3.7.4.2FieldsStretch

Force on a Moving Charge

A current is just charge in motion, so it should be no surprise that a lone moving charge — not confined to a wire at all — feels a force in a magnetic field too. But because that force is always perpendicular to the charge’s velocity, it never speeds the charge up or slows it down — it just steers it, bending its path into a perfect circle.

40 min Video by Science Shorts 3.7.4 Magnetic fields

Charges in Magnetic Fields — A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the equation $F = B Q v sin θ$ for the force on a moving charge
Explain why the magnetic force on a moving charge never does work on it
Explain why a charged particle moving perpendicular to a uniform field follows a circular path
Derive and use the radius equation $r = \frac{m v}{B Q}$
Describe an application of charged particles moving in magnetic fields

Force on a moving charge

A charge $Q$ moving with speed $v$ through a magnetic field of flux density $B$ experiences a force, whose size depends on the angle $θ$ between the velocity and the field. As with the force on a current-carrying wire, the force is greatest when velocity is perpendicular to the field and zero when the charge moves exactly along the field lines.

F = B Q v sin θ

Force on a moving charge (θ = angle between velocity and field).

1Perpendicular means

θ = 90°

, so

sin θ = 1

2Use

F = B Q v

F = 0.20 \times 1.60 \times 1 0^{- 19} \times 2.0 \times 1 0^{6}

Answer

F = 6.4 \times 1 0^{- 14}

Tip — The direction is again found with Fleming’s left-hand rule, using the direction of conventional current — for a negative charge like an electron, this means using the direction OPPOSITE to its actual velocity as your "current" direction.

Circular motion in a magnetic field

Because the magnetic force on a moving charge is always perpendicular to its velocity, it can never do any work on the charge — it changes the $direction$ of motion but never the $speed$ . A force that is always perpendicular to velocity, with constant magnitude (true here for motion perpendicular to a uniform field, since $v$ and $θ$ stay constant through the turn), is exactly the condition for $circular motion$ — so the magnetic force provides the centripetal force needed, and the charge follows a perfect circular path.

B Q v = \frac{m v ^{2}}{r} \Rightarrow r = \frac{m v}{B Q}

Equating magnetic force to centripetal force gives the radius of the circular path.

1Use

r = \frac{m v}{B Q}

r = \frac{1.67 \times 1 0 ^{- 27} \times 3.0 \times 1 0 ^{6}}{0.40 \times 1.60 \times 1 0 ^{- 19}}

3Numerator:

5.01 \times 1 0^{- 21}

. Denominator:

6.4 \times 1 0^{- 20}

Answer

r \approx 0.078

m (7.8 cm)

Tip — Since r = mv/(BQ), a more massive (or faster) particle curves in a LARGER circle for the same field and charge — heavier particles are "harder to turn," exactly as intuition suggests.

Applications

This principle underlies devices from mass spectrometers (which sort ions by the radius of their circular path, revealing their mass-to-charge ratio) to particle detectors in accelerators, where the curvature of a charged particle’s track in a known magnetic field reveals its momentum, and the direction of curvature reveals the sign of its charge.

Equation recap

F = B Q v sin θ

Force on a moving charge.

r = \frac{m v}{B Q}

Radius of circular path in a uniform field (v ⊥ B).

Common mistakes to avoid

Thinking the magnetic force changes a charged particle’s speed.

The magnetic force is always perpendicular to velocity, so it does no work and never changes speed — only direction.

Using the actual velocity direction of a negative charge (like an electron) directly in Fleming’s left-hand rule.

Fleming’s left-hand rule uses the direction of CONVENTIONAL current — for a negative charge, this is opposite to its actual direction of travel.

Assuming a heavier particle curves in a smaller circle than a lighter one, for the same speed, charge and field.

r = mv/(BQ) increases with mass — a heavier particle actually curves in a LARGER circle (it is harder to deflect).

Key takeaways

Force on a moving charge: F = BQv sinθ, maximum when velocity is perpendicular to the field, zero when parallel.
The magnetic force on a moving charge is always perpendicular to its velocity, so it changes direction but never speed (does no work).
A charge moving perpendicular to a uniform field follows a circular path, radius r = mv/(BQ).
This principle is used in mass spectrometers and particle detectors to determine mass-to-charge ratio, momentum and charge sign.

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Force on a Moving Charge

40 min Video by Science Shorts 3.7.4 Magnetic fields