1.602000e-16
N
1.602000e-14
N
1.618020e-14
N
1.7763e+16
m/s²
101,000
N/C
0.000057
m
1.602000e-16
N
1.602000e-14
N
1.618020e-14
N
1.7763e+16
m/s²
101,000
N/C
0.000057
m
The Lorentz Force Calculator computes the total electromagnetic force acting on a charged particle moving through combined electric and magnetic fields. The Lorentz force is the fundamental force law of classical electrodynamics, governing the motion of charged particles in everything from cathode ray tubes to particle accelerators and the auroras in Earth's magnetosphere.
This calculator resolves the electric and magnetic force components separately and combines them to give the total force magnitude, along with the resulting acceleration assuming an electron mass for reference.
The Lorentz force on a particle with charge $$q$$ moving with velocity $$\mathbf{v}$$ in an electric field $$\mathbf{E}$$ and magnetic field $$\mathbf{B}$$ is:
$$\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}$$
The magnitude of each component is:
$$F_E = |q|E \quad \text{(electric force)}$$
$$F_B = |q|vB\sin\theta \quad \text{(magnetic force)}$$
where $$\theta$$ is the angle between the velocity vector and the magnetic field. When $$\theta = 90°$$, the magnetic force is maximum. When $$\theta = 0°$$ (particle moves parallel to B), the magnetic force vanishes.
The total force magnitude in the worst case (both forces aligned) is:
$$F_{total} = F_E + F_B = |q|(E + vB\sin\theta)$$
Key characteristics of the two force components differ fundamentally: the electric force acts along $$\mathbf{E}$$ regardless of the particle's motion and can do work (change kinetic energy), while the magnetic force is always perpendicular to $$\mathbf{v}$$, meaning it changes the direction but not the speed of the particle — it does zero work. This is why magnetic fields cause circular or helical trajectories.
The acceleration is computed as $$a = F/m$$ using the electron rest mass $$m_e = 9.109 \times 10^{-31}$$ kg for reference.
Electric Force (F_E) is the component from the electric field — it acts on the charge whether moving or stationary. Magnetic Force (F_B) only acts on moving charges and is zero when the velocity is parallel to B. Total Lorentz Force is the combined magnitude assuming worst-case alignment. Acceleration gives the force divided by electron mass for reference — for other particles (protons, ions), divide the force by the appropriate mass. Note that at velocities approaching the speed of light, relativistic corrections become important.
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At v = 10⁶ m/s in a 0.1 T field, the magnetic force dominates by a factor of 100 over the electric force from 1000 V/m.
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A stationary charge experiences no magnetic force regardless of B. Only the electric field exerts a force.
The magnetic force $$q\mathbf{v} \times \mathbf{B}$$ is always perpendicular to the velocity. Since work is $$W = \mathbf{F} \cdot \mathbf{v}$$ and the dot product of perpendicular vectors is zero, the magnetic force cannot change a particle's kinetic energy — only its direction.
When the velocity is parallel to the magnetic field, $$\sin(0°) = 0$$, so the magnetic force vanishes. The particle continues in a straight line along B with only the electric force acting on it.
A velocity selector uses crossed E and B fields where $$F_E = F_B$$, meaning $$qE = qvB$$, so $$v = E/B$$. Only particles with this specific velocity pass through undeflected. It is used in mass spectrometers.
Electrons are the most commonly studied charged particles in electromagnetic fields. For protons, divide the force by $$m_p = 1.673 \times 10^{-27}$$ kg. For ions, use the appropriate ionic mass.
When only a magnetic field acts (E = 0, θ = 90°), the particle moves in a circle of radius $$r = mv/(|q|B)$$, called the cyclotron or Larmor radius. This is the basis of cyclotron particle accelerators.
Yes, but the equation of motion must use relativistic momentum: $$\mathbf{F} = \frac{d}{dt}(\gamma m \mathbf{v})$$ where $$\gamma = 1/\sqrt{1-v^2/c^2}$$. The force law itself remains valid.
Roboculator Team
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