Magnetic Force on Moving Charge Calculator

Calculate the magnetic force on a moving charged particle using F = qv × B.

Proton/electron: ±1.6×10⁻¹⁹ C
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How a Magnetic Field Deflects a Moving Charge

A magnetic field exerts force only on the component of a charged particle's motion that is perpendicular to the field. That is why the magnitude contains sinθ. Motion parallel or antiparallel to B gives zero magnetic force, while motion at 90° gives the maximum magnitude qvB. At intermediate angles, only v sinθ contributes to the sideways deflection.

The force direction is perpendicular to both velocity and magnetic field. For a positive charge, use the right-hand rule for v × B; for a negative charge, reverse that direction. Because the magnetic force is perpendicular to the instantaneous velocity, it changes the direction of motion without directly changing the particle's speed or kinetic energy.

F = |q|vB sinθ
SymbolMeaningWhy it appears / units
qParticle chargeCoulombs; the sign determines force direction.
vParticle speedm/s; only the perpendicular component contributes.
BMagnetic flux densityTesla (T); stronger fields produce stronger deflection.
θAngle between v and BUse the angle between the two vectors, not an angle to a diagram axis.

When velocity is exactly perpendicular to a uniform magnetic field, the force acts as a centripetal force and the particle follows a circular path. If the velocity also has a parallel component, the perpendicular component produces circular motion while the parallel component continues unchanged, creating a helix around the magnetic-field direction.

Worked Examples

Example 1: Proton in MRI: q=1.6e-19, v=1e6, B=3T, θ=90°
F=1.6e-19×1e6×3×1
Result: 4.8×10⁻¹³ N
Circular motion in MRI magnetic field
Example 2: Electron in Earth's field: B=5e-5 T
F=1.6e-19×1e7×5e-5×sin90°
Result: 8×10⁻¹⁷ N
Causes aurora when deflected to poles
Example 3: Force at a 30° angle
|q|=3×10−6C, v=2000m/s, B=0.40T, θ=30° → F=|q|vB sin30°
Result: 1.20×10−3 N
Since sin30°=0.5, the force is half the value it would have for perpendicular motion.
Example 4: Motion parallel to the field
θ=0° → sin0°=0
Result: F=0 N
A charged particle can move through a magnetic field without magnetic deflection when its velocity is exactly parallel to B.

Common Mistakes

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Using cosθ instead of sinθ

The magnetic force depends on the perpendicular velocity component, v sinθ. Using cosine reverses the zero-force and maximum-force cases.

⚠️
Using charge sign inside the magnitude calculation

Use |q| for the magnitude. The positive or negative sign is needed when determining the force direction with the right-hand rule.

⚠️
Assuming magnetic force speeds the particle up

A purely magnetic force is perpendicular to velocity, so it bends the trajectory but does no work on the particle.

Frequently Asked Questions

When is magnetic force maximum?
When velocity is perpendicular to B (θ=90°). Force is zero when particle moves parallel to field (θ=0°).
Does magnetic force do work?
No — magnetic force is always perpendicular to velocity. It changes direction of motion but not speed or kinetic energy.
What happens if the particle velocity has both parallel and perpendicular components?
The perpendicular component experiences magnetic force and produces circular motion, while the parallel component is unaffected by the magnetic field. Combining those motions gives a helical path around the field lines. The helix becomes tighter for stronger B and wider for larger perpendicular momentum.
Why does a negative charge bend in the opposite direction?
The vector law is F=q(v×B). The cross product v×B gives the force direction for a positive charge. Multiplying by a negative q reverses that vector, so an electron and a positive particle moving with the same velocity in the same field curve oppositely.
Can a stationary charge feel magnetic force?
Not from the magnetic qvB force, because v=0 makes the force zero. A stationary charge can still experience an electric force if an electric field is present. This difference is why the full Lorentz force combines electric and magnetic contributions as F=q(E+v×B).
What units should magnetic field B use?
Use tesla when applying F=|q|vB sinθ with SI units. If a field is given in gauss, convert first: 1 T = 10,000 G. Earth's magnetic field is only tens of microtesla, while laboratory magnets and MRI systems can reach values measured in tesla.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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