Question

If a charged particle goes unaccelerated in a region containing electric and magnetic fields, then $($more than one may be correct$)$

• A. $\overrightarrow{E}$ must be perpendicular to $\overrightarrow{B}$
• B. $\overrightarrow{v}$ must be perpendicular to $\overrightarrow{B}$
• C. $\overrightarrow{v}$ must be perpendicular to $\overrightarrow{E}$
• D. $\overrightarrow{E}$ must be parallel to $\overrightarrow{B}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\overrightarrow{E}$ must be perpendicular to $\overrightarrow{B}$
B $\overrightarrow{v}$ must be perpendicular to $\overrightarrow{B}$
C $\overrightarrow{v}$ must be perpendicular to $\overrightarrow{E}$

Charge particle in a region goes unaccelerated only if magnetic force and electric force action on it cancle each other.
therefore, $q\overrightarrow{E}=q(\overrightarrow{v}\times \overrightarrow{B})$
therefore, $\overrightarrow{E}$ is definitely perpendicular to $\overrightarrow{B}$
there is no necessity of $\overrightarrow{v}$ being perpendicular to $\overrightarrow{B}$ but $\overrightarrow{v}$ should not be parallel to $\overrightarrow{B}$ otherwise magnetic force will be zero leading to imbalancing the effect of electric force and particle will accelerate therefore, option(A) is true but option (B) is not true from this point of view which may be a mistake.
But since magnetic force will always perpendicuar to the direction of motion of charge particle therefore, Electric force should also be perpendicular and opposite to the magnetic force to balance the magnetic force therefore, $\overrightarrow{v}$ should be perpendicular to $\overrightarrow{E}$
and since $\overrightarrow{E}$ and $\overrightarrow{B}$ are perpendicular to each other
therefore, $\overrightarrow{v},\overrightarrow{E},and\overrightarrow{B}$ all are mutually perpendicular to each other.