Magnetic Force on a Moving Charge Calculator
Calculate the magnetic (Lorentz) force on a charged particle moving in a magnetic field. Enter the charge, speed, flux density and the angle between velocity and field to get the force in newtons.
Find the Force on a Charged Particle in a Magnetic Field
Magnetic Force on a Charge
Where:
- \(F\) — magnetic force magnitude (N)
- \(|q|\) — magnitude of the particle's charge (C)
- \(v\) — speed of the particle (m/s)
- \(B\) — magnetic flux density, i.e. field strength (T)
- \(\theta\) — angle between the velocity vector and the magnetic field (degrees or radians)
This is the magnetic part of the Lorentz force: it gives the magnitude of the push a magnetic field exerts on a moving charge. The force is largest when the particle moves perpendicular to the field (\(\theta = 90^\circ\)) and vanishes when it moves parallel to it (\(\theta = 0^\circ\) or \(180^\circ\)).
Magnetic Force on a Charge – Calculation Example
Example: a particle with a charge of 1 µC moves at 500 m/s through a 10 mT magnetic field, perpendicular to the field lines (\(\theta = 90^\circ\)).
\[ F = |q|\, v\, B\, \sin(\theta) = (1\times10^{-6}\ \text{C})(500\ \text{m/s})(10\times10^{-3}\ \text{T})\sin(90^\circ) \]
\[ F = (1\times10^{-6})(500)(0.01)(1) = 5\times10^{-6}\ \text{N} \]
Result: the magnetic force is \(5\times10^{-6}\) N, or 5 µN, directed perpendicular to both the velocity and the field.
How to use the calculator
- Enter the Charge (q) — the magnitude of the particle's charge — and choose its unit (nC, µC, mC or C).
- Enter the Speed (v) of the particle and select the unit (m/s, km/s, km/h or ft/s).
- Enter the Magnetic Flux Density (B) and pick the unit (mT, T, G or µT).
- Enter the Angle (θ) in degrees between the velocity vector and the magnetic field (0° to 180°).
- Read the Magnetic Force Magnitude in newtons, along with the step-by-step calculation details.
What is the magnetic force on a moving charge?
A magnetic field exerts no force on a charge that sits still, and none on a charge that moves exactly along the field lines. But a charge moving at an angle to the field experiences a sideways push known as the magnetic (Lorentz) force. Its magnitude depends on three quantities multiplied together: how much charge is moving, how fast it is going, and how strong the field is — scaled down by \(\sin(\theta)\), which accounts for the angle between the velocity and the field.
The direction of the force is what makes magnetism unusual: it is always perpendicular to both the velocity of the particle and the magnetic field. You find it with the right-hand rule — point your fingers along the velocity, curl them toward the magnetic field, and your thumb gives the force direction for a positive charge. For a negative charge such as an electron, the direction is reversed.
Because the force is always perpendicular to the velocity, it does no work on the particle: it cannot speed the particle up or slow it down. Instead, it continuously bends the path. A charge moving perpendicular to a uniform field travels in a circle; at an arbitrary angle it follows a helix, spiraling around the field lines. This is the principle behind particle accelerators, mass spectrometers, cyclotrons and the magnetic confinement of plasma in fusion reactors.
For comparison, an electric field pushes a charge along the field lines regardless of its motion — see the Point Charge Electric Field calculator. The full Lorentz force combines both effects, \( F = q(E + v \times B) \); this calculator covers only the magnetic term \( qv \times B \).
Assumptions and limitations
- Only the magnetic part of the Lorentz force is computed. Any electric field present is ignored, as is the radiation a rapidly accelerating charge may emit.
- The field is treated as uniform at the particle's location. In strongly non-uniform fields the force varies along the trajectory.
- Relativistic effects are neglected. The formula \( F = |q|vB\sin(\theta)\) holds exactly for the coordinate speed at any velocity, but Newtonian trajectory analysis breaks down near the speed of light.
- Only magnitudes are entered. The calculator returns the force magnitude; the sign of the charge merely reverses the force direction, so use the right-hand rule to find it.
