Lorentz force calculator
Choose a moving charge or a wire. For a charge, pick the particle, its speed, the magnetic field and the angle between them to get the force, the radius of the circle it draws and how often it goes round. For a wire, enter the current, the length in the field and the field strength.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose a moving charge or a wire.
- For a charge, pick the particle (or enter a charge in units of e and a mass in u) and its speed.
- For a wire, enter the current and the length inside the field.
- Enter the field in tesla, millitesla or gauss and the angle (90° when the motion is across the field).
- Charge: F = |q| × v × B × sin θ.
- Wire: F = B × I × L × sin θ.
- Radius: r = m v sin θ ÷ (|q| B). With an angle other than 90° the path is a helix of this radius.
- Cyclotron frequency: f = |q| B ÷ (2π m), period T = 1 ÷ f.
Constants: e = 1.602176634 × 10⁻¹⁹ C, electron mass 9.1093837 × 10⁻³¹ kg, proton mass 1.6726219 × 10⁻²⁷ kg.
Worked example
An electron at 10⁶ m/s crosses a 10 mT field at 90°. F = 1.602177 × 10⁻¹⁹ × 10⁶ × 0.01 = 1.60218E-15 N. It circles with r = 9.109384 × 10⁻³¹ × 10⁶ ÷ (1.602177 × 10⁻¹⁹ × 0.01) = 0.5686 mm, at 279,924,898.72 Hz (about 280 MHz).
A wire carrying 10 A over 0.5 m in a 0.2 T field at 90° feels F = 0.2 × 10 × 0.5 = 1 N.
Tips
- Direction. For a positive charge use the right-hand rule: fingers along v, curl toward B, thumb gives F. An electron is pushed the other way.
- Along the field, no force. At θ = 0° or 180° the force is zero and the particle keeps going straight.
- Gauss. 1 T = 10,000 G and 1 mT = 10 G, so the default 10 mT is 100 G.
Frequently asked questions
What is the Lorentz force?
The force on a charge q moving with velocity v through a magnetic field B is F = q v × B; its size is |q| v B sin θ, where θ is the angle between v and B. It is always perpendicular to both, so it bends the path but does not change the speed. With an electric field the full Lorentz force adds qE.
How is the force on a wire related?
A current is many moving charges, and their forces add up to F = B I L sin θ for a straight wire of length L in a uniform field. This is the force that turns electric motors.
Why does the particle move in a circle?
The force is perpendicular to the velocity, so it acts as a centripetal force, m v² ÷ r = |q| v B, which gives r = m v ÷ (|q| B). The time for one turn, 2π m ÷ (|q| B), does not depend on the speed; that is how a cyclotron works.
Is this valid at very high speeds?
The force formula is exact, but the radius and frequency here use the rest mass. Near the speed of light the momentum grows by the Lorentz factor γ, so the real radius is γ times larger and the frequency γ times lower. Keep speeds well below 3 × 10⁸ m/s, or correct by γ from the time dilation calculator.