Lorentz Force Formula: F = q(E + v × B) with Solved Examples

The Lorentz force formula, $\vec F = q(\vec E + \vec v \times \vec B)$ , gives the complete force felt by a charge q travelling at velocity v through an electric field E and a magnetic field B at the same time. The first term, qE, acts whether the charge moves or not. The second term, the magnetic part, needs motion and always acts sideways to it.

The calculator on this page handles the common exam setup where v is at right angles to B and both pushes point the same way, so their sizes simply add: $F = q(E + vB)$ . For opposite pushes, enter E as a negative number. Each half also has its own page: the electric field formula and the magnetic force formula.

Quick facts

Formula
$\vec F = q(\vec E + \vec v \times \vec B)$
Electric part
$q\vec E$ , along E for positive q
Magnetic part
$qvB\sin\theta$ , at right angles to v and B
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
Balance condition
$v = E/B$ in crossed fields
Type
Vector

Parts of the formula of Lorentz force

Terms in F = q(E + v × B)
Symbol	What it is	SI unit
$\vec F$	Net electromagnetic force on the particle	N
$q$	Charge, with its sign	C
$\vec E$	Electric field at the particle	N/C = V/m
$\vec v$	How fast and which way the charge moves	m/s
$\vec B$	Magnetic field at the particle	T
$\vec v \times \vec B$	Cross product, size $vB\sin\theta$	T m/s

Keep the sign of q inside the bracket. For an electron q is negative, so both the electric and the magnetic parts reverse compared with a proton in the same fields. A useful unit check: tesla times m/s equals N/C, so vB and E can be added directly.

How to apply it step by step

Draw axes. Mark the directions of E, B and v on a sketch.
Find qE. Size qE. Direction along E for a positive charge, against E for a negative one.
Find the magnetic part. Size qvB sin θ. Direction from the right hand rule for v × B, reversed for negative q.
Add as vectors. If the two parts are parallel, add sizes. If opposite, subtract. If at right angles, use Pythagoras.
State size and direction. Give the answer in newtons with a direction in words.

Lorentz force problems solved

Problem 1: both pushes in the same direction

A proton travels at $5.0 \times 10^5$ m/s, crossing a 0.020 T field perpendicularly. An electric field of 4000 V/m points the same way as the magnetic push. Find the total force.

Magnetic term: $vB = (5.0 \times 10^5)(0.020) = 1.0 \times 10^4$ V/m. Add E: 4000 + 10 000 = 14 000 V/m.

$F = qE_{\text{total}} = (1.602 \times 10^{-19})(1.4 \times 10^4) \approx 2.2 \times 10^{-15}\ \text{N}$

Problem 2: a velocity selector

In crossed fields E = $3.0 \times 10^4$ V/m and B = 0.15 T, which speed passes straight through?

No deflection means qE = qvB, so the charge cancels:

$v = \frac{E}{B} = \frac{3.0 \times 10^4}{0.15} = 2.0 \times 10^5\ \text{m/s}$

Slower particles bend toward the electric push; faster ones bend toward the magnetic push.

Problem 3: radius in a pure magnetic field

With E switched off, a proton ( $m = 1.673 \times 10^{-27}$ kg) enters a 0.50 T field square on, at $1.0 \times 10^6$ m/s. How big is its circle?

$r = \frac{mv}{qB} = \frac{(1.673 \times 10^{-27})(1.0 \times 10^6)}{(1.602 \times 10^{-19})(0.50)} \approx 0.021\ \text{m}$

That is about 2.1 cm.

Special cases of the Lorentz force formula

What the motion looks like
Situation	Force	Path
Only E, charge at rest or moving	$q\vec E$	Straight line along E, or a parabola if v is across E
Only B, v at 90° to B	$qvB$	Circle of radius mv/(qB)
Only B, v along B	Zero	Straight line at constant speed
Only B, v at an angle	$qvB\sin\theta$	Helix around the field lines
Crossed E and B with v = E/B	Zero net force	Undeflected straight line

Mass spectrometers use that final row: first a velocity selector lets through one speed, then a magnetic field alone sorts particles by mass using the radius.

Work, energy and dimensions

Only the electric part can change a particle's speed. The magnetic part is always at right angles to v, so its power $\vec F \cdot \vec v$ is zero and it does no work. A charge accelerated through a potential difference V gains kinetic energy qV from the electric field alone; the voltage formula page explains that link.

Both terms must have the dimensions of force. For qE: [I T] times [M L T⁻³ I⁻¹] gives [M L T⁻²]. For the magnetic term, charge [I T] times speed [L T⁻¹] times tesla [M T⁻² I⁻¹] again leaves [M L T⁻²]. A matching pair like this is a quick way to confirm you have written the formula correctly.

Errors to watch for

Adding E and B directly. They have different units. Add E to vB, not to B.
Forgetting to flip directions for electrons. A negative q reverses both terms.
Writing B × v. The order is v × B. Swapping it reverses the magnetic force.
Claiming a magnetic field speeds charges up. It only turns them.

See the electricity and magnetism formula list for the field formulas used inside this one.

Frequently asked questions

What is the formula of Lorentz force?

The Lorentz force is F = q(E + v × B). It is the sum of the electric force qE and the magnetic force qv × B acting on a charge q moving with velocity v.

Which unit is the Lorentz force measured in?

Being a force, the Lorentz force is quoted in newtons (N). Each part, qE and qvB, also comes out in newtons when SI units are used.

Is the Lorentz force only magnetic?

Some Class 12 books use the name for the magnetic part qvB alone. The full Lorentz force includes the electric term too. Read the question to see which one it wants.

When is the Lorentz force zero?

The total vanishes when the electric and magnetic parts cancel, as in a velocity selector where v = E / B. With no electric field, it is also zero for a charge at rest or moving along B.

Does the Lorentz force do work?

The electric part can do work and change the speed. The magnetic part never does work because it acts at right angles to the velocity.