ablacdot(sigma ablaphs{varphi} + misigmaomegaphsvect{A}) label{eq:gauss-law-for-d-aphi-harmo-final} cr
& ext{Coulomb gauge: } ablacdot(varepsilonphsvect{A}) = 0. label{eq:coulomb-gauge-aphi-harmo-final}
end{align}
]

(b) According to section 2(b), for the time-harmonic case, Ampère's law implies the continuity of total current in conducting domain (Omega_c) and Gauss's law for (vect{D}) in non-conducting domain (Omega_I). Therefore, this property may also be applicable here for the time-harmonic (vect{A}-varphi) formulation. Apply divergence to Equation eqref{eq:ampere-law-aphi-harmo-final} and also use the Coulomb gauge in Equation eqref{eq:coulomb-gauge-aphi-harmo-final}, we have

[
egin{aligned}
& ablacdotleft[ (sigma+ miomegavarepsilon) ablaphs{varphi} ight] + miomega ablacdotleft[ (sigma+ miomegavarepsilon)phsvect{A} ight] = 0 cr
& miomega ablacdot(varepsilon ablaphs{varphi}) = - ablacdot(sigma ablaphs{varphi}) - ablacdot( misigmaomegaphsvect{A}) + omega^2 ablacdot(varepsilonphsvect{A}) cr
& miomega ablacdot(varepsilon ablaphs{varphi}) = - ablacdot(sigma ablaphs{varphi} + misigmaomegaphsvect{A}),
end{aligned}
]
which is just Equation eqref{eq:gauss-law-for-d-aphi-harmo-final}. Therefore, Ampère's law implies Gauss's law for (vect{D}) under the precondition (omega>0). When (omega=0), Gauss's law for (vect{D}) should be explicitly enforced.

(c) Summarize the above, the (vect{A}-varphi) formulation for time-harmonic Maxwell equations is given as below.

When (omega>0),

[
egin{align}
& ext{Ampère's law: } abla imesleft(frac{1}{mu} abla imesphsvect{A} ight) + (sigma+ miomegavarepsilon) ablaphs{varphi} + miomega(sigma+ miomegavarepsilon)phsvect{A} = 0 cr
& ext{Coulomb gauge: } ablacdot(varepsilonphsvect{A}) = 0.
end{align}
]

When (omega=0),

[
egin{align}
& ext{Ampère's law: } abla imesleft(frac{1}{mu} abla imesphsvect{A} ight) + sigma ablaphs{varphi} = 0 cr
& ext{Gauss's law in $Omega_I$ with $phs{ ho}=0$: } - ablacdot(varepsilon ablaphs{varphi}) = 0 cr
& ext{Gauss's law in $Omega_c$: } - ablacdot(sigma ablaphs{varphi}) = 0 cr
& ext{Coulomb gauge: } ablacdot(varepsilonphsvect{A}) = 0.
end{align}
]