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Magnetic Force on Charges & Currents

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Force on a moving charge

A charge moving through a magnetic field feels a force — but only if it moves across the field lines. The magnitude is F = qvB sinθ, where θ is the angle between the velocity and the field. The force is maximum when the charge moves perpendicular to the field (θ = 90°) and zero when it moves parallel (θ = 0°). The direction is the strangest part: the force is perpendicular to both the velocity and the field, given by a right-hand rule (fingers point along v, curl toward B, thumb gives the force on a positive charge — reverse it for a negative charge).

Why it makes circles — and does no work

Because the magnetic force is always perpendicular to the velocity, it can never speed a particle up or slow it down — it only bends the path. A force perpendicular to motion does zero work and changes only direction, not speed. In a uniform field a charged particle therefore travels in a circle (or a helix), with the magnetic force supplying the centripetal force. A parallel current-carrying wire feels a related force, F = BIL, where L is the length of wire in the field.

Magnetic force on a charge and on a wire
F = q · v · B · sinθ · F = B · I · L
θ is the angle between v and B. Maximum force at θ = 90° (sinθ = 1); zero force at θ = 0°. For a wire, L is the length within the field and I the current.
Worked example

A proton (q = 1.6 × 10⁻¹⁹ C) moves at 2.0 × 10⁶ m·s⁻¹ perpendicular to a 0.50 T magnetic field. Find the magnetic force on it.

  1. 1.Perpendicular motion means θ = 90°, so sinθ = 1 and F = qvB.
  2. 2.Substitute: F = (1.6 × 10⁻¹⁹ C)(2.0 × 10⁶ m·s⁻¹)(0.50 T).
  3. 3.Multiply the first two: 1.6 × 10⁻¹⁹ × 2.0 × 10⁶ = 3.2 × 10⁻¹³.
  4. 4.Multiply by B: 3.2 × 10⁻¹³ × 0.50 = 1.6 × 10⁻¹³ N.
Answer: 1.6 × 10⁻¹³ N, directed perpendicular to both the velocity and the field (curving the proton into a circular path).
Checkpoint

A proton moves at 2.0 × 10⁶ m·s⁻¹ perpendicular to a 0.50 T magnetic field. What is the magnitude of the magnetic force on it? (q = 1.6 × 10⁻¹⁹ C)

Watch out

A magnetic field can change a charged particle’s direction but never its speed or kinetic energy, because the force is always perpendicular to the velocity. If a problem claims a magnetic field “speeds up” a particle, be suspicious — that job belongs to an electric field.

Checkpoint

A charged particle moves exactly parallel to a uniform magnetic field. What magnetic force does it experience?

On the exam

For any magnetic-force problem, first find the angle between v and B. Parallel → no force; perpendicular → maximum force qvB. The direction always comes from a right-hand rule, remembering to flip it for a negative charge.

Answer the 2 checkpoints as you read.

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