A proton moving with a constant velocity passes through a region of space without any change in its velocity. If E⃗ and B⃗ represent the electric and magnetic fields respectively, then the region of space may have :
(A) E=0, B=0<?p>
(B) E=0, B ≠ 0
(C) E ≠ 0, B=0
(D) E ≠ 0, B ≠ 0
Choose the most appropriate answer from the options given below :

Solution

Physics sir · 2 steps

No change in velocity means zero net force, so check each case for a way to make the total force vanish.

1. Condition. Velocity stays constant only if the net force on the proton is zero. The total force is the Lorentz force, so qE⃗ + q v⃗ × B⃗ must equal zero.

   F⃗ = q(E⃗ + v⃗ × B⃗) = 0

   Case	Electric force	Magnetic force	Can net force be zero?
   (A) E = 0, B = 0	0	0	Yes, no field at all
   (B) E = 0, B ≠ 0	0	qvB sin θ, zero if v⃗ ∥ B⃗	Yes, when the proton moves along B⃗
   (C) E ≠ 0, B = 0	qE, never zero	0	No, nothing cancels qE⃗
   (D) E ≠ 0, B ≠ 0	qE	qvB	Yes, when E⃗ = -v⃗ × B⃗ (velocity selector)

2. So. Cases (A), (B) and (D) are possible, while (C) is not. The answer is option (c).

The idea to keep. An electric field always pushes a charge, moving or not. A magnetic field pushes only when v⃗ is not along B⃗. For zero net force, an electric force needs a magnetic force to cancel it.

Answer from the NTA final answer key. Solution by PrepWiser.