Magnetic Force on a Moving Charge
Calculate the force on a charged particle moving through a magnetic field.
The basis of electric motors and particle accelerators.
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The Formula
F = qvB × sin(θ)
A charged particle moving through a magnetic field experiences a force perpendicular to both its velocity and the field. This force changes the particle's direction but not its speed.
Variables
| Symbol | Meaning |
|---|---|
| F | Magnetic force (Newtons) |
| q | Electric charge of the particle (Coulombs) |
| v | Velocity of the particle (m/s) |
| B | Magnetic field strength (Tesla) |
| θ | Angle between velocity and magnetic field |
Example 1
A proton (q = 1.6 × 10⁻¹⁹ C) moves at 2 × 10⁶ m/s perpendicular to a 0.5 T field
θ = 90°, so sin(90°) = 1
F = 1.6 × 10⁻¹⁹ × 2 × 10⁶ × 0.5 × 1
F = 1.6 × 10⁻¹³ N
Example 2
An electron (q = 1.6 × 10⁻¹⁹ C) moves at 3 × 10⁵ m/s at 30° to a 0.2 T field
F = 1.6 × 10⁻¹⁹ × 3 × 10⁵ × 0.2 × sin(30°)
F = 1.6 × 10⁻¹⁹ × 3 × 10⁵ × 0.2 × 0.5
F = 4.8 × 10⁻¹⁵ N
When to Use It
Use the magnetic force formula when:
- Calculating forces on current-carrying wires in motors
- Understanding how particle accelerators bend charged beams
- Analyzing the motion of charged particles in magnetic fields
- Designing mass spectrometers and cyclotrons
Key Notes
- Formula: F = qvB sinθ: F is the magnetic force (N), q is the charge (C), v is the speed (m/s), B is the magnetic field (T), and θ is the angle between the velocity and the field.
- Direction from the right-hand rule: Point fingers in the direction of v, curl them toward B, and the thumb points in the direction of F for positive charges. For negative charges, the force is reversed.
- Maximum and zero force: Force is maximum (F = qvB) when v ⊥ B (θ = 90°). Force is zero when v ∥ B (θ = 0° or 180°). A charge moving parallel to a magnetic field experiences no force.
- Magnetic force does no work: The magnetic force is always perpendicular to the velocity, so it changes the direction of motion but never the speed or kinetic energy of a charged particle.
- Circular motion in a uniform field: A charged particle moving perpendicular to a uniform magnetic field travels in a circle with radius r = mv/(qB). Particle accelerators and mass spectrometers use this principle.