Free Lorentz Force Calculator

What Is the Lorentz Force?

The Lorentz Force Calculator — also called a Magnetic Force Calculator or qvB Calculator — provides a quick way to evaluate the magnetic force exerted on a moving charged particle. This tool applies the Lorentz force law, which describes the interaction between electromagnetic fields and electric charges. While the full law includes both electric and magnetic components, the focus here is exclusively on the magnetic part. (The electric part can be explored with separate tools like a Coulomb's law calculator.)

Key Principles of the Lorentz Force Law

The Lorentz force law states that a magnetic field produces a force on a charged particle only when the particle is in motion relative to the field. The magnitude of that force depends on four factors: the particle's electric charge ($q$), its speed ($v$), the magnetic flux density ($B$), and the angle ($\alpha$) between the velocity vector and the magnetic field direction.

One unique aspect of the Lorentz force is its orientation: it always acts perpendicular to both the velocity and the magnetic field. Consequently, the particle's trajectory is deflected in a curved path, unless the motion is exactly parallel to the field lines — in that case, the force becomes zero.

The Lorentz Force Equation

The mathematical expression for the magnetic component of the Lorentz force is:

$$F = q \, v \, B \, \sin(\alpha)$$

where:

• $F$ – resulting force in newtons (N)
• $q$ – particle charge in coulombs (C)
• $v$ – particle speed in meters per second (m/s)
• $B$ – magnetic field strength in teslas (T)
• $\alpha$ – angle between the velocity and the magnetic field

In this calculator, the default angle is set to $90^\circ$ for simplicity, giving the maximum possible force. If you adjust the angle to $0^\circ$, the force drops to zero. Any intermediate angle produces a force proportional to $\sin(\alpha)$.

Determining the Direction of the Force

Because the Lorentz force is a vector product (cross product) of the velocity and magnetic field, its direction can be found using the right-hand rule. For a positively charged particle:

1. Point your right thumb in the direction of the particle's velocity.
2. Point your right index finger along the magnetic field.
3. Your right middle finger, extended perpendicular to the palm, will indicate the direction of the force.

For a negatively charged particle, simply reverse the direction obtained by the right-hand rule.

Practical Applications of the Lorentz Force

The Lorentz force appears in numerous scientific and everyday settings. In research, it is harnessed to accelerate particles in cyclotrons and synchrotrons, enabling discoveries in particle physics. It also drives mass spectrometers, which separate ions by their mass-to-charge ratio for chemical analysis.

Common consumer technologies also rely on the Lorentz force. Electric motors and loudspeakers use current-carrying wires (moving charges) inside magnetic fields to produce motion or sound. Railguns, though less common in households, are another example where the Lorentz force accelerates projectiles.

Example Calculation with the Lorentz Force

To illustrate the use of the equation, consider an electron moving at one‑tenth the speed of light through a uniform magnetic field. The data are:

• $q = 1.602 \times 10^{-19}\ \text{C}$ (magnitude of electron charge)
• $v = 0.1 \times c = 2.998 \times 10^{7}\ \text{m/s}$
• $B = 0.5\ \text{T}$
• $\alpha = 90^\circ$ (field perpendicular to motion)

Plugging into the formula:

$$F = (1.602 \times 10^{-19})\ \text{C} \times 0.5\ \text{T} \times (2.998 \times 10^{7})\ \text{m/s} \times \sin(90^\circ)$$ $$F \approx 2.402 \times 10^{-12}\ \text{N}$$

This is the magnitude of the Lorentz force acting on the electron. The tool can perform such calculations instantly for any set of inputs, making it an easy way to explore the relationship between charge, speed, field strength, and angle.

Using the Lorentz Force Calculator

Simply enter the charge (in coulombs), the velocity (in meters per second), the magnetic field (in teslas), and optionally the angle. The Electromagnetic Force Calculator will display the resulting force in newtons. It also shows how varying the angle changes the force magnitude, helping you grasp the role of $\alpha$ in the Lorentz force equation.