Let's say we have an electromagnetic problem where $\epsilon = \epsilon(x,y)$, $\rho = \rho(x,y)$, and $J = J(x,y)$. The physicist argument is that by the symmetry of the problem we also have $E = E(x,y)$ and $H = H(x,y)$. However, I am a terrible physicist so usually arguments from symmetry are not obvious to me. Therefore I was trying to prove this from Maxwell's equations. Someone told me to take $\partial z$ of both sides of each equation.
Ok, so here was my attempt (assuming $\epsilon$ is scalar)...
$$ \nabla \cdot D = \rho$$ $$ \nabla \cdot \epsilon E = \rho$$ $$ \partial_z(\nabla \cdot \epsilon E) = 0$$ $$ \nabla \cdot \partial_z(\epsilon E) = 0$$ $$ \nabla \cdot \epsilon \partial_z E = 0$$ $$ \partial_x(\epsilon \partial_z E_x) + \partial_y(\epsilon \partial_z E_y) + \partial_z(\epsilon \partial_z E_z)= 0$$ $$ \partial_x(\epsilon \partial_z E_x) + \partial_y(\epsilon \partial_z E_y) + \epsilon \partial^2_z E_z= 0$$
Didn't see where to go so I tried Ampere's Law
$$ \nabla \times H = J + \partial_t D $$ $$ \partial_z (\nabla \times H) = \partial_t \partial_z D $$ $$ \nabla \times \partial_z H = \partial_t \partial_z \epsilon E $$ $$ \nabla \times \partial_z H = \partial_t (\epsilon \partial_z E) $$
Again, not clear to me how to conclude $d_zE = 0$.
In addition, I can think of a simple counter example to this symmetry argument. In empty space we have invariance in all directions. But, for any direction you choose I can form a plane wave traveling in that direction, so $E$ varies in that direction.
So when and how can we really conclude that translational invariance of sources/materials implies translational invariance of fields?
EDIT:(Attempting Mark's comment)
$$ \nabla \times (\nabla \times E) = -\partial_t \nabla \times B $$ $$ \nabla(\nabla \cdot E) - \nabla^2 E = -\mu (\partial_t J + \partial_t \epsilon E) $$ $$ \nabla(\nabla \cdot E) + \mu \partial_t J = \nabla^2 E - \mu \epsilon \partial_t E $$
so we get the three scalar equations
$$ \partial_x(\nabla \cdot E) + \mu \partial_t J_x = \nabla^2 E_x - \mu \epsilon \partial_t E_x $$ $$ \partial_y(\nabla \cdot E) + \mu \partial_t J_y = \nabla^2 E_y - \mu \epsilon \partial_t E_y $$ $$ \mu \partial_t J_z = \nabla^2 E_z - \mu \epsilon \partial_t E_z $$