JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Magnetic Effects of Current

For the velocity to stay constant, the net force must vanish: $q\vec{E}+q\vec{v}\times \vec{B}=0$. This holds for:

(A) $\vec{E}=0,\vec{B}=0$ — trivially.

(B) $\vec{E}=0,\vec{v}\parallel \vec{B}$ (so $\vec{v}\times \vec{B}=0$).

(D) $\vec{E}=0,\vec{v}\times \vec{B}\ne 0$, replaced instead by $\vec{E}=-\vec{v}\times \vec{B}$ with both fields non-zero (the velocity-selector condition).

Case (C), $\vec{E}\ne 0,\vec{B}=0$, is impossible on its own: with no magnetic force to cancel it, a non-zero electric field would accelerate the proton. So only (A), (B) and (D) are possible, matching option (3) of the source, i.e. option C here.