A negative charge moving with a constant velocity V enters a uniform magnetic field directed out of the page. What is the direction of the Lorentz force on the charge?

Question

BlackTom AI · Accepted Answer

F = q (v × B). For a charged particle moving in a magnetic field the force is perpendicular to both v and B and given by that cross product; for a negative charge the sign of q reverses the direction found by the right‑hand rule.

1) Relevant concepts
- Lorentz (magnetic) force: F = q(v × B).
- Direction of v × B found with the right‑hand rule: thumb along v, index along B, middle finger gives v × B for a positive q.
- If q is negative, the actual force is opposite to v × B.
- Magnitude: |F| = |q| v B sinθ; here v ⟂ B so sinθ = 1 and |F| = |q| v B.

2) Step‑by‑step reasoning
- Define axes: take the page plane as x–y, with up the page = +y and right = +x. Out of the page is +z.
- v vector: upward = +ŷ.
- B vector: out of the page = +ẑ.
- Compute v × B: ŷ × ẑ = x̂ (standard cyclic relations x×y=z, y×z=x, z×x=y).

3) Intermediate calculation
- v × B = v B (ŷ × ẑ) = v B x̂.
- For a positive charge q this would give F = q v B x̂ (i.e., to the right).
- For a negative charge q = −|q|, F = q(v × B) = −|q| (v B x̂) = −|q| v B x̂ = (−x̂) |q| v B, i.e., to the left.

4) Why the correct answer is correct
- The cross product gives a rightward direction for a positive charge; because the charge is negative the force reverses and points left. The magnitude is nonzero because v and B are perpendicular.

5) Why other options are incorrect
- a (top): that would be along v, but magnetic force is perpendicular to v, so incorrect.
- b (no force): incorrect because v ⟂ B so magnetic force is maximal, not zero.
- c (bottom): opposite of v; force is perpendicular to v so bottom is wrong.
- d (right): would be correct for a positive charge, but here the charge is negative so direction is reversed to left.

6) Conclusion
- Therefore the Lorentz force on the negative charge is to the left, matching the provided answer e.

A negative charge moving with a constant velocity V enters a uniform magnetic field directed out of the page. What is the direction of the Lorentz force on the charge?Single choice

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