Suppose you know that in a particular inertial frame neither the electric field $\EE$ nor the magnetic field $\BB$ has an $x$ component, but neither $\EE$ nor $\BB$ is zero. Consider another inertial frame moving with respect to the first one with velocity $v$ in the $x$-direction, and denote the electric and magnetic fields in this frame by $\EE{}'$ and $\BB{}'$, respectively.

1. What are the conditions on $\EE$ and $\BB$, if any, and the value(s) of $v$, if any, such that $\EE{}'$ vanishes for some value of $v$?
2. What are the conditions on $\EE$ and $\BB$, if any, and the value(s) of $v$, if any, such that $\BB{}'$ vanishes for some value of $v$?
3. What are the conditions on $\EE$ and $\BB$, if any, and the value(s) of $v$, if any, such that both $\EE{}'$ and $\BB{}'$ vanish for the same value of $v$?
4. Is it possible that $\EE{}'$ and $\BB{}”$ vanish for different values of $v$? (We write $\BB{}”$ rather than $\BB{}'$ to emphasize that $\EE{}'$ and $\BB{}”$ are with respect to different reference frames.)

1. Inserting $\EE{}'=0$ into (18) of §11.2. we get \begin{equation} v = |\vv| = \frac{|\EE|}{|\BB|} \end{equation} since $\vv$ is perpendicular to $\BB$. This is only possible if $|\EE|<\cc|\BB|$, which also follows immediately from the invariance of (19) of §11.6.

2. Inserting $\BB{}'=0$ into (17) of §11.2. we get \begin{equation} v = |\vv| = \frac{c^2|\BB|}{|\EE|} \end{equation} which is only possible if $\cc|\BB|<|\EE|$. This condition also follows immediately from the invariance of (19) of §11.6.

3. This is not possible; if the electric and magnetic fields are both zero in any frame, they are zero in all frames.

4. No; the conditions in the first two problems can not both be satisfied.