A charged particle enters a region with mutually perpendicular electric and magnetic fields E and B with velocity v = E / B, but also has a non-zero velocity component v_parallel along the direction of B. What path does the particle follow?

Answer and explanation

Correct answer: D. A uniform helix with axis along B

  1. Step 1: In the plane perpendicular to B, the electric force qE and magnetic force qv_perp*B balance because v_perp = E/B, so there is no net transverse acceleration.
  2. Step 2: Parallel to B, the magnetic force is zero because v_parallel is parallel to B.
  3. Step 3: If there is no parallel electric field, the parallel velocity v_parallel remains constant.
  4. Step 4: Constant motion parallel to B combined with circular or uniform transverse progression produces a uniform helical trajectory along B.
Answer
A uniform helix with axis along B.
Why not the others: a pure circle requires zero velocity parallel to B; a parabola requires an unbalanced electric force; a straight line requires zero transverse circular frequency.

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