In a region of free space, where a uniform and constant electric field of intensity $\vec{E}$ and a magnetic field of induction $\vec{B}$ coexist, an electron projected with speed v in the positive $x -$ direction, moves undeviated without change in speed. In addition, an electron projected with velocity v in the positive y-direction in this region; also, moves undeviated without change in speed. Which of the following conclusions can you make?

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$| \vec{E} | = v | \vec{B} |$

$| \vec{E} | < v | \vec{B} |$

The magnetic field is either at $45^{0}$ or $135^{0}$ with the positive x-direction

The electric field is perpendicular to the x-y plane

answer is A, C, D.

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Detailed Solution

$\bar{E} + \bar{V} \times \bar{B} = \bar{0} \Rightarrow \bar{E}$ is $⊥^{r}$ to plane of $\bar{V} & \bar{B}$

$\begin{array}{l} is a long X - axis is in Y - Z plane \\ is a long Y - axis is in X - Z plane \end{array}} \Rightarrow \bar{E} i s a l o n g z - a x i s (⊥^{r} t o x - y p l a n e)$

$∴ \tan θ = 1 \Rightarrow θ = \frac{π}{4}, \frac{3 π}{4}$

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