- Special angles: at \(\theta = 0^\circ\) or \(180^\circ\) the particle moves parallel to the field and the force is zero; at \(\theta = 90^\circ\) the force is at its maximum, \( F = |q|vB \).
Typical field strengths for reference
| Source | Approximate B |
|---|---|
| Earth's magnetic field at the surface | 25–65 µT |
| Refrigerator magnet | ~5 mT |
| Neodymium magnet near its pole | ~0.5 T |
| MRI scanner | 1.5–3 T |
| Strongest laboratory pulsed magnets | up to ~100 T |
With such a wide range of field strengths, unit choice matters — always check that the selected unit (Gauss, mT, µT or T) matches your data. 1 T = 10,000 G. If you are studying other forces on a moving object, try the Drag Force calculator, the Gravitational Force calculator, or the Projectile Motion calculator for trajectories driven by constant forces.
Frequently Asked Questions
How do I calculate the magnetic force on a moving charge by hand?
Multiply the three quantities together and scale by the sine of the angle: \( F = |q|\,v\,B\sin(\theta) \). First convert everything to SI units (coulombs, meters per second, tesla), find the angle between the velocity vector and the field lines, then multiply. Example: \( 2\,\mu\text{C} \) at \( 100\ \text{m/s} \) in \( 0.5\ \text{T} \) at \( 30^\circ \) gives \( F = (2\times10^{-6})(100)(0.5)(0.5) = 5\times10^{-5}\ \text{N} \).
How do I convert between tesla, gauss, millitesla and microtesla?
The base unit is the tesla (T). 1 T = 10,000 G = 1,000 mT = 1,000,000 µT. So Earth's field of about 50 µT equals 0.05 mT or 0.5 G, and a 0.5 T neodymium magnet equals 5,000 G. A frequent mistake is mixing up gauss and tesla — a factor of 10,000 error — so always double-check which unit your data source uses before entering values.
Why is the force zero when the charge moves parallel to the field?
Because of the \( \sin(\theta) \) factor: at \( \theta = 0^\circ \) or \( 180^\circ \), \( \sin(\theta) = 0 \), so \( F = 0 \) no matter how large the charge, speed or field is. Physically, the magnetic force comes from the component of velocity that crosses the field lines. A charge sliding exactly along the field lines crosses none of them and feels no push, while motion at \( 90^\circ \) gives the maximum force \( F = |q|vB \).
How do I find the direction of the magnetic force?
Use the right-hand rule: point your fingers along the velocity \( v \), curl them toward the magnetic field \( B \), and your thumb points along the force for a positive charge. For a negative charge such as an electron, reverse the direction. The force is always perpendicular to both \( v \) and \( B \), which is why the calculator only returns a magnitude — the direction depends on the sign of the charge and the geometry, not the numbers.
Can I use this calculator for an electron or a proton?
Yes. The elementary charge is \( e = 1.602\times10^{-19}\ \text{C} \); enter that as the charge magnitude (0.1602 nC) and the sign only flips the force direction. Because the charge is tiny, forces on individual particles are small — but so are their masses, so the resulting acceleration can be huge. Note that this tool gives the force only; to get the radius of the circular path, use \( r = mv/(|q|B) \) separately.
Does the angle have to be in degrees or radians?
Enter the angle in degrees between 0° and 180°. The formula itself works with radians too, and \( \sin(\theta) \) gives the same value either way, but a common mistake in hand calculations is computing \( \sin \) of a degree value with a calculator set to radians — at \( 90^\circ \) that gives \( \sin(1.57) \approx 1 \) only by luck. Also remember the angle is between \( v \) and \( B \), so 30° and 150° give the same force magnitude.
Why can't a magnetic force speed a particle up or slow it down?
Because the force is always perpendicular to the velocity, and a perpendicular force does no work: \( W = F\,d\cos(90^\circ) = 0 \). The magnetic force changes only the direction of motion, never the speed or kinetic energy. In a uniform field, a charge moving perpendicular to the field travels in a circle; at an angle it follows a helix around the field lines. Only an electric field can change a particle's speed — see the full Lorentz force \( F = q(E + v\times B) \).