I am trying to answer to the following question:

"Based on Maxwell equations explain how and in what circunstances electrical fields and magnetic fields can generate each other

mutually."

I have recently started to study Maxwell equations so i am still trying to figure out many things about the set of the 4 equations.

I know that the Faraday Law is what shows that there is another source of electrical fields besides the distribution of charges, and that

other source is directly linked to the magnetic fields, since the Faraday Law shows that the variations in magnetic fields

are able to produce an electromotive force and therefore an electrical field.

Maxwell has able to prove without any experiment that if a magnetic field which varies in time is able to produce an electrical field an electrical field wich varies is also able to produce a magnetic field.

To prove this i probably need to start by proving the law of conservation of charges in order to prove that the ampere equation was incomplete, and then go throug maxwell equations untill i get there right?

So the law of conservation of charges is given by:

\( (\partial p /\partial t) = - \nabla * j \) where j is the specific current density

Now calculating the divergence of the equation \( (\nabla * B) = uo * j \) we get :

\( \nabla * (\nabla * B) = uo * j \) and since the divergence of any rotational is zero we get \( \nabla * j = 0 \)

So this shows that the Ampere equations was incomplete.

And now from here? Do i need to add the term \( uo*e0 * (\partial E /\partial t) \) and calculate the divergence of the Ampere modified law which would be

\( \nabla * (\nabla * B) = uo *\nabla * j + u0*e0*(\partial (\nabla*E)/(\partial t)\) and now form here?

Is there any way of explaining this effect using the Maxwell equations, that does not involve such a high degree of mathematization? I suppose not...

Thanks