Magnetic Force on a Moving Charge and on a Current-Carrying Wire

F = qvB sin θ and F = BIL sin θ are the same equation twice, because qv and IL have the same units. This returns both, the radius and period of the circular path a charge would follow, and the one number that is always exactly zero — the work a magnetic force does.

Last updated 11 Nov 2026

Lorentz force on a charge and on a wire

qvB sin θ and BIL sin θ

Magnetic flux density (T)In tesla. For scale: the Earth’s field is 5×10⁻⁵ T, a fridge magnet about 5×10⁻³ T at its surface, a laboratory electromagnet a few tenths of a tesla, and a clinical magnetic resonance scanner 1.5 to 3 T. If your question gives milliteslas or gauss, convert first: 1 mT = 10⁻³ T and 1 gauss = 10⁻⁴ T.

Angle between the velocity (or the wire) and the field, in degreesIn DEGREES — the page converts for you, because the engine’s trigonometry works in radians and that conversion is one of the standard ways to get a confident wrong answer. The two limits are worth knowing cold: at 90° the force is at its maximum, and at 0° or 180° — motion along the field — the force is exactly zero, which is why a charge moving parallel to a field goes straight on as if the field were not there.

Charge in motion (nC)In nanocoulombs. For a single electron or proton enter 1.602176634×10⁻¹⁰ nC, which is the elementary charge in these units; the row below converts whatever you enter into a number of elementary charges so you can check. The sign changes the DIRECTION of the force, which Fleming’s left-hand rule gives you and no scalar formula can.

Speed of the charge (m/s)In metres per second. Drift speeds in a wire are fractions of a millimetre per second, thermal speeds in a gas are hundreds of metres per second, and an electron accelerated through a few hundred volts is doing some millions. This formula is non-relativistic; above about a tenth of the speed of light the momentum is no longer mv and the circular-radius rows below stop being right.

Current in the wire (A)For the second half of the page. The wire case is the same physics as the charge case: a current is charge in motion, and IL in the wire formula plays exactly the part that qv plays in the charge formula. Check the units if you doubt it — an ampere-metre and a coulomb-metre-per-second are the same thing.

Length of wire inside the field (m)Only the length actually IN the field counts, which is a favourite of examiners: a one-metre wire lying across the 5 cm gap of a magnet has L = 0.05 m, not 1 m. The rest of the wire is carrying the same current through no field and feels no force.

1.250000mNExample

A 40 nC charge at 2.5 × 10⁵ m/s, crossing a 0.25 T field at 30° to it

One equation written twice, and the sin θ that goes to zero

F = qvB sin θ · F = BIL sin θ · at θ = 90°: r = mv / qB and T = 2πm / qB · W_magnetic = 0, always

θ: the angle between the velocity (or the wire) and the FIELD, not between the velocity and the force — the force is at right angles to both. At 90° the force is greatest; at 0° it is exactly zero and the charge sails straight through as though there were no field. Convert to radians before you take the sine, or let a page do it: the engine behind this one has no degree function at all, so its expression contains an explicit ×π/180
qv against IL: the two formulae are the same equation. A current I in a length L carries charge past a point at exactly the rate that a charge q moving at speed v does, and the units confirm it: A·m = C·m/s. If you can do one of these you can do the other, and any question that looks like a new formula is the same one
r = mv/qB: the radius of the circle a charge moves in when its velocity is perpendicular to the field, from setting qvB equal to mv²/r. It needs a MASS, which the force formula does not — so this is the row where electrons and protons part company, by a factor of 1836
T = 2πm/qB: the period, and the remarkable thing about it is what is missing: no v and no r. Every charge of a given mass-to-charge ratio goes round in the same time whatever its speed, because a faster one travels a proportionally bigger circle. That independence is what makes a cyclotron possible
W = 0: the work done by any magnetic force on any charge over any path. Exactly zero, not approximately: the force is always perpendicular to the displacement, so F·ds vanishes at every instant. A magnetic field cannot speed a particle up, and anything that does — a cyclotron, a betatron — is using an electric field to do it

