Explain why a charged particle moving parallel to a uniform magnetic field experiences no magnetic force, whilst the same particle moving perpendicular to the field experiences a force that does no work on the particle.

IB DP Physics Higher Level (2023 syllabus) — D.2 Electric and magnetic fields · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

Model answer (4 marks)

The magnetic force on a charged particle is given by \(\mathbf F=q\mathbf v\times\mathbf B\). The magnitude is \(F=qvB\sin\theta\), where \(\theta\) is the angle between the velocity and the field.

1. If the particle moves parallel to \(\mathbf B\) then \(\theta=0°\) (or 180°), so \(\sin\theta=0\) and \(\mathbf F=0\). Hence no magnetic force.

2. If the particle moves perpendicular to \(\mathbf B\) then \(\theta=90°\) and \(\sin\theta=1\), giving a force of magnitude \(qvB\). The direction of \(\mathbf F\) is perpendicular to \(\mathbf v\) (centripetal).

3. Because the force is always perpendicular to the displacement, the work done is \(W=\mathbf F\cdot\mathbf d=Fd\cos90°=0\).

4. Therefore the kinetic energy and speed of the particle remain unchanged.

Examiner tips

  • Use the cross‑product formula and state the angle dependence; mention sin θ=0 for parallel motion and sin θ=1 for perpendicular motion.
  • Explain that the force is perpendicular to velocity, so work is zero – this shows understanding of work–force relationship.

Common mistakes

  • Confusing parallel motion with zero field; forgetting that sin 0=0.
  • Claiming the force is zero for perpendicular motion or that work is non‑zero; mis‑applying the dot product.

Mark scheme (4 marks)

  1. The magnetic force on a charged particle is given by F = qv × B, so the force depends on the component of velocity perpendicular to the field.
  2. When the particle moves parallel to the field, the angle between v and B is 0° (or 180°), so sin θ = 0 and the force is zero.
  3. When the particle moves perpendicular to the field, the magnetic force is always directed perpendicular to the velocity (centripetal direction).
  4. Because the force is perpendicular to the displacement (velocity), the work done W = F · d = Fd cos 90° = 0, so the particle's kinetic energy and speed remain constant.

Key terms in this question

magnetic force · perpendicular · work

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