The Lorentz force calculator above gives the force acting on a charged particle moving through electric and magnetic fields together. It reports the two contributions separately, because they behave in completely different ways: the electric force is the same whether the charge is stationary or moving at half the speed of light, while the magnetic force exists only while the charge is in motion and vanishes entirely if that motion is parallel to the field.
Arb Digital builds free calculators that each own one job. This page owns the velocity-dependent case — a charge in motion, with an angle between that motion and the magnetic field. The site's electric field calculator covers the electrostatic side on its own, where nothing is moving and only the field strength and charge matter. If you have a field strength in gauss or amperes per metre and simply need it in tesla, the magnetic field converter rescales the units, and the magnetic field of a wire calculator produces the B value that this page consumes.
What This Lorentz Force Calculator Does
The full Lorentz force has two terms. The electric term, qE, points along the field for a positive charge and against it for a negative one, and does not care about velocity. The magnetic term, qv × B, is a cross product, which means its magnitude depends on the sine of the angle between the velocity and the field, and its direction is perpendicular to both of them at once.
That perpendicularity is the whole character of magnetic force. A force always at right angles to the velocity cannot speed the particle up or slow it down; it can only turn it. A charge moving perpendicular to a uniform magnetic field therefore travels in a circle at constant speed, and the tool reports the radius of that circle and the time it takes to go round once.
The angle input is the one that trips people up, because the answer swings the whole way from maximum to zero across it. At ninety degrees the sine is one and the magnetic force is at its largest. At zero or one hundred and eighty degrees — a charge running straight along a field line — the sine is zero and there is no magnetic force whatsoever, no matter how fast the particle is going or how strong the field is.
How to Use It
- Pick or enter the charge. The elementary charge is the default; an alpha particle carries twice it, and an ion carries its charge state times it.
- Enter the speed in metres per second. This is a non-relativistic treatment, so keep the speed well below the speed of light for the radius and period figures to hold.
- Set the magnetic flux density in tesla. A fridge magnet is a few hundredths of a tesla; a medical scanner bore is one and a half to three; a laboratory superconducting magnet reaches into the tens.
- Set the angle between the velocity and the field. Leave it at ninety for the maximum-force case, and drop it toward zero to see the force fall away as the sine.
- Add an electric field if there is one. Its direction relative to the magnetic force has to be stated, because unlike the magnetic term it is not derived from the velocity.
The Formula: How the Lorentz Force Is Calculated
The magnitude of the magnetic force is F = qvB sin θ, where θ is the angle between the velocity vector and the magnetic field vector. OpenStax University Physics Volume 2, section 11.2 on magnetic fields and lines, states the magnitude of the force as F = qvB sin θ and describes how it depends on the charge, the speed, the field strength and the sine of the angle between velocity and field.
The electric force is F = qE and is independent of motion. Adding the two vectors gives the total. Where they are perpendicular the tool combines them with Pythagoras; where they are along the same line it adds or subtracts them, which is the arrangement used in a velocity selector.
For a charge moving perpendicular to a uniform field, setting the magnetic force equal to the centripetal force needed for circular motion gives qvB = mv²/r, so the radius is r = mv/(qB). The period follows as T = 2πm/(qB), and the striking feature of that expression is that the speed has cancelled out: a fast particle traces a bigger circle but takes exactly the same time to complete it. The elementary charge used in the presets is the exact defined value listed by the NIST CODATA fundamental constants database, which also gives the vacuum magnetic permeability as 1.25663706127 × 10−6 N A−2.
Work the defaults as a check. An electron with charge 1.602177 × 10−19 C moving at 2 × 106 m/s across a 0.5 T field at ninety degrees. The force is 1.602177 × 10−19 × 2 × 106 × 0.5 × 1 = 1.602 × 10−13 N. The radius is (9.109 × 10−31 × 2 × 106) ÷ (1.602 × 10−19 × 0.5) = 2.275 × 10−5 m, about 23 micrometres.
Why the Magnetic Force Does No Work
This is the single most useful thing to understand about the Lorentz force, and it is easy to state and easy to forget. Work is force multiplied by displacement along the direction of that force. The magnetic force is perpendicular to the velocity, and the displacement is along the velocity, so the component of force along the displacement is exactly zero at every instant. The magnetic field therefore cannot change a particle's kinetic energy.
The consequences are everywhere. A particle in a mass spectrometer enters and leaves at the same speed, however sharply it was bent. Electrons in a cathode-ray tube are steered by magnetic coils but accelerated by electric fields, because only the latter can add energy. A cyclotron uses a magnetic field to bring the particle back round again and an oscillating electric field in the gap to actually speed it up — the magnet is the racetrack, the electric field is the engine.
Where this catches people is in explanations that seem to show magnetic fields doing work, such as a magnet pulling a nail. The energy in those cases comes from the field's interaction with the internal magnetic moments of the material rather than from the magnetic force on a free moving charge, and unpicking that properly requires more than the free-charge expression on this page. For the free charge itself the rule holds exactly: the magnetic term bends, and only the electric term energises. The electric potential calculator covers the energy side of the electric term.
