Lorentz Force Interactive Calculator

If you’re trying to control the path of charged particles in a system—be it a mass spectrometer, a motor, or any plasma setup—you need to get the Lorentz force calculations sorted before fabricating anything. This Lorentz Force Calculator lets you work out the total force, look at electric and magnetic force components separately, find the needed field strengths, or solve for particle velocity using charge, velocity, field data, and field orientation. It’s a necessity when working on accelerator design, tuning Hall sensors, or any technique where the control of charged particle paths really matters. On this page, you’ll find the main Lorentz force formula, a worked example of a velocity selector, plain-language theory, and a FAQ covering outliers like relativistic corrections.

What is the Lorentz Force?

The Lorentz force tells you what a charged particle will feel when it’s inside electric and magnetic fields together. You get two main effects: The electric force pushes the particle no matter if it’s moving or still, while the magnetic force only comes into play if the particle’s actually in motion.

Simple Explanation

A charged particle is like a ball with “+” or “–” written on it. If you put it in an electric field, it gets straightforwardly pushed in one direction, similar to a ball rolling downhill. Bring in a magnetic field, and now the ball gets nudged sideways, but only when it's moving—and the faster it goes, the stronger that sideways force gets. The Lorentz force is just both of these pushes, happening together.

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How to Use This Calculator

1. Select your calculation mode from the dropdown — total force, magnetic force, electric force, required field strength, or particle velocity.
2. Enter the particle charge in coulombs (C), and fill in the visible input fields: velocity (m/s), magnetic field (T), electric field (V/m), and angle θ (degrees) as required by your chosen mode.
3. If you're solving for a required field or velocity, enter your target force value in the Target Force field.
4. Click Calculate to see your result.

Lorentz Force Diagram

Lorentz Force Calculator

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Lorentz Force Equations

The following formula is used for finding the Lorentz force acting on a charged particle subjected to electric and magnetic fields.

Simple Example

An electron (q = 1.602 × 10^-19 C) travels at 1.0 × 10⁶ m/s through a 0.5 T magnetic field at 90° to it, with a 1000 V/m electric field also applied.

• Magnetic force: F_B = (1.602 × 10^-19)(1.0 × 10⁶)(0.5) sin(90°) = 8.01 × 10^-14 N
• Electric force: F_E = (1.602 × 10^-19)(1000) = 1.602 × 10^-16 N
• Total force: |F| = √((8.01 × 10^-14)² + (1.602 × 10^-16)²) ≈ 8.01 × 10^-14 N

Theory & Practical Applications of the Lorentz Force

Fundamental Physics of Electromagnetic Forces on Charged Particles

The Lorentz force tells you how a charged particle will behave if you put it in electric or magnetic fields—not just one, but both, and especially if the particle is moving. Unlike gravity, which only depends on mass, or static electricity, which only cares about charge and position, the Lorentz force makes things depend on particle velocity as soon as a magnetic field is involved. This makes particle paths look fundamentally different compared to forces that depend only on position, and is key for real-world devices from accelerators to fusion reactors.

The electric part (F_E = qE) points with or against the electric field and speeds up or slows down the particle, actually changing its energy. The magnetic part (F_B = q(v × B)) is always perpendicular to both the velocity and the magnetic field, which means it turns the particle without increasing its speed—so it’s great for steering, not energy transfer. This “sideways only” effect is why, for example, cyclotrons make particles go around in circles, and why “magnetic bottles” can trap plasma without automatically making it hotter.

The key term in the magnetic force is sin(θ): if a particle’s velocity lines up exactly with the magnetic field (θ = 0° or 180°), the magnetic force is zero. Devices like magnetic mirrors use this property—for certain velocity directions, particles are free to escape. But when θ = 90°, you get maximum bending, leading to the tightest possible circles (the gyroradius or Larmor radius: r = mv/(qB)).

Industrial Applications Across Multiple Sectors

Mass Spectrometry and Chemical Analysis: Mass spectrometers rely on Lorentz forces to split ions of different mass-to-charge ratios. When ions get accelerated by a fixed electric potential, their speed ends up depending on their mass. Once they fly into a magnetic field, how much they get bent tells you directly about their mass and charge. With well-tuned equipment, you can reliably measure isotope variations for work like radiometric dating or finding trace chemicals.

Particle Accelerators and Medical Physics: Facilities like the LHC or proton therapy centers use the Lorentz force to tightly steer and focus high-speed charged particles. Superconducting magnets keep proton beams circling massive rings. Medical devices use magnetic beam steering to aim treatment beams precisely. Because the force depends on particle speed, accelerating a particle demands continuous magnetic adjustment—modern setups sync magnet power ramps with accelerating RF cavities.

Hall Effect Sensors and Current Measurement: When current moves through a conductor in a perpendicular magnetic field, charge carriers drift to one side. That side develops a Hall voltage, which lets you measure current without making contact. This approach is common in power meters, brushless motor position sensors, and even geophysical field mapping. High-sensitivity Hall sensors can detect magnetic fields so weak they pick up signals from the human brain or heart.

Plasma Processing and Semiconductor Manufacturing: Magnetron sputtering uses both E and B fields to drive plasma and deposit thin films. Crossed fields trap electrons in spiral paths near the target, massively raising plasma density and efficiency—practical if you want even coatings when making chips or solar panels.

Worked Example: Velocity Selector Design for Ion Beam Purification

Suppose you need a velocity selector for a mixed ion beam to pick out calcium-40 ions (q = +1.602 × 10^-19 C, m = 6.64 × 10^-26 kg) moving at v = 3.75 × 10⁵ m/s. The design puts electric and magnetic fields at right angles, so only ions at the right velocity go straight; all others are deflected.

Part A: Calculate the required field strengths

Set the magnetic field using what you have, B = 0.218 T. For straight-through motion, electric and magnetic forces need to cancel perfectly. Calculate the magnetic force at the target velocity, and set the electric field to give an equal and opposite force. For your plate separation, you can also back-calculate the voltage needed.

Part B: Determine the deflection of calcium-44 contaminant ions

Other ions—like calcium-44—will have different velocities (even with the same energy). Calculate their new velocity, see what magnetic force they feel, and find the leftover net force. Then work out how far off-path they’ll be at the exit. In this example, a contaminant ion ends up 2 mm out of line, which is enough to block it with a small physical aperture and get high-purity beam selection.

Part C: Evaluate non-ideal behavior at field boundaries

Ideal formulas assume abrupt field boundaries, but in real hardware, fields taper off (“fringe fields”). As ions enter or leave the selector, these zones can slightly kick their direction off. Any asymmetry—like a 0.1 mm plate misalignment—can introduce a nontrivial error at the end. That’s why alignment and field shaping are critical in high-resolution selectors.

Critical Engineering Considerations and Limitations

The Lorentz force formula works well for isolated particles in uniform, “textbook” fields. In real-world systems, things get complicated: fields may vary, particles may go fast enough to require special relativity, and if particle density is high, their fields interact (“space charge” effects). In dense plasmas, you can’t just track single particles—the group acts collectively, so you need plasma physics and magnetohydrodynamics to get meaningful results.

When particle speed gets close to 10% of light (v > 3 × 10⁷ m/s), you can’t ignore relativistic mass increase—replace mass m with γm (where γ handles the relativity correction). This alters the way particles curve in a field, and for electrons, substantial energy can be lost as synchrotron radiation while circling in strong fields. These corrections become critical if you're designing accelerators or plasma devices working at very high energies.

See our engineering calculator collection for related electromagnetic tools.

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