Magnetic force on a moving charge

Three forces that all come from the same magnetic field

A magnetic field pushes on anything carrying charge that moves through it. The push has three everyday faces: on a wire full of current, on a single particle in flight, and between two wires running side by side. This calculator handles all three, in the units the parts and instruments are actually marked in, and for the moving particle it also returns the radius of the curve, the time for one turn and the cyclotron frequency.

10 N

on 1 m of wire at 10 A in a 1 T loudspeaker gap

500 µN

the same wire in the Earth's field of 50 µT

11.37 µm

circle an electron makes at 1 Mm/s in 0.5 T

2.088 cm

a proton in the same place, 1836 times wider

How the three formulas relate to one another

They are one formula wearing three coats. A current is charge in motion, so the force on a wire, F = BIl sinα, is nothing but the sum of the forces on every charge inside it, each of them obeying F = qvB sinα. The third case folds the first two together: the current in one wire makes a field, and the second wire sits in it, which gives F = μ₀I₁I₂l / (2πd).

The sine matters more than people expect. Run the current straight along the field and the force is zero, no matter how strong the magnet. Tip it to 30° and you get half the maximum; at 45° you get 70.7%. Only at a right angle does the field give everything it has, which is why motor windings and speaker voice coils are wound to cross the field squarely rather than to run with it.

The direction is the part that catches people out. The force is perpendicular to both the current and the field at once, so it points out of the plane the two of them define. Reverse either one and the push reverses; reverse both and nothing changes. On a moving charge that perpendicularity has a consequence worth holding on to: the force never speeds the particle up or slows it down, it only bends the path. A magnetic field does no work.

Which boxes each mode asks for

Which force - on a wire, on a moving charge, or between two parallel wires. Nothing is assumed if you skip it; the calculator asks.
Flux density B with its unit - teslas, millitesla or gauss. Gauss is what most magnet datasheets and cheap field meters print, and 10 000 G is 1 T.
Current with its unit, for a wire - milliamps, amps or kiloamps.
Length in the field with its unit, for a wire or a parallel pair - millimeters, centimeters or meters. Only the part actually inside the field counts.
Angle between the motion and the field, from 0 to 180 degrees. There is no hidden default here: a blank box is a question, not a right angle.
Particle, for a moving charge - electron, proton, alpha particle, or something else whose charge and mass you type in. The choice sets the mass, and the mass sets the radius of the path.
Speed with its unit - meters per second, kilometers per second, or straight as a fraction of the speed of light.
Two currents, the separation and their direction, for a parallel pair. Same direction pulls the wires together, opposite directions pushes them apart, and the size of the force is identical either way.
Read the answer. The headline tile carries the force, the small tiles carry what follows from it, and the table underneath moves one variable so you can see which way the answer travels.

Reference values for flux density, from a fridge door to a scanner

The right-hand column is the same test run through the calculator each time: one meter of wire carrying ten amps, crossing the field squarely. It turns an abstract field strength into a push you can picture.

Where you meet it	Flux density	In gauss	Force on 1 m at 10 A
The Earth's field at the surface	25 to 65 µT	0.25 to 0.65	500 µN at 50 µT
A fridge magnet, at its face	about 5 mT	50	50 mN
The gap of a loudspeaker motor	about 1 T	10 000	10 N
The face of a strong neodymium block	up to about 1.4 T	14 000	14 N
A clinical magnetic resonance scanner	1.5 to 3 T	15 000 to 30 000	30 N at 3 T
The strongest steady laboratory magnet	about 45 T	450 000	450 N

The jump from the first row to the last is a factor of nearly a million, and the force follows it exactly, because F = BIl sinα is linear in every one of its terms. Double the current and you double the push; halve the length inside the field and you halve it.

Three particles at one speed in one field

Send an electron, a proton and an alpha particle through 0.5 T at 1 000 000 m/s, all crossing the field squarely. The electron and the proton feel exactly the same force, because they carry the same size of charge. What separates them is mass, and mass shows up only once the path starts bending.

