Lorentz Force Calculator
Find the total force on a moving charge from electric and magnetic fields, and see the circle or helix it follows in a magnetic field.
🧲 What is the Lorentz Force?
The Lorentz force is the total electromagnetic force on a charged particle. A charge q at rest feels only the electric force qE. Once it moves with velocity v, a magnetic field B adds a second force q v × B that is perpendicular to both the velocity and the field. The complete law, F = q(E + v × B), was assembled in the 1890s and is named after Hendrik Lorentz. It is the bridge between electromagnetism and mechanics, because it says how fields push on matter, while Maxwell's equations say how charges and currents produce the fields.
Everything from small devices to large machines relies on it. A cathode ray tube steers electrons with deflecting fields. A mass spectrometer separates ions by the radius of their circular paths. Cyclotrons and synchrotrons bend particle beams with magnets and push them with electric fields. Hall sensors in cars and phones detect the sideways push on moving charge carriers, and electric motors use the force on currents, which is the Lorentz force summed over many charges. In space, the same law shapes the paths of cosmic rays around planets and stars.
Several features surprise newcomers. The magnetic force depends on the velocity, so a charge at rest feels none. It is perpendicular to the velocity, so it never changes the speed and does no work, it only bends the path. When the velocity is exactly along B the force is zero, and at right angles it is largest. Also, the electric and magnetic parts can cancel exactly for one speed, which is the principle of a velocity selector. For fast particles you replace the mass with the relativistic mass γ m when finding the radius and frequency.
This calculator gives you three views. The angle form computes the magnetic force from speed, field and angle, and charts it against the angle. The vector form computes the complete force from the three components of v, E and B, with a table of the electric, magnetic and total parts. The orbit form turns a speed and a field into a radius, a frequency and a pitch for the helical path.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Proton Across a Magnetic Field
A proton at 2,000,000 m/s moves at 90 degrees to a 0.5 T field
Example 2 — Electron at 30 Degrees
An electron at 10,000,000 m/s enters a 0.01 T field at 30 degrees
Example 3 — A Velocity Selector
A proton at v = (100000, 0, 0) m/s with E = (0, 2000, 0) V/m and B = (0, 0, 0.02) T
Example 4 — Electron in a Magnetic Field Only
An electron with v = (300000, 400000, 0) m/s in B = (0, 0, 0.1) T and no electric field
Example 5 — Alpha Particle in Crossed Fields
An alpha particle with v = (0, 2000000, 1000000) m/s, E = (500, 0, 0) V/m and B = (0.1, 0, 0) T
Example 6 — Proton Orbit in a 1 T Field
A proton with v⊥ = 1,000,000 m/s and no parallel motion in a 1 T field
Example 7 — A Fast Electron on a Helix
An electron with v⊥ = 59,000,000 m/s and v∥ = 20,000,000 m/s in a 0.001 T field
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Lorentz force formula?
The Lorentz force on a charge q moving with velocity v in an electric field E and a magnetic field B is F = q(E + v × B). The first term is the electric force along E, and the second is the magnetic force, perpendicular to both v and B.
How do you calculate the magnetic force on a moving charge?
Use F = q v B sin θ, where θ is the angle between v and B. For a proton at 2e6 m/s perpendicular to a 0.5 T field, F = 1.602e-19 × 2e6 × 0.5 = 1.6022e-13 N.
What is the direction of the Lorentz force?
The magnetic part points along v × B for a positive charge and opposite to it for a negative charge. By the right hand rule, point your fingers along v, curl them toward B, and your thumb gives the direction for a positive charge.
Why does the magnetic force do no work?
The force is always perpendicular to the velocity, so the power F · v is zero. The magnetic field changes the direction of motion of a particle but not its speed. Only the electric field can change the kinetic energy.
What is a velocity selector?
A velocity selector uses crossed electric and magnetic fields so that the two forces cancel for one speed, v = E/B. With E = 2000 V/m and B = 0.02 T, particles at 100000 m/s go straight and every other speed is deflected.
How do I find the radius of a charged particle in a magnetic field?
For motion perpendicular to B, the radius is r = m v / (|q| B), and with relativity r = γ m v / (|q| B). A proton at 1e6 m/s in 1 T has a radius of 0.0104 m, which is 10.4 mm.
What is the cyclotron frequency?
The frequency of circular motion is f = |q| B / (2π γ m), independent of the speed at low speeds. A proton in 1 T circles at 1.5245e7 Hz, about 15.2 MHz, and an electron in 1 mT circles near 28 MHz.
What path does a charged particle follow in a magnetic field?
A circle if its velocity is perpendicular to B, a straight line if it is parallel, and a helix otherwise. The helix has radius set by the perpendicular speed and pitch equal to the parallel speed times the orbit period.
Does the Lorentz force depend on the mass of the particle?
The force does not, but the resulting motion does. The force q v B is independent of mass, while the acceleration F/m and the orbit radius m v/(qB) are larger for heavier particles. A heavier alpha particle has a larger orbit than a proton at the same speed.
What happens if the electric and magnetic forces are in the same direction?
They add as vectors. In vector mode the calculator gives the electric and magnetic parts as separate rows and the total, with their magnitudes, so you can see whether they add, partly cancel or cancel exactly.