Magnetic Force Formula: F = qvB sin θ and F = BIL sin θ

The magnetic force formula for a charged particle is $F = qvB\sin\theta$ , where θ is the angle between the velocity and the field. For a straight wire carrying current I through a length L of field, the same physics gives $F = BIL\sin\theta$ . Both answers come out in newtons.

The force only acts on moving charge. A charge sitting still in a magnet's field feels nothing, and a charge sliding exactly along the field lines feels nothing either. This page covers the size of the force, how to find its direction, and the exam slips that come with it. The topic sits inside the electricity and magnetism formulas group, next to the magnetic field formula.

Quick facts

Formula
$F = qvB\sin\theta$
Wire form
$F = BIL\sin\theta$
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
Type
Vector, at right angles to both v and B
Maximum
θ = 90°, so F = qvB
Zero
θ = 0° or 180°, or v = 0

Symbols in the magnetic force formula

Letters in F = qvB sin θ
Letter	Stands for	SI unit
$F$	Magnitude of the sideways push	N
$q$	Charge on the particle (use the size, then fix direction separately)	C
$v$	Speed of the particle	m/s
$B$	Magnetic field strength, also called flux density	T (tesla)
$\theta$	Angle between the velocity and the field	degree or radian
$I,\ L$	Current in a wire and the length inside the field	A, m

In vector form the rule is $\vec F = q\,\vec v \times \vec B$ . The cross product explains the sine: only the part of the velocity that crosses the field lines counts. A 1 T field exerts 1 N on 1 C crossing it squarely at 1 m/s, so 1 T = 1 N s/(C m) = 1 N/(A m).

Finding the direction of the force

The size comes from the formula, but the direction needs a hand rule.

Right hand rule (US books). Point your fingers along v, curl them toward B, and your thumb gives the force on a positive charge.
Fleming's left hand rule (Indian books, for wires). First finger along the field, second finger along the current, thumb along the force, all three at right angles.
Negative charges. Find the direction for a positive charge, then reverse it. An electron is pushed the opposite way to a proton moving with it.

Because the force is always perpendicular to the velocity, it changes the direction of motion and never the speed. It does no work on the particle, so the kinetic energy stays the same.

Three solved magnetic force problems

Case 1: a wire between magnet poles

A straight wire carries 5 A. A 0.40 m length of it lies at right angles to a 0.30 T field. Find the force.

Given: I = 5 A, L = 0.40 m, B = 0.30 T, θ = 90°.

$F = BIL\sin\theta = 0.30 \times 5 \times 0.40 \times 1 = 0.60\ \text{N}$

Case 2: a proton crossing at an angle

A proton moves at $2.0 \times 10^6$ m/s at 30° to a 0.50 T field. Find the force on it.

Given: $q = 1.602 \times 10^{-19}$ C, $v = 2.0 \times 10^6$ m/s, B = 0.50 T, sin 30° = 0.5.

$F = (1.602 \times 10^{-19})(2.0 \times 10^6)(0.50)(0.5) = 8.0 \times 10^{-14}\ \text{N}$

Case 3: an electron with the field in millitesla

An electron travels at $3.5 \times 10^6$ m/s at right angles to a 40 mT field. Find the force.

Convert first: 40 mT = 0.040 T.

$F = (1.602 \times 10^{-19})(3.5 \times 10^6)(0.040) = 2.2 \times 10^{-14}\ \text{N}$

The size is the same as for a proton at that speed. Only the direction flips.

Charged particles moving in circles

When v is at right angles to a uniform B, the magnetic force acts as the centripetal force. Setting $qvB = \frac{mv^2}{r}$ gives the radius of the path:

$r = \frac{mv}{qB}$

Faster or heavier particles swing in wider circles. A stronger field or a bigger charge tightens the circle. The time for one turn, $T = \frac{2\pi m}{qB}$ , does not depend on the speed at all, which is the idea behind the cyclotron in Class 12. If an electric field is also present, add its push qE to get the full Lorentz force formula.

If the velocity has a part along B, the path becomes a helix: a circle from the crossing part, and steady drift from the part along the field.

Units and dimensions check

Force is mass times acceleration, giving the bracket form [M¹ L¹ T⁻²] whatever produced it. You can test the formula: charge is [I T], speed is [L T⁻¹] and the tesla works out to [M T⁻² I⁻¹]. Multiplying gives [I T][L T⁻¹][M T⁻² I⁻¹] = [M L T⁻²], which matches. The sine is a pure number and adds no dimensions. For the wire form, current times length is [I L], and [I L][M T⁻² I⁻¹] gives the same answer.

This also explains why moving charge in a wire and a free electron obey one rule. A current is just charge per second, so $I L$ plays the same role as $q v$ . For more on field strength see the magnetic flux density formula.

Mistakes students make with magnetic force

Using cos instead of sin. The force is largest at 90°, so the sine is correct. Check by putting θ = 0: the force must vanish.
Measuring θ from the wrong line. θ is between v (or I) and B, not between v and the force.
Forgetting millitesla and microtesla. Divide mT by 1000 and μT by a million before you substitute.
Giving the electron a negative force size. Write the size as positive and state the reversed direction in words.
Saying the force speeds a charge up. It cannot. It only bends the path, because it is always sideways to the motion.

Frequently asked questions

What is the formula of magnetic force?

For a moving charge it is F = qvB sin θ. For a current carrying wire it is F = BIL sin θ. In both, θ is the angle between the motion (or current) and the field.

In which unit is magnetic force measured?

Like every force, it is measured in newtons (N). The tesla is defined so that 1 C moving at 1 m/s across a 1 T field feels 1 N.

When is the magnetic force zero?

It is zero when the charge is at rest, when it moves parallel or antiparallel to the field (θ = 0° or 180°), or when the particle has no charge, such as a neutron.

Does the magnetic force do work on a charge?

No. Since the push stays perpendicular to the motion, it bends the path while the speed stays fixed. The kinetic energy of the particle stays constant.

What is the difference between magnetic force and magnetic field?

The field B, in tesla, describes the region around a magnet or current. The force F, in newtons, is what a moving charge or a current actually feels when it is placed in that field.