# Do Maxwell's Equations overdetermine the electric and magnetic fields?

It isn't a problem because two of the eight equations are constraints and they're not quite independent from the remaining six.

The constraint equations are the scalar ones, $${\rm div}\,\,\vec D = \rho, \qquad {\rm div}\,\,\vec B = 0$$ Imagine $\vec D=\epsilon_0\vec E$ and $\vec B=\mu_0\vec H$ everywhere for the sake of simplicity.

If these equations are satisfied in the initial state, they will immediately be satisfied at all times. That's because the time derivatives of these non-dynamical equations ("non-dynamical" means that they're not designed to determine time derivatives of fields themselves; they don't really contain any time derivatives) may be calculated from the remaining 6 equations. Just apply ${\rm div}$ on the remaining 6 component equations, $${\rm curl}\,\, \vec E+ \frac{\partial\vec B}{\partial t} = 0, \qquad {\rm curl}\,\, \vec H- \frac{\partial\vec D}{\partial t} = \vec j.$$ When you apply ${\rm div}$, the curl terms disappear because ${\rm div}\,\,{\rm curl} \,\,\vec V\equiv 0$ is an identity and you get $$\frac{\partial({\rm div}\,\,\vec B)}{\partial t} =0,\qquad \frac{\partial({\rm div}\,\,\vec D)}{\partial t} =-{\rm div}\,\,\vec j.$$ The first equation implies that ${\rm div}\,\,\vec B$ remains zero if it were zero in the initial state. The second equation may be rewritten using the continuity equation for $\vec j$, $$\frac{\partial \rho}{\partial t}+{\rm div}\,\,\vec j = 0$$ (i.e. we are assuming this holds for the sources) to get $$\frac{\partial ({\rm div}\,\,\vec D-\rho)}{\partial t} = 0$$ so ${\rm div}\,\,\vec D-\rho$ also remains zero at all times if it is zero in the initial state.

Let me mention that among the 6+2 component Maxwell's equations, 4 of them, those involving $\vec E,\vec B$, may be solved by writing $\vec E,\vec B$ in terms of four components $\Phi,\vec A$. In this language, we are left with the remaining 4 Maxwell's equations only. However, only 3 of them are really independent at each time, as shown above. That's also OK because the four components of $\Phi,\vec A$ are not quite determined: one of these components (or one function) may be changed by the 1-parameter $U(1)$ gauge invariance.

I) Let us just for fun generalize OP's question to $n$ spacetime dimensions, and check how the counting of eqs. and degrees of freedom (d.o.f.) work out in this general setting. We shall use Lubos Motl's answer as a template for this part. Also we shall use a special relativistic $(-,+,\ldots,+)$ notation with $c=1$, where $\mu,\nu\in\{0,\ldots,n-1\}$ denote spacetime indices, while $i,j \in\{1,\ldots,n-1\}$ denote spatial indices. Maxwell eqs. are the following.

1. Source-free Bianchi identities: $${\rm d}F~=~0 \qquad\qquad \Leftrightarrow \qquad\qquad \sum_{\rm cycl.~\mu,\nu,\lambda} d_{\lambda} F_{\mu\nu} ~=~0, \qquad\qquad F~:=~\frac{1}{2} F_{\mu\nu}~ {\rm d}x^{\mu} \wedge {\rm d}x^{\nu}.$$ Here $$\left(\begin{array}{c} n \cr 3\end{array}\right) {\rm~Bianchi~identities} ~=~ \left(\begin{array}{c} n-1 \cr 3\end{array}\right) {\rm~constraints}~+~ \left(\begin{array}{c} n-1 \cr 2\end{array}\right) {\rm~dynamical~eqs.}$$ $$~=~ ({\rm No~magnetic~monopole~eqs.})~+~ ({\rm Faraday's~law}).$$

