Faraday law, third Maxwell's equation in Mathematica

Question

Three question about this equation:

$ \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}]

Answer 1

1. I suppose you know the electric field as you mentioned in 2. : In this case if you solve the equation with Mathematica (or in any other way) you will get the magnetic field.

2. The calculation of the electric field (as for the magnetic field) depend on your given data (do you know the charge density? the magnetic field?, how some particle effected by this field?...) . For example if you know the charge density in the space as a function of time you can use Maxwell's equation: ${\displaystyle {\vec {\nabla }}\cdot {\vec {\mathbf {E} }}={\frac {\rho }{\epsilon _{0}}}} $ that equivalent to it's integral form: ${\displaystyle \oint _{S}{\vec {\mathbf {E} }}\cdot d\mathbf {S} ={\frac {q}{\epsilon _{0}}}}$. Note that this kind of calculation can get very complicated and only analytical solvable in specific cases.

3. If you know both electric and magnetic field you simply know the electromagnetic field. There is no specific common wat to write their combination (EDIT: you can use the Electromagnetic field tensor, as @Joe suggest). If you want you can define vector in $ℝ^6$ that describe all the components of electric and magnetic fields. In this notion, the electromagnetic field will be EM: $ℝ^4→ℝ^6$