An electron moving in the positive y direction, at right angles to a magnetic field, experiences a magnetic force in the negative x direction. What is the direction of the magnetic field?

Question

BlackTom AI · Accepted Answer

Magnetic force: F = q(v × B). For an electron q = −e, so the direction of v × B is opposite the direction of the observed force.

1. Relevant concepts
- Lorentz force for magnetic part: F = q(v × B).
- Cross product direction: use right-hand rule for v × B (for a positive charge). For an electron (negative), the actual force is opposite to v × B.
- Coordinate unit vectors: +x = i, +y = j, +z = k.

2. Step-by-step reasoning
- Given: v is along +y (v = +j). Observed magnetic force F_obs is along −x (F_obs = −i).
- For the electron: F_obs = q(v × B) = −e (v × B). Rearranging, v × B = F_obs / (−e) = (−i)/(−e) = (+i)/e, so v × B must point in the +x direction.

3. Calculations and intermediate steps
- Let B = a i + b j + c k. Compute v × B:
  v × B = j × (a i + b j + c k) = a (j × i) + b (j × j) + c (j × k)
  Using cross-product identities: j × i = −k, j × j = 0, j × k = i
  So v × B = a(−k) + 0 + c(i) = c i − a k.
- We require v × B = +i, so c = 1 (positive) and a = 0. The b component is irrelevant (j × j = 0), so B = 0 i + b j + 1 k. Any j component doesn’t affect the cross product; the essential part is a positive k component.

4. Why the correct answer is correct
- The positive z-component (k) of B produces j × k = i, giving v × B in +x. The electron’s negative charge reverses the sign, producing the observed −x force. Therefore B must have a +z direction.

5. Why other options are incorrect
- −x or +x: would give v × B with k components or zero but not produce the required +x before sign flip.
- −y: B along −y is parallel to v, so j × (±j) = 0; no force would result.
- “None of these”: incorrect because +z matches the required direction.

6. Conclusion
B must point in the +z direction. This confirms the provided answer: +z direction.

An electron moving in the positive y direction, at right angles to a magnetic field, experiences a magnetic force in the negative x direction. What is the direction of the magnetic field?单项选择题

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