Questions: Magnetic Force on Moving Charges (Lorentz Force)

Question 1

A proton moves through a uniform magnetic field. What happens to its kinetic energy?

• A. It increases because the magnetic force does positive work in the direction of motion
• B. It remains constant — the magnetic force is always perpendicular to the velocity and does no work
• C. It decreases because the field redirects the proton and steals kinetic energy
• D. It oscillates, alternately increasing and decreasing as the proton curves

Answer: (B)

Work equals force dotted with displacement: W = F⃗ · dr⃗. The magnetic force F⃗ = q(v⃗ × B⃗) is always perpendicular to v⃗ (a property of the cross product), and since the particle's displacement is always along v⃗, the dot product is always zero. No work is done, so kinetic energy — and therefore speed — cannot change. The magnetic force is a perpetual steering mechanism: it deflects the particle's direction without ever accelerating or decelerating it.

Question 2

A particle accelerator must both increase a charged particle's energy and bend its path in a circle. How must it use electric and magnetic fields?

• A. Use magnetic fields for energy gain; use electric fields for bending the circular path
• B. Use only magnetic fields — they can do both since they exert force on moving charges
• C. Use electric fields for energy gain; use magnetic fields for bending the path
• D. Use only electric fields — magnetic fields cannot exert force on charged particles

Answer: (C)

Electric fields do work on charges (force parallel to motion is possible), so they can increase kinetic energy. Magnetic fields do no work, so they cannot increase kinetic energy — they can only change the direction of motion. This is why modern accelerators like synchrotrons use radiofrequency electric cavities to accelerate particles and strong dipole magnets to bend the beam in a closed circular path. Magnetic fields are perfect for steering precisely because they don't alter the particle's energy.

Question 3

A stationary charged particle placed in a strong magnetic field will be accelerated by the magnetic force.

• T. True
• F. False

Answer: False

The Lorentz magnetic force is F⃗ = q(v⃗ × B⃗). If the particle is stationary, v⃗ = 0, so F⃗ = 0 regardless of how strong B⃗ is. The magnetic force requires motion to exist at all — it is velocity-dependent. This is why magnetic fields alone cannot start a particle from rest; an electric field is needed to provide the initial impulse. Once the particle is moving, the magnetic force can redirect it but still cannot change its speed.

Question 4

A charged particle moving exactly parallel to a magnetic field (velocity vector parallel to B⃗) experiences no magnetic force.

• T. True
• F. False

Answer: True

The cross product v⃗ × B⃗ equals |v||B|sin θ, where θ is the angle between v⃗ and B⃗. When v⃗ ∥ B⃗, θ = 0° and sin 0° = 0, so the force is zero. The particle continues in a straight line at constant velocity — completely unaffected by the field. This is why charged particles spiral along magnetic field lines: the component of velocity parallel to B⃗ is unaffected, while the perpendicular component undergoes circular motion, combining to produce the helix.

Question 5

Why does the magnetic force on a moving charge produce circular (or helical) motion rather than linear acceleration? Explain using the direction of the force relative to velocity.

Model answer: The magnetic force F⃗ = q(v⃗ × B⃗) is always perpendicular to the velocity (by the definition of the cross product). A force perpendicular to velocity changes the direction of motion but not its magnitude — this is precisely the condition for circular motion, where the centripetal force points radially inward while velocity remains tangential. Setting qvB = mv²/r gives the cyclotron radius r = mv/(qB). If there is also a velocity component along B⃗, that component is unaffected (force is zero for it), so the particle traces a helix.

This perpendicularity is the key geometric fact underlying all magnetic force effects. It also explains why magnetic forces do no work: power = F⃗ · v⃗ = 0 when F⃗ ⊥ v⃗. The cyclotron radius formula r = mv/(qB) has immediate practical applications: larger mass m means harder to deflect (larger radius), stronger field B means tighter curve (smaller radius). Mass spectrometers exploit this — different isotopes with different m/q ratios follow arcs of different radii, separating them spatially.