Magnetic Force — The Complete Physics Guide
Unlike gravity and the electric force, which act on stationary and moving objects alike, the magnetic force has a peculiar requirement: it only acts on moving charges. A stationary electron sitting inside a strong magnetic field feels absolutely nothing — but the instant it starts moving, it experiences a force that depends on its speed, its charge, the field strength, and — most unusually — the angle between its velocity and the field, following a very different mathematical form to any other force in classical mechanics.
This velocity-dependence, combined with the fact that magnetic force always acts perpendicular to velocity (so it changes direction but never speed), makes magnetism behave unlike anything else in introductory physics — and yet it underlies technologies as different as the electric motor, the mass spectrometer, and the aurora borealis.
What is Magnetic Force?
The magnetic force on a moving charge is given by F = qvB sinθ, where θ is the angle between the velocity vector and the magnetic field vector. When a charge moves exactly perpendicular to the field (θ = 90°), the force is maximised. When it moves parallel or antiparallel to the field (θ = 0° or 180°), the force is exactly zero — a charge moving directly along field lines feels no magnetic push at all. This angular dependence is fundamentally different from gravity or the electric force, neither of which cares about the direction an object happens to be moving.
A current-carrying wire experiences an analogous force, since current is just a coordinated flow of many moving charges: F = BIL sinθ, where I is the current and L is the length of wire within the field. This is the direct physical basis of the electric motor — a current-carrying coil placed in a magnetic field experiences a force (and therefore a torque) that causes it to rotate.
The Formulas Explained
B, magnetic field strength (also called magnetic flux density), is measured in tesla (T) — a large unit; Earth's own magnetic field is only about 0.00005 T (50 μT), while a strong laboratory electromagnet might reach several tesla. q and v are the charge and speed of the moving particle. I and L are the current and length of wire exposed to the field. In every case, θ is the angle between the direction of motion (or current) and the field direction — always use sinθ, never cosθ, a very common point of confusion carried over from other physics topics where cosine appears more often.
The direction of magnetic force is found using the right-hand rule: point your fingers in the direction of velocity (or current), curl them toward the field direction, and your thumb points in the direction of the force (for a positive charge; reverse it for a negative charge). This calculator computes the magnitude of the force; the direction must be determined separately using the right-hand rule appropriate to your specific geometry.
How to Use This Calculator
Use "Charge in a field" to find the force on a moving charged particle, given its charge, speed, the field strength, and the angle between velocity and field. Use "Wire in a field" to find the force on a current-carrying wire. Use "Find B from a wire" to work backwards from a measured force to determine an unknown magnetic field strength — a technique used in devices called current balances to measure field strength precisely.
Worked Example 1 — Force on a Fast Electron
Problem: An electron moves at 1×10⁶ m/s perpendicular to a 0.5 T magnetic field. Find the force on it.
F = qvB sinθ = (1.602×10⁻¹⁹)(1×10⁶)(0.5)(sin 90°)
F = 8.01×10⁻¹⁴ N
Worked Example 2 — Force on a Current-Carrying Wire
Problem: A 0.3 m wire carrying 5 A sits perpendicular to a 0.8 T field. Find the force on it.
F = BIL sinθ = (0.8)(5)(0.3)(sin 90°)
F = 1.2 N
Worked Example 3 — Measuring an Unknown Field
Problem: A 0.2 m wire carrying 3 A experiences a force of 0.15 N when perpendicular to an unknown field. Find B.
B = F/(IL sinθ) = 0.15/[(3)(0.2)(1)]
B = 0.25 T
Why Magnetic Force Never Does Work
Because the magnetic force is always perpendicular to a charge's velocity, it can never speed up or slow down the charge — it can only change the direction of motion, never the speed. This means magnetic force does zero work on a moving charge, no matter how strong the field or how long the charge remains within it. A charged particle moving through a uniform magnetic field perpendicular to its velocity traces out a perfect circle, continuously redirected but never accelerated or decelerated along its path.
This circular motion is the operating principle behind mass spectrometers (which separate particles by mass using the radius of their curved path) and cyclotron-style particle accelerators, and it's also why charged particles from the solar wind spiral around Earth's magnetic field lines rather than crashing straight through the atmosphere, funnelling toward the poles to create the aurora.
Common Mistakes
Using cosθ instead of sinθ: the single most frequent error. Maximum force occurs at 90° (perpendicular), where sin 90° = 1, not at 0°, which would be the case if cosine were used instead.
Forgetting force is perpendicular to velocity: students sometimes treat magnetic force like other forces and assume it acts along the direction of motion — it never does; it's always perpendicular to both velocity and field.
Mixing up tesla and other field units: older texts and some regions still use gauss (1 T = 10,000 G); always confirm units before substituting into SI-based formulas.
Real-World Applications
Electric motors: current-carrying coils in a magnetic field experience a torque that produces continuous rotation — the fundamental mechanism behind nearly every electric motor ever built.
Mass spectrometers: charged particles are deflected into circular paths whose radius depends on their mass-to-charge ratio, allowing extremely precise identification of unknown chemical or isotopic samples.
Particle accelerators: from hospital cyclotrons used to produce medical isotopes to the Large Hadron Collider, magnetic fields steer and focus beams of charged particles along precisely controlled circular or spiral paths at close to the speed of light, with the field strength required scaling directly with the particle's momentum via r = mv/(qB).
The Lorentz Force — Combining Electric and Magnetic Effects
In general, a charged particle moving through a region containing both electric and magnetic fields experiences a combined force known as the Lorentz force: F = qE + qv×B. The electric part (qE) acts along the field direction regardless of the particle's motion, while the magnetic part (qv×B) depends entirely on velocity and acts perpendicular to it. This unified formula is one of the most important equations in all of classical electromagnetism, describing everything from the deflection of a television cathode ray tube's electron beam to the behaviour of plasma in a fusion reactor.
Velocity selectors — devices that allow only particles of a specific speed to pass through — exploit precisely this combination: crossed electric and magnetic fields are arranged so their forces exactly cancel only for one particular velocity, letting particles of that speed travel in a straight line while all others are deflected away. This technique is a standard component in mass spectrometers and other precision particle-selection instruments.
Circular Motion in a Magnetic Field
When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force provides exactly the centripetal force needed for circular motion: qvB = mv²/r, which rearranges to give the radius of the circular path, r = mv/(qB). This relationship is the working principle of the cyclotron, one of the earliest types of particle accelerator, where charged particles spiral outward in ever-increasing circles as they're repeatedly accelerated by an alternating electric field timed precisely to their circular motion.
The same physics explains the trapping of charged particles in Earth's Van Allen radiation belts and the spectacular auroras at the poles — particles from the solar wind spiral along Earth's magnetic field lines, funnelling toward the magnetic poles where they collide with atmospheric gases and produce the glowing curtains of light seen in polar skies.