Worked example

A 40 nC charge at 2.5 × 10⁵ m/s, crossing a 0.25 T field at 30° to it
CONVERT THE CHARGE AND TAKE THE SINE. q = 40 nC = 4.0×10⁻⁸ C, and sin 30° = 0.5 exactly. The degrees are the trap: a calculator left in radian mode returns sin 30 = −0.988, so the answer comes out with the wrong magnitude AND the wrong sign, and nothing about it looks obviously wrong
MULTIPLY THE FOUR FACTORS. F = qvB sinθ = (4.0×10⁻⁸)(2.5×10⁵)(0.25)(0.5). Take it in two steps: qv = 1.0×10⁻² C·m/s, and then ×0.25×0.5 = 1.25×10⁻³ N, which is 1.250000 mN
CHECK THE ANGLE BY REMOVING IT. At 90° the same charge and speed would feel 2.5 mN — exactly double, because sin 90°/sin 30° = 2. If your answer does not scale that way when you change the angle, the sine is in the wrong place
NOW THE WIRE, WHICH IS THE SAME SUM. A 6 A current through 0.35 m of wire at the same 30° in the same field: F = BIL sinθ = 0.25 × 6 × 0.35 × 0.5 = 0.2625 N, or 0.75 N per metre. Note that IL = 2.1 A·m here against qv = 0.01 C·m/s for the charge — a 6 A current is an enormous amount of charge flow compared with one 40 nC speck, which is why the wire force is 210 times larger
THE CIRCULAR PATH NEEDS A MASS. If this 2.5×10⁵ m/s belonged to a proton crossing the field at 90°, its radius would be mv/qB = 1.044 cm and its period 2.624×10⁻⁷ s, a cyclotron frequency of 3.81 MHz. For an electron at the same speed the radius is 5.686 µm, 1836 times smaller, because the mass is 1836 times smaller
AND THE NUMBER THAT IS ALWAYS ZERO. The work done by this magnetic force is exactly 0 J, whatever the path and however long the particle spends in the field. The proton above goes round for ever at 2.5×10⁵ m/s and never gains or loses a joule. That is not a limitation of the model, it is a consequence of the force being perpendicular to the velocity at every instant

What the angle does, at this page’s other defaults

The two zeros are the rows worth remembering. A charge moving exactly along a magnetic field, in either direction, feels no magnetic force at all and travels in a straight line as if the field were absent — which is why a charge entering a field at some general angle spirals: the component of its velocity along the field is untouched and the perpendicular component goes in a circle. Note also that 45° gives 0.707 of the maximum rather than half of it; the force is not linear in the angle, and the half-maximum is at 30°.
Angle to the field	sin θ	Force on the charge	Force on the wire
0° — along the field	0.000	0.000000 mN	0.000000 N
30°	0.500	1.250000 mN	0.262500 N
45°	0.707	1.767767 mN	0.371231 N
60°	0.866	2.165064 mN	0.454663 N
90° — across the field	1.000	2.500000 mN	0.525000 N
180° — against the field	0.000	0.000000 mN	0.000000 N

Two formulae that are one formula, and the force that never does any work

F = qvB sinθ and F = BIL sinθ are not two things to learn. A current is charge in motion; IL in the second formula sits exactly where qv sits in the first, and the units agree — an ampere-metre is a coulomb-metre per second. Learn one, check the units, and you have both. This page computes both from the same field and the same angle so that the correspondence is in front of you.

What neither formula tells you is the direction, and that is where the marks are. The force is perpendicular to BOTH the velocity and the field, which leaves two possibilities, and the one that is right comes from Fleming’s left-hand rule — first finger field, second finger current, thumb motion — or from the cross product if you have met it. Cambridge 9702 names the rule in the outcome itself, 20.2.2. Two things catch people out: for a NEGATIVE charge the force is opposite to the rule’s answer for conventional current, and the rule gives nonsense if the velocity and field are not perpendicular, in which case you resolve first.

The angle is in degrees and the trigonometry is in radians. That mismatch is responsible for more wrong answers on this topic than the physics is. A calculator left in radian mode turns sin 30° into −0.988, which is wrong in magnitude and in sign, and the result still looks like a plausible force. The engine behind this page has no degree function at all — its expression contains an explicit multiplication by π/180 — which is the same conversion you have to do by hand.

And then there is the zero. The work done by a magnetic force is exactly zero, always, over any path, because the force is perpendicular to the displacement at every instant. This page prints it as a row rather than leaving it out, because it is a result and not an omission. Its consequences are large: a magnetic field can bend a charged particle’s path into a circle, and the radius mv/qB is how a mass spectrometer separates isotopes and how a bubble chamber measures momentum — but the particle’s speed never changes. Anything that actually accelerates particles, a cyclotron included, does the accelerating with an electric field and uses the magnetic field only for steering. For the fields that a current itself produces, see the field of a wire, a coil and a solenoid.

Frequently asked questions

What happens when a charge moves parallel to the magnetic field?

Nothing at all: sin 0° = 0, so the force is exactly zero and the charge travels in a straight line as though no field were there. This page returns 0.000000 mN for an angle of zero, which is the correct answer rather than a refusal. It is also why a charge entering a field at a general angle moves in a helix — the velocity component along the field is completely unaffected and carries on, while the perpendicular component goes round in a circle.

Why does the period of the circular motion not depend on the speed?

Because the radius grows in proportion to the speed. r = mv/qB, so a particle going twice as fast travels a circle twice as big, and the time round is unchanged: T = 2πm/qB, with no v in it. That independence is the entire operating principle of the cyclotron — the accelerating voltage can be reversed at a fixed frequency and stay in step with the particle however much energy it has gained, until relativity spoils it at high speed.

