# Faraday law, third Maxwell's equation in Mathematica

$\displaystyle\nabla\times\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

1 If I solve this equation with Mathematica, I find the magnetic field $b(x,y,z,t),b:\mathbb{R}^4\rightarrow\mathbb{R}^3$ right?

2 I have put an arbitrary function $e:\mathbb{R}^4\rightarrow\mathbb{R}^3$ as electric field for this experiment, but how can I calculate that function for a real case; what I need to do this?

3 One time that I have both the electric and magnetic field how can I compose the electromagnetic field?

Needs["VectorAnalysis"]

(*Electric field e : R^4->R^3 *)
e[x_, y_, z_, t_] := {x - 3 y, 4 y + t, y + z + t};

Maxwell = Curl[e[x, y, z, t]] == -D[b[x, y, z, t], t];

DSolve[Maxwell, b[x, y, z, t], {x, y, z, t}]
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As a first step, you need definitely boundary conditions for your differential equations to be solved and to get real life case as you whant. –  TMS Dec 2 '12 at 16:49
Both Electric and Magnetic fields must satisfy the four Maxwell's equations, not only one. That is the meaning of a "real case". Doing that, you'll have the electromagnetic field. –  Anuar Dec 2 '12 at 16:57