Formula & Calculator
Magnetic Force on a Moving Charge
Calculates the magnetic force on a charged particle moving through a magnetic field, based on its charge, speed, field strength, and angle of motion.
Interpretation
Magnetic force on a moving charge: F = q·v·B·sinθ, where q is charge, v is velocity, B is magnetic field, θ is angle between v and B. It is perpendicular to both v and B. Example: q=1.6e-19 C, v=3e6 m/s, B=0.5 T, θ=90° → F = 2.4e-13 N.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| F | Magnetic force on the charge | N |
| q | Magnitude of the electric charge | C |
| v | Speed of the charge | m/s |
| B | Magnetic field strength | T |
| theta | Angle between the velocity and the magnetic field | degrees |
What it means
The magnetic force on a moving charge is given by the Lorentz force law component: F = q (v × B), with magnitude F = qvB sinθ. This force is always perpendicular to the velocity and the magnetic field, causing the charge to move in a circular or helical path. This principle is used in mass spectrometers, cyclotrons, and particle accelerators. It also explains the Hall effect and the deflection of electrons in cathode ray tubes. In engineering, magnetic forces are used in motors and actuators. Understanding this force is essential for designing devices that manipulate charged particles and for studying plasmas.
Worked example
Magnetic Force on a Charge – Two Examples
Real‑World| Parameter | Value |
|---|---|
| q | 1.6×10⁻¹⁹ C |
| v | 1×10⁶ m/s |
| B | 0.5 T |
| θ | 90° |
| Parameter | Value |
|---|---|
| v | 2×10⁶ m/s |
| B | 1 T |
Common mistakes
- Charge q: In coulombs (C) – could be positive or negative.
- Velocity v: Vector – direction relative to B matters.
- Magnetic field B: In teslas (T).
- Angle θ: The angle between v and B – force is zero if parallel (θ=0 or 180°).
- Force direction: Perpendicular to both v and B (right‑hand rule) – not along v or B.
Applications
The magnetic force on a moving charge, F = q·v·B·sinθ, is the Lorentz force. It is the basis for many devices, including cathode ray tubes, mass spectrometers, and particle accelerators. Engineers and physicists use it to design magnetic confinement in fusion reactors, to guide charged particles in cyclotrons, and to develop magnetic sensors. In aerospace, it is used in ion propulsion. In materials science, it helps characterise materials via magnetoresistance. By understanding the magnetic force, professionals can control charged particle beams and exploit the interaction between charges and magnetic fields for technological innovation.
- Design of cathode ray tubes and electron microscopes
- Mass spectrometry and isotope separation
- Particle accelerator and cyclotron design
- Ion thrusters for spacecraft propulsion
- Magnetic field measurement devices (Hall effect sensors)
Frequently Asked Questions
A charge q moving with velocity v in a magnetic field B experiences a force F = q·v·B·sinθ, where θ is the angle between v and B. The direction is given by the right‑hand rule (perpendicular to both v and B).
Forgetting that a charge moving parallel to the field (θ=0°) experiences zero force, regardless of speed. Also, using the wrong sign for the charge (positive or negative).
The force is in newtons (N).
The magnetic force is always perpendicular to the velocity, so it changes the direction of motion but not the speed. This results in circular or helical motion.
For a particle in a uniform magnetic field, the angular frequency is ω = qB/m, and the radius is r = mv/(qB). The frequency is independent of speed.
Electric force is parallel to the electric field and does work (changes speed). Magnetic force is perpendicular to the field and velocity, does no work (only changes direction).
Ions are accelerated and then bent by a magnetic field. The radius of curvature depends on the mass‑to‑charge ratio, allowing separation of isotopes.
The total force on a charge in electric and magnetic fields is F = q(E + v×B). This is the Lorentz force law, combining both effects.