2. Maxwell eqs. with source terms: $$d_{\mu}F^{\mu\nu}~=~-j^{\nu} .$$ Here $$n {\rm~source~eqs.}~=~1 {\rm~constraint} ~+~ (n-1) {\rm~dynamical~eqs.}$$ $$~=~({\rm Gauss'~law}) ~+~ ({\rm Ampere's~law~with~displacement~term}).$$

We have used the terminology that a dynamical eq. contains time derivatives, while a constraint does not. So the number of dynamical eqs. is

$$\left(\begin{array}{c} n-1 \cr 2\end{array}\right)~+~(n-1)~=~ \left(\begin{array}{c} n \cr 2\end{array}\right),$$

which precisely matches

$${\rm the~number~} \left(\begin{array}{c} n \cr 2\end{array}\right) {\rm~of~} F_{\mu\nu} {\rm~fields}$$ $$~=~\left(\begin{array}{c} n-1 \cr 2\end{array}\right){~\rm magnetic~fields~} F_{ij} ~+~(n-1) {\rm~electric~fields~}F_{i0} .$$

Maxwell eqs. with source terms imply the continuity eq.

$$d_{\nu}j^{\nu} ~=~-d_{\nu}d_{\mu}F^{\mu\nu}~=~0,\qquad\qquad F^{\mu\nu}~=~-F^{\nu\mu},$$

so one must demand that the background sources $j^{\nu}$ obey the continuity eq.

For consistency, the time derivative of each of the constraints should vanish. In the case of the no-magnetic-monopole-eqs., this follows from Faraday's law. In the case of Gauss' law, this follows from the modified Ampere's law and the continuity eq.

II) The previous section (I) made the counting in terms of the $\left(\begin{array}{c} n \cr 2\end{array}\right)$ field strengths $F_{\mu\nu}$. In terms of the $n$ gauge potentials $A_{\mu}$, the counting goes as follows. The Bianchi identities are now trivially satisfied,

$$F~=~{\rm d}A\qquad\qquad A~:=~A_{\mu}~ {\rm d}x^{\mu}.$$

There are still the $n$ Maxwell eqs. with source terms

$$(\Box\delta^{\mu}_{\nu}-d^{\mu}d_{\nu})A^{\nu}~=~-j^{\mu} , \qquad\qquad \Box~:=~d_{\mu}d^{\mu}.$$

There is a single gauge d.o.f. because of gauge symmetry $A \to A + {\rm d}\Lambda$ and $F \to F$. If one gauge-fixes using the Lorenz gauge condition

$$d_{\mu}A^{\mu}~=~0,$$

the Maxwell eqs. become $n$ decoupled wave equations

$$\Box A^{\mu}(x)~=~-j^{\mu}(x).$$

By a spatial Fourier transformation, these become decoupled linear second-order ODEs with constant coefficients,

$$(d^2_t+\vec{k}^2) \hat{A}^{\mu}(t;\vec{k})~=~\hat{j}^{\mu}(t;\vec{k}) ,$$

which, starting from some initial time $t_0$, may be solved for all times $t$, cf. OP's question. [One should check that the solution

$$\hat{A}^{\mu}(t;\vec{k}) ~=~\int {\rm d} t^{\prime} ~G(t-t^{\prime};\vec{k})~\hat{j}^{\mu}(t^{\prime};\vec{k}), \qquad\qquad (d^2_t+\vec{k}^2)G(t-t^{\prime};\vec{k})~=~\delta(t-t^{\prime}),$$

satisfies the Lorenz gauge condition. This follows from the continuity eq.]

III) It is interesting to derive the complete solution $\tilde{A}^{\mu}(k)$ in $k^{\nu}$-momentum space without gauge-fixing. The Fourier-transformed Maxwell eqs. read

$$M^{\mu}{}_{\nu}~\tilde{A}^{\nu}(k)~=~\tilde{j}^{\mu}(k), \qquad\qquad M^{\mu}{}_{\nu}~:=~k^2\delta^{\mu}_{\nu} -k^{\mu}k_{\nu}.$$

To proceed one must analyze the matrix $M^{\mu}{}_{\nu}$ for fixed $k^{\lambda}$. There are three cases.

