Derivations of Maxwell equations In my book of electrodynamics, the Maxwell equations are always used for specific conditions (electrostatics, magnetostatics, …). But nowhere I see a complete derivation of the equations. Maybe it will sound compendiously: how can we derive this equations?
 A: You can't. You can justify them from physical experiments and observations, complemented with some mathematical handling for consistency, but the Maxwell equations are ultimately axioms - they are postulates that we put forth from the beginning, and they get their validity from the predictive and explicative ability of the EM theory we build from those equations to match and explain experiments.
(That said, there does exist a "deeper" set of axioms, in classical field theory, from which the Maxwell equations can be derived, but (i) it will likely look outlandish to you, and (ii) it raises exactly the same questions about how you 'derive' those axioms as the Maxwell equations raise.) 
A: I guess you can do a charge balance using continuity equation
$$-\nabla \cdot I/Area=- \nabla \cdot J=\frac{\partial \rho}{\partial t}$$
We know V=IR so J=Conductivity×E substitute to get a relation that might be used later with a partial of time.
Better way to get Gauss first law Q=CV C=epsilon A/D, V=ED does mean the first law $$ \epsilon \nabla \cdot E = \rho$$. Or use Coulombs law F=EQ=KQ1Q2/R^2 . E=KQ/R^2 charge density is del dot E
For magnetic fields in=out number of lines in= number of lines out
$$\nabla \cdot B=0 $$
We also know F=EQ+QV × B so F/Q= E+ V × B
Maybe you can use the material derivative along a stream line to be constant E and B
$$\frac{DE}{Dt}=\frac{\partial{E}}{\partial{t}}+ c \cdot \nabla c $$
The partial is a sine wave
You can try the rest of the derivativion yourself and I'll check it out