Why is only the length of wire inside the field used?

Because the rest of the wire has no field to interact with. F = BIL sinθ is really a sum over the current elements, and the elements outside the field region contribute nothing. A one-metre wire crossing the 4 cm gap of a magnet has L = 0.04 m. Questions that quote a long wire and a small magnet are usually checking exactly this.

How do I get the direction of the force?

Fleming’s left-hand rule: first finger along the FIELD, second finger along the CONVENTIONAL current, thumb then points along the force. For a moving negative charge, work out the direction for conventional current — that is, for motion the opposite way — and the force is as the rule gives it. No scalar formula can supply this, which is why every one of these questions has a diagram.

Can a magnetic field speed up a charged particle?

No. The magnetic force is always perpendicular to the velocity, so it does no work, so the kinetic energy and the speed cannot change. This page prints the work as a row reading exactly zero to make the point. A magnetic field can change a particle’s direction as much as you like — that is what a mass spectrometer and a bending magnet do — but every machine that actually raises a particle’s energy does it with an electric field.

Which is bigger, the force on a 40 nC charge or on a 6 A wire?

The wire, by a factor of about 200 at this page’s defaults, and the reason is the comparison that makes a current feel large. The charge contributes qv = 0.01 C·m/s; the wire contributes IL = 2.1 A·m, which is the same unit. Six amperes through a third of a metre moves charge past a point far faster than one 40 nC speck at a quarter of a million metres per second. That ratio is the most useful single number on this page for getting a feel for how much charge a current really is.

Related calculators

References

Syllabus: Cambridge International AS & A Level Physics 9702, for examination in 2025, 2026 and 2027 (Version 1), Cambridge Assessment International Education, cambridgeinternational.org/Images/664565-2025-2027-syllabus.pdf (read 11 November 2026). Cited for the outcomes a Cambridge candidate is examined on, which are worded differently from the Indian board’s: 18.3.2 “recall and use Coulomb’s law F = Q1Q2 /(4πε0r2) for the force between two point charges in free space”, 18.5.3 “use V = Q/(4πε0r) for the electric potential in the field due to a point charge”, 9.1.4 “use, for a current-carrying conductor, the expression I = Anvq, where n is the number density of charge carriers”, 10.1.5 “understand the effects of the internal resistance of a source of e.m.f. on the terminal potential difference”, 19.1.3 “derive, using C = Q/V, formulae for the combined capacitance of capacitors in series and in parallel”, 20.2.2 “recall and use the equation F = BIL sin θ, with directions as interpreted by Fleming’s left-hand rule” and 20.3.2 “recall and use F = BQv sin θ”. The copy read here was truncated before the data-and-formulae sheet, so what is on THAT sheet could not be established from it.
Annual Syllabus Session 2023-24, Class XII Physics (Code 042), Central Board of Secondary Education, edustud.nic.in/edu/SYLLABUS_2023_24/12/12_physics_eng_2023_24.pdf (read 11 November 2026). The board’s own document, cited and not endorsed by it. Unit I Chapter 1 asks for “Coulomb’s law-force between two point charges; forces between multiple charges, superposition principle”; Chapter 2 for “electrical potential energy of a system of two point charges” and “capacitance of a parallel plate capacitor with and without dielectric medium between the plates, energy stored in a capacitor.(No derivation, formulae only)”. Unit II asks for “drift velocity, mobility and their relation with electric current”, “Internal resistance of a cell, potential difference and emf of a cell, combination of cells in series and in parallel, kirchhoff’s rules, Wheatstone bridge”. Unit III asks for “Biot – Savart law and its application to the current carrying circular loop”, “Ampere’s law and its applications to infinitely long straight wire. Straight solenoid (only qualitative treatment)” and “force between two parallel current-carrying conductors-definition of ampere”.
2022 CODATA Recommended Values of the Fundamental Physical Constants (wallet card), NIST Committee on Data for Science and Technology, physics.nist.gov/cuu/pdf/wallet_2022.pdf (read 11 November 2026). Every constant on this page is taken from here, and the reason to read the card rather than a textbook is the bracket. The elementary charge is printed as 1.602 176 634 × 10⁻¹⁹ C with no bracket at all, because since the 2019 revision of the SI it is exact by definition — the coulomb is now defined from it, not the other way round. The same is true of c, h, k and N_A. But the vacuum electric permittivity is printed 8.854 187 8188(14) × 10⁻¹² F/m and the vacuum magnetic permeability 1.256 637 061 27(20) × 10⁻⁶ N/A²: both now have an uncertainty, about 1.6 parts in 10¹⁰, where before 2019 μ₀ was exactly 4π × 10⁻⁷ and ε₀ exactly 1/μ₀c². Almost every school book still prints the old exact forms. At school precision nothing changes; the logic of which constants are definitions changed completely.