Particle	Force	Radius of the circle	One full turn	Cyclotron frequency
Electron	8.011 x 10^-14 N	11.37 µm	7.145 x 10^-11 s	14 GHz
Proton	8.011 x 10^-14 N	2.088 cm	131.2 ns	7.623 MHz
Alpha particle	1.602 x 10^-13 N	4.147 cm	260.6 ns	3.838 MHz

The proton's circle is 1836 times wider than the electron's, which is the ratio of their masses and nothing else. The alpha particle is heavier still but carries twice the charge, so its radius comes out only about twice the proton's rather than four times.

Things worth knowing about magnetic force

The cyclotron frequency does not care how fast the particle goes. Speed the electron up and its circle grows in exactly the same proportion as the distance it has to cover, so it still comes round 14 billion times a second in a 0.5 T field. That coincidence is what made the cyclotron possible at all.

Two wires a meter apart carrying one amp each pull with 2 x 10^-7 N per meter. That sentence used to be the definition of the ampere. It no longer is: the ampere is now fixed through the elementary charge, 1.602176634 x 10^-19 C exactly, and this force became a measured consequence instead of the starting point.

A magnetic field cannot change a particle's energy. The force sits at a right angle to the motion at every instant, so it does no work. Anything that genuinely accelerates charged particles, a cyclotron included, uses an electric field for the pushing and the magnetic field only for the steering.

Above a tenth of the speed of light the classical radius is wrong. At 0.5 c the Lorentz factor is 1.1547, so the real circle is 15.5% wider than the textbook formula predicts. This calculator applies that factor and says so; it also refuses any speed at or above light speed rather than printing a number.

The same force runs a loudspeaker and a rail launcher. A voice coil at 1 A in a 1 T gap with 5 m of wire wound into it gets 5 N, enough to move a cone at audio rates. Scale the current into the kiloamps and the identical formula throws a projectile.

Five situations that send someone here

Physics coursework is the obvious one, since the three modes match the three cases every syllabus covers. Beyond that: sizing a homemade linear actuator, where the question is whether 50 mN from a hobby magnet is worth building around; checking whether an instrument cable run beside a busy feeder is pushed at all, and finding it is a matter of micronewtons; and working a mass spectrometer problem, where the radius column above is the whole answer.

Questions about magnets, currents and charges

Why do an electron and a proton feel the same force but follow completely different paths?

Because the force depends on charge and the path depends on mass. Both carry the same 1.602176634 x 10^-19 C, so both feel 8.011 x 10^-14 N in the example above. The proton is 1836 times heavier, so the same push bends it far less and its circle comes out at 2.088 cm against the electron's 11.37 µm.

How do I convert gauss to tesla?

Divide by 10 000. A magnet advertised at 5000 G is 0.5 T, and the Earth's field of around 0.5 G is 50 µT. You do not have to do it by hand here, because the field box takes gauss directly, which matters given how many magnet listings and hand-held meters use them.

Which way does the force actually point?

At a right angle to both the current and the field. Point the fingers of your right hand along the conventional current, curl them towards the field, and your thumb gives the push. For a negative charge such as an electron the result flips, which is the most common slip in this topic. The size of the force, which is what this calculator returns, is the same either way.

What angle should I enter if the wire is not straight?

Break it into straight pieces and add the results, because F = BIl sinα assumes one direction along the whole length. A coil is the everyday case: only the sides crossing the field contribute, which is why motor windings are laid out as they are. A piece running along the field at 0° contributes nothing.

Do two parallel wires attract or repel?

Currents running the same way attract, currents running opposite ways repel. It surprises people, because two magnets with like poles facing push apart while two wires doing the matching thing pull together. The force is the same size in both cases, so the calculator asks for the direction rather than guessing, and prints the answer in words next to the number.

How close to the real world is the two-wire formula?

It treats both wires as infinitely long, infinitely thin and perfectly parallel. That holds well when the run is much longer than the gap and the gap is much wider than the wire itself. Push the wires almost into contact, or use a run of a few centimeters, and the real force drifts away from the figure, because the ends of a short wire carry field lines the model ignores.

Can I use this for the pull between two permanent magnets?

No, and that is worth being clear about. These three formulas describe forces on moving charge. The attraction between two magnets, or between a magnet and a steel plate, comes from magnetized material rather than a current, and it falls off far more steeply with distance than 1/d. Use the wire mode only when there is a real current in a real conductor.

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