1. Constant mode $k^{\mu}=0$. Then the matrix $M^{\mu}{}_{\nu}=0$ vanishes identically. Maxwell eqs. are only possible to satisfy if $\tilde{j}^{\mu}(k=0)=0$ is zero. The gauge potential $\tilde{A}_{\mu}(k=0)$ is not restricted at all by Maxwell eqs., i.e., there is a full $n$-parameter solution.

2. Massive case $k^2\neq 0$. The matrix $M^{\mu}{}_{\nu}$ is diagonalizable with eigenvalue $k^2$ (with multiplicity $n-1$), and eigenvalue $0$ (with multiplicity $1$). The latter corresponds to a pure gauge mode $\tilde{A}^{\mu}~\propto~k^{\mu}$. The complete solution is a $1$-parameter solution of the form $$\tilde{A}^{\mu}(k) ~=~\frac{\tilde{j}^{\mu}(k)}{k^2}~+~ik^{\mu}\tilde{\Lambda}(k).$$ Apart from the source term, this is pure gauge.

3. Massless case $k^2=0$ and $k^{\mu}\neq 0$. The matrix $M^{\mu}{}_{\nu}$ is not diagonalizable. There is only eigenvalue $0$ (with multiplicity $n-1$). Maxwell eqs. are only possible to satisfy if the source $\tilde{j}^{\mu}(k)=\tilde{f}(k)k^{\mu}$ is proportional to $k^{\mu}$ with some proportionality factor $\tilde{f}(k)$. In that case Maxwell eqs. become $$-k_{\mu}\tilde{A}^{\mu}(k)~=~\tilde{f}(k).$$ Let us introduce an $\eta$-dual vector$^1$ $$k^{\mu}_{\eta}~:=~(-k^0,\vec{k})\qquad {\rm for}\qquad k^{\mu}~=~(k^0,\vec{k}).$$ Note that $$k_{\mu}~k^{\mu}_{\eta}~=~(k^0)^2+\vec{k}^2$$ is just the Euclidean distance square in $k^{\mu}$-momentum space. The complete solution is an $(n-1)$-parameter solution of the form $$\tilde{A}^{\mu}(k) ~=~-\frac{k^{\mu}_{\eta}}{k_{\nu}~k^{\nu}_{\eta}}\tilde{f}(k) ~+~ik^{\mu}\tilde{\Lambda}(k)~+~\tilde{A}^{\mu}_{T}(k).$$ The term proportional to $k_{\mu}$ is pure gauge. Here $\tilde{A}^{\mu}_{T}(k)$ denote $n-2$ transversal modes, $$k_{\mu}~\tilde{A}^{\mu}_{T}(k)~=~0, \qquad\qquad k_{\mu}^{\eta}~\tilde{A}^{\mu}_{T}(k)~=~0.$$ The $n-2$ transversal modes $\tilde{A}^{\mu}_{T}$ are the only propagating physical d.o.f. (electromagnetic waves, photon field).

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$^1$ Longitudinal and timelike polarizations are in the massless case proportional to $k^{\mu}\pm k^{\mu}_{\eta}$, respectively.

Equations are written for any time $t$ and there is no need to "prove" their validity at any time. These equations are the experimental laws and are, of course, consistent at any time. The constraints are imposed here not to the fields, but to the electric and magnetic charges. The charges do not have sources/sinks so the derived equations like $\partial\rho/\partial t + \rm div \vec{j}=0$ say namely that and are called the charge conservation laws. (They are are an experimental fact.) The charge conservation laws do not determine the charge dynamics; for the latter the "mechanical" equations exist. In case of one elementary charge $q$, its conservation means its time-independence: $\frac{dq}{dt}=0$ which is not usually written as an additional equation, but used as its solution $q=const$ in the "mechanical" equations.

So you have six equations for fields and two as conservation laws for charges.