The Velocity Selector and Why It Works
Put an electric field and a magnetic field at right angles to each other, send a charged particle through perpendicular to both, and the two forces oppose. The electric force is qE regardless of speed; the magnetic force is qvB and grows with speed. There is therefore exactly one speed at which they cancel, and it is v = E/B — a value that does not depend on the charge or the mass at all.
A beam of mixed particles entering such an arrangement is sorted by speed alone. Anything slower is pushed one way by the dominant electric force, anything faster is pushed the other way by the dominant magnetic force, and only particles at E/B travel straight through the slit. This is the standard front end of a mass spectrometer, because it hands the analysing stage a beam of known speed, which is precisely what is needed to turn a measured radius into a mass.
The tool's velocity-selector preset sets this up so you can watch the cancellation. Change the speed slightly in either direction and the total force stops being zero, and the sign of the residual tells you which way the particle deflects. It is worth noting that the balance is a knife edge rather than a trap: there is no restoring force pulling a slightly-off particle back to the correct speed, which is why a physical slit is needed to select the ones that made it through undeflected.
What Happens When the Angle Is Not Ninety Degrees
The circular-motion result assumes the velocity is entirely perpendicular to the field. When it is not, the sensible move is to split the velocity into two components: one along the field and one across it. The component along the field feels no magnetic force at all and simply continues unchanged. The component across the field does everything described above and circles.
Superimposing the two gives a helix. The particle spirals along the field line, with the radius set by the perpendicular component of the velocity and the forward drift set by the parallel one. The period of the spiral is unchanged, because the period never depended on speed in the first place. This is how charged particles from the sun travel along the Earth's magnetic field lines toward the poles, and it is why aurorae appear in rings around the magnetic poles rather than uniformly across the sky.
The radius and period figures in the result grid are computed from the perpendicular component of the velocity, so they remain correct for the helical case. When the angle is zero or a hundred and eighty degrees there is no perpendicular component and no circle, and the tool says so in words rather than reporting an infinite or undefined radius.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the angle between the force and the field — the sine term takes the angle between the velocity and the field. The force is perpendicular to both and is not part of that angle.
- Assuming a magnetic field speeds particles up — it never does. It bends the path at constant speed, and only an electric field can change the kinetic energy.
- Adding the two forces arithmetically — they are vectors, and unless they lie along the same line they must be combined by components rather than simply summed.
- Applying the radius formula at relativistic speeds — above a few per cent of light speed the momentum is no longer mv, and the circle is larger than the simple expression predicts.
- Forgetting that a parallel velocity gives zero force — a charge running exactly along a field line feels nothing from it, however strong the field.
Related Free Tools From Arb Digital
For the static case with no motion, use the electric field calculator and the electric potential calculator. Get the field around a conductor from the magnetic field of a wire calculator, and rescale units with the magnetic field converter. The Faraday's law calculator covers the induced-voltage side of moving charges and changing fields, and the relativistic kinetic energy calculator takes over where the non-relativistic momentum used here stops being valid. Everything is indexed on the free online tools hub.
Frequently Asked Questions
The total force is the electric term qE plus the magnetic term qv cross B. The magnetic part has magnitude qvB sin theta, where theta is the angle between the velocity and the magnetic field, and it points perpendicular to both of them.
Because the magnetic force is always perpendicular to the velocity, and work requires a force component along the direction of motion. The field can bend the path as sharply as you like, but the particle leaves at exactly the speed it entered.
Nothing. The sine of zero is zero, so the magnetic force vanishes entirely regardless of how fast the particle travels or how strong the field is. Only a velocity component across the field lines produces any magnetic force.
No, only its direction. A positive and a negative charge moving identically through the same fields feel forces of equal magnitude in opposite directions, which is why they curve in opposite senses in the same field.
Because a faster particle travels a proportionally larger circle. The period works out as 2 pi m divided by qB, in which the speed has cancelled entirely, so every particle of the same charge-to-mass ratio circulates at the same rate.
Perpendicular electric and magnetic fields push a charge in opposite directions. The electric force is constant while the magnetic force grows with speed, so they cancel at exactly one speed, E divided by B, which is independent of charge and mass.
The force expression itself stays valid, but the circular radius and period assume momentum equals mass times velocity. Above a few per cent of the speed of light the relativistic momentum is larger, so the real radius exceeds what this page reports.
A helix. The velocity component along the field carries on unaffected while the perpendicular component circles, and the two together spiral along the field line. The radius and period shown here use the perpendicular component, so they remain correct.
This tool is provided for educational and estimating use. It applies the classical non-relativistic Lorentz force expression to a single point charge in uniform fields, ignoring radiation from the accelerating charge, any field the particle itself produces, and collisions with surrounding matter. The circular radius and period assume the speed is small compared with the speed of light.
