Are Maxwell's equations linearly dependent?

[cianfa72]

TL;DR Summary

About the linearly independence of Maxwell's
PDE equations

HI,
consider the 4 Maxwell's equations in microscopic/vacuum formulation as for example described here Maxwell's equations (in the following one assumes charge density ##\rho## and current density ##J## as assigned -- i.e. they are not unknowns but are given as functions of space and time coordinates).

Two of the equations are scalar (divergence based equations) while the other two give rise to 6 equations in 6 unknowns (curl based equations).

Therefore it seems there are 8 equations in 6 unknowns (##E## and ##B## field components).

Are the above partial differential equations (PDEs) actually linearly dependent ? Thanks.

 

[anuttarasammyak]

Maxwell equations with EM potential is
[tex]\square A^\mu - \partial^\mu(\partial_\nu A^\nu)=-\mu_0 j^\mu[/tex]
where ##\mu,\nu##=0,1,2,3. Four equations for Four variables. Does this meet your point ?

 

[cianfa72]

Four equations for Four variables. Does this meet your point ?

Ok, from the solution of the above four PDEs one then get the EM fields.

BTW is the EM potential ##A^{\mu}## solution given in a specific gauge ?

 

[cianfa72]

Coming back to the original question, the 4 Maxwell equations give conditions on divergence and curl of fields ##E## and ##B##. Perhaps the point is that to uniquely define each field such two conditions are actually necessary and sufficient.