Maxwell's equations PDE interdependence and solutions

[cianfa72]

TL;DR Summary

About the interdependence of Maxwell's equations from the point of view of PDE theory

Hi, as in this thread Are maxwells equations linearly dependent I would like to better understand from a mathematical point of view the interdependence of Maxwell's equations.

Maxwell's equations are solved assuming as given/fixed the charge density $\rho$ and the current density $J$ as functions of $(x,y,z,t)$. Therefore one can freely assign both $\rho(x,y,z,t)$ and $J(x,y,z,t)$ as long as the continuity condition is fulfilled: $$\nabla \cdot J = - \frac {\partial \rho} {\partial t}$$ The 4 Maxwell's PDE equations put conditions on divergence and curl of $E$ and $B$ vector fields.

From Partial Derivatives Equations (PDE) theory are the above conditions actually necessary and sufficient to uniquely define an unique solution?

Thanks.

 

[anuttarasammyak]

In considering your problem by using EM potential
$$B=\nabla \times A$$
we see obviously
$$\nabla \cdot B = 0$$
[EDIT] Similarly Faraday's law holds with EM potential. These two equations in Maxwell equations and introduction of EM potential seem equivalent. The 4 equations with $j^\mu$ source terms survive.

 

[cianfa72]

Yes, but why one needs to introduce to EM potential $A$ ? If we stick to the equations/conditions for the divergence and curl of $E$ and $B$ alone, are they necessary and sufficient to get an unique solution for both the fields ?

 

[pasmith]

Yes, but why one needs to introduce to EM potential $A$ ? If we stick to the equations/conditions for the divergence and curl of $E$ and $B$ alone, are they necessary and sufficient to get an unique solution for both the fields ?

Yes. Maxwell's equations are linear, so the difference $(E, B)$ between two solutions $(E_1,B_1)$ and $(E_2,B_2)$ with identical sources and subject to identical initial and boundary conditions must satisfy the same equations with the sources set to zero, and with homogenous initial and boundary conditions.

You can then show directly (by taking the curls of the equations for the curls) that $E$ and $B$ satisfy homogenous wave equations with homogenous boundary and initial conditions, so that the only solution is zero (since we have an existence and uniqueness theorem for the wave equation).

 

[cianfa72]

From Wikipedia

A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).

You can then show directly (by taking the curls of the equations for the curls) that $E$ and $B$ satisfy homogenous wave equations with homogenous boundary and initial conditions, so that the only solution is zero (since we have an existence and uniqueness theorem for the wave equation).

So in case of Maxwell's PDEs we can have both boundary and initial conditions associated to the independent variables $(x,y,z,t)$.

What is an homogenous boundary and initial conditions for the homogenous wave equation ? (Yes I know a linear PDE is homogenous whether it has not a constant term, possibly a given function).

 

[cianfa72]

Therefore specifying the divergence and curl is actually necessary and sufficient to identify a unique vector field.