# The lab have a constant electric field and a constant magnetic field, what is the electric and magnetic field inside a conductor and far from it

The following is an old question from an exam in a Physics $2$ course I am taking, I have tried to solve the question and after I thought I got the answer I looked at the solution and saw it isn't correct.

The solution explains what the answer is, but doesn't really explains the facts used.

In the lab frame there is an electric field $E=E_{y}$ and a magnetic field $B_{z}=\beta E$.

We place a cube made of a conducting material, point $a$ is at the center of the cube while point $b$ is far from it.

Calculate the following: $$E_{y}(a),E_{y}(b),B_{z}(a),B_{z}(b)$$ and $$E'_{y}(a),E'_{y}(b),B'_{z}(a),B'_{z}(b)$$ as they are seen in a frame of reference moving at a speed $c\beta\hat{x}$.

My efforts:

I wrote that $$E_{y}(a)=E_{y}(b)=E_{y}=E$$

because the electric field is constant (in the lab frame).

Similarly, I wrote that $$B_{y}(a)=B_{y}(b)=B_{z}=B$$

because the magnetic field is constant (in the lab frame).

Then I used field transformation $$E'_{\perp}=\gamma(E_{\perp}+\beta\times B)$$ $$B'_{\perp}=\gamma(B_{\perp}+-\beta\times E)$$

to calculate the other four values requested.

I then looked at the solution, the solution claims that $$E_{y}(a)=0,\, E_{y}(b)=E$$

$$B_{z}(a)=\beta E,\, B_{z}(b)=\beta E$$

and the other field were calculated using the above transformations.

I then remembered that in a conductor the electric field is $0$, this explains why $E_{y}(a)=0$.

I don't understand this situation completely, Please help me understand it by answering the following questions (that goes a bit beyond the question, but I find interesting and am having difficulties answering myself):

1) Why does $b$ have to be far from the conductor so we can say that the electric is $E$ there ? what happens near the conductor ? (I wonder if we can say something about the charge distribution on it boundary, $\sigma$)

2) Why does the magnetic field does not change in space, inside or outside the conductor ? (I know that the electric field does change, at least inside the conductor, why shouldn't the magnetic field change as well ?)

3) What can we say about the work done while moving the conductor in the lab ?