Electromagnetic Force Calculator
Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer
Last updated: August 20, 2026Two Forces, One Underlying Field
A charged particle feels a push from an electric field whether it's moving or not, but it only feels a magnetic force when it's in motion — and that magnetic push acts sideways, perpendicular to both its velocity and the field. Combined, these two effects make up the Lorentz force, the complete description of how electric and magnetic fields act on charge. This calculator handles all three cases: pure electric force, pure magnetic force, and the combined Lorentz force.
The Formulas
Magnetic force: F = q × v × B × sin(θ)
Lorentz force: F = qE + qvB × sin(θ)
q is the particle's charge, E is electric field strength, v is velocity, B is magnetic field strength, and θ is the angle between the velocity and magnetic field vectors — the magnetic term is maximized at θ = 90°, where velocity and field are perpendicular, and vanishes when they're parallel.
Where This Governs Real Systems
- Particle accelerators — magnetic fields steer charged particles around a ring using exactly this sideways magnetic force, while electric fields do the accelerating.
- Electric motors and generators — the force on current-carrying wires in a magnetic field (a direct extension of the v×B term) is what converts electrical energy to mechanical motion and back.
- Mass spectrometry — ions are deflected by known magnetic fields, and the resulting curved path reveals their mass-to-charge ratio.
- Cathode ray and electron-beam devices — electric fields accelerate electrons while magnetic fields steer or focus the beam.
Two Reference Calculations
A proton (charge 1.602×10&supminus;¹&sup9; C) moving at 2×10&sup6; m/s perpendicular to a 0.5 T magnetic field experiences:
A 2 µC charge sitting in a 5,000 V/m electric field feels a purely electric force of:
Note how much larger the electric force is here despite the smaller charge — the relative strength of each term depends entirely on the specific charge, speed, and field values involved.
How to Use This Calculator
- Choose a mode: Electric Force, Magnetic Force, or Lorentz Force (Combined).
- Enter the Charge q (C) — required for every mode.
- For Electric Force, enter Electric Field E (V/m).
- For Magnetic Force, enter Velocity v (m/s), Magnetic Field B (T), and the Angle between v and B (defaults to 90°).
- For Lorentz Force, enter all of the above: Electric Field, Velocity, Magnetic Field, and Angle.
- Select Calculate to see the resulting force in newtons along with the worked formula.
Related Calculations
For the force between two point charges rather than a charge in a field, see the Coulomb's Law Calculator, or compare against the much weaker gravitational pull between masses with the Gravitational Force Calculator.
Principles of Classical Electrodynamics and the Lorentz Force Law
An electromagnetic force calculator computes electrostatic Coulomb interactions, magnetic field deflection forces, and total combined vector forces acting on charged particles under Classical Electrodynamics and the Lorentz Force Law. In plasma physics, particle accelerator engineering (CERN LHC), and electrical engineering, electromagnetic forces dictate electron beam trajectories, mass spectrometry, and electric motor torque.
The Fundamental Electromagnetic Force Formulas
Where ke = 1 / ( 4πε0 ) = 8.98755 × 109 N·m2/C2 | q = Charge in Coulombs (e = 1.6022 × 10-19 C)
Magnetic Lorentz Force: Fm = q · ( v × B ) &implies; Magnitude: Fm = | q | · v · B · sin(θ)
Where v = Particle velocity (m/s) | B = Magnetic flux density (Tesla) | θ = Angle between velocity and magnetic field vectors
Total Lorentz Vector Force: F = q · [ E + ( v × B ) ]
Cyclotron Gyroradius: rg = ( m · v⊥ ) / ( | q | · B )
Right-Hand Rule Vector Orientation in Magnetic Deflection
- Index Finger: Points in the direction of positive charge velocity vector (v).
- Middle Finger: Points in the direction of the magnetic field vector (B).
- Thumb: Indicates the direction of the resulting magnetic deflecting force vector (Fm).
- Note: For negatively charged electrons, the resulting magnetic force vector points in the exact opposite direction.
Step-by-Step Worked Calculation Example
Example: Calculating Magnetic Deflection and Cyclotron Radius of a Proton
Problem: A high-energy proton (mass m = 1.6726 × 1027 kg, charge q = +1.6022 × 10-19 C) enters a uniform magnetic field of B = 1.50 Tesla at a perpendicular velocity of v = 3.00 × 106 m/s (θ = 90.0°, sin θ = 1.0). Calculate: (1) Magnetic Lorentz force; (2) Centripetal acceleration; and (3) Cyclotron circular orbit radius.
Step 1: Calculate Magnetic Lorentz Force (Fm = q · v · B):
Fm = ( 1.6022 × 10-19 C ) × ( 3.00 × 106 m/s ) × 1.50 T = 7.2099 × 10-13 Newtons
Step 2: Calculate Acceleration (a = F / m):
a = 7.2099 × 10-13 N / 1.6726 × 10-27 kg = 4.3106 × 1014 m/s2
Step 3: Calculate Cyclotron Gyroradius (r = m v / q B):
r = ( 1.6726 × 10-27 × 3.00 × 106 ) / ( 1.6022 × 10-19 × 1.50 )
r = ( 5.0178 × 10-21 ) / ( 2.4033 × 10-19 ) = 0.02088 Meters (2.088 cm)
Conclusion: The proton executes circular cyclotron orbits with a tight radius of 2.09 cm inside the 1.5 Tesla field.
The Hall Effect in Solid-State Physics
When an electrical current I flows through a conducting semiconductor slab immersed in a transverse magnetic field B:
The magnetic Lorentz force deflects charge carriers toward one lateral edge, generating a measurable transverse Hall Voltage (VH):
Where n is charge carrier density, q is carrier charge, and d is semiconductor slab thickness.
Hall effect sensors serve as ubiquitous solid-state non-contact magnetic proximity switches, brushless DC motor encoders, and automotive crankshaft position sensors.
Magnetic Torque on Current Loops in Electric Motors
In electromagnetic machinery (induction motors, automotive EV drivetrains), a rectangular coil of N wire turns carrying current I in a uniform magnetic field B experiences a mechanical Torque (τ):
Where A is the coil cross-sectional area and θ is the angle between the magnetic field and the coil's normal vector.
Relativistic Magnetic Force Transformations
In Einstein's Special Relativity, magnetic forces are revealed as the relativistic electrostatic Coulomb forces of moving charges under Lorentz Length Contraction.
When an observer moves relative to a current-carrying wire, relativistic contraction changes linear charge density, transforming magnetic forces into pure electrostatic interactions.
Synchrotron Radiation in High-Energy Accelerators
When charged particles accelerate along curved circular magnetic paths at relativistic velocities approaching the speed of light, they emit intense directional Synchrotron Radiation, utilized in advanced materials science research.
MHD Magnetohydrodynamics in Fusion Energy
In magnetic confinement tokamak reactors (ITER), powerful superconducting magnetic field coils confine 100-million-degree deuterium-tritium plasma away from reactor chamber walls via magnetic Lorentz pinch forces.