Consider an inertial system $\mathcal{O}$ and a Lorentz boosted system $\mathcal{O}'$, moving with a velocity $\vec{v}$ with respect to $\mathcal{O}$. Then we have expressions for the electromagnetic fields as follows: $$\vec{B}=\gamma\vec{B'}+\frac{\vec{v}}{v^2}(\vec{v}\cdot\vec{B'})(1-\gamma)+\gamma\frac{\vec{v}}{c}\times\vec{E'}$$ $$\vec{E}=\gamma\vec{E'}+\frac{\vec{v}}{v^2}(\vec{v}\cdot\vec{E'})(1-\gamma)-\gamma\frac{\vec{v}}{c}\times\vec{B'} $$ Now, I want to find the condition that $\vec{E'}$ and $\vec{B'}$ have to satisfy such that there exists a $\vec{v}$ such that $\vec{E}=0$. I reckoned that the third term $\vec{v}\times\vec{B'}$ is perpendicular to the second term $\vec{v}$, so those two cannot cancel each other. However, how can I formulate these conditions in terms of $\vec{E'}$ and $\vec{B'}$?
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