Force on a Current-Carrying Wire

In one sentence: A wire that carries electric current is pushed sideways by a magnetic field, and the push grows with the current, the length of wire in the field and the field strength.

$F = B\, I\, L \sin\theta$

Read it out loud: “F equals B times I times L times the sine of theta. The force F equals the magnetic field B, times the current I, times the length L of the wire inside the field, times the sine of the angle theta between the wire and the field.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Electric current $I$ and the ampere (A): Current is the flow of electric charge past a point. Its unit is the ampere (A), and one ampere means one coulomb of charge passes a point every second. A 100 W light bulb on a 120 V supply draws about 0.83 A, because 100 watts divided by 120 volts gives about 0.83 amperes.
Magnetic field $B$ , the tesla (T) and the gauss: A magnetic field pushes on moving charges and on wires that carry current. Its strength is measured in tesla (T), where one tesla is one newton per ampere per meter. An older unit, the gauss (G), is smaller: one tesla equals $10^{4}$ gauss, which is 10,000 gauss. So a field of 2000 gauss is 0.2 tesla, because 2000 divided by 10,000 is 0.2.
Powers of ten and centimeters: A power of ten like $10^{4}$ is ten multiplied by itself four times, which is 10,000. A negative power divides, so $10^{-2}$ is one hundredth. The prefix centi (c) means one hundredth, so 1 cm equals 0.01 m. A length of 30 cm is $30 \times 0.01 = 0.30$ m. Convert lengths to meters before using a formula that wants SI units.
Sine and the angle between wire and field: The sine, written $\sin$ , turns an angle into a number from $-1$ to $1$ . It is 0 for an angle of 0 degrees, 0.5 at 30 degrees and 1 at 90 degrees. Here it measures how much of the wire lies across the magnetic field. A wire running exactly along the field has a sine of 0, so it feels no push.
Electrons in a wire: Current in a metal wire is a flow of electrons. Each electron is a tiny charge that moves through the metal, and it drifts slowly, much more slowly than a walking pace, while the electric signal travels along the wire much faster. The force on the whole wire is the sum of the forces on all of its moving electrons.

What each symbol means

Symbol	Meaning
$F$	Force on the wire, the sideways push, measured in newtons (N). One newton is roughly the weight of a 100 gram apple. In the example the push on the wire is 0.30 N, which is about the weight of 31 grams.
$B$	Magnetic field strength, measured in tesla (T). One tesla is one newton per ampere per meter. The example field is 0.2 T, which is 2000 gauss and much stronger than the Earth’s magnetic field at its surface.
$I$	Current in the wire, measured in amperes (A). One ampere is one coulomb of charge passing a point every second. The example wire carries 5 A, the same current that a 600 W heater draws from a 120 V supply.
$L$	Length of the wire that lies inside the magnetic field, measured in meters (m). Only the part of the wire inside the field counts. The example wire is 0.30 m long, which is 30 cm, and all of it lies in the field.
$\theta$	The angle between the wire and the magnetic field, measured in degrees. The force is biggest when the wire is at right angles to the field, where the sine is 1, and zero when the wire lies along the field, where the sine is 0.

The formula piece by piece

$L$: Start with the length of the wire inside the field, in meters. The longer the piece in the field, the more moving charges are pushed at once, so the force grows with the length.
$I\, L$: Multiply the current by the length. The current counts the charges that pass each second, and the length counts how much of the wire is in the field. Their product measures how much moving charge the field is pushing.
$B\, I\, L$: Multiply by the field strength. Each moving charge feels a push that is proportional to the field, so the total push is the field times the current times the length. Doubling any one of the three doubles the force.
$\sin\theta$: The angle factor, a number from 0 to 1. It tells you how much of the wire lies across the field, and only that part gets pushed. At right angles it is 1, and along the field it is 0.
$F = B\, I\, L \sin\theta$: The whole formula in one sentence: the force on a straight wire in a magnetic field equals the field strength times the current times the length of wire in the field, times the sine of the angle between the wire and the field.
$\vec{F} = I\,\vec{L} \times \vec{B}$: The same law in vector form. The vector $\vec{L}$ points along the wire in the direction of the current, and the cross product gives a force at right angles to both the wire and the field. The right-hand rule gives its direction.

What the formula is doing, and why

A current is a stream of moving charges, so each charge in the wire feels the magnetic force described on the moving-charge page. Add up the forces on all the charges along the wire and you get the force on the whole wire. The formula $F = B I L \sin\theta$ gives that total in newtons (N), with $B$ in tesla (T), $I$ in amperes (A) and $L$ in meters (m).

Why multiply the three parts? Each factor counts something that adds to the push. Double the current and the force doubles, because twice as much charge passes through the field each second. Double the length inside the field and the force doubles, because twice as many charges are pushed at once. Double the field strength and the force doubles as well. Each factor scales the push in direct proportion, so the three multiply.

Why the sine, and which way? The sine sets how much of the wire is pushed. A wire at right angles to the field feels the full push, and a wire that lies along the field feels none. The force is at right angles to both the wire and the field. Point the fingers of your right hand along the current, curl them toward the field, and your thumb points along the force.

The electric motor. This is the basis of the electric motor. A coil carrying current sits between magnets. The current in the two sides of the coil flows in opposite directions, so the two sides feel forces in opposite directions. Those forces make a turning effect, called torque, that spins the shaft. Each side of the coil obeys this formula.

When it applies. The formula is exact for a straight wire in a uniform field, meaning a field with the same strength and direction everywhere. A curved wire, or a field that changes from place to place, needs the forces on short pieces of wire added up along its length. The formula also assumes the whole wire lies in the field, or else uses only the part that does.

An everyday picture

Think of a crowd of walkers in a long hallway, all moving in the same direction, while a wind blows sideways across the hall. The total sideways push depends on how many people walk past each second, how long the section of hallway is and how hard the wind blows. A wire is a hallway full of moving charges, and the magnetic field is the wind.

Worked example: Force on a 30 cm wire carrying 5 A

A straight wire 30 cm long carries a current of 5 A. It sits in a uniform magnetic field of 0.2 T. Find the force when the wire is at right angles to the field, then when it is tilted to 30 degrees from the field direction.

Convert the length to meters. The wire is 30 cm long. Since 1 cm is 0.01 m, 30 cm is $30 \times 0.01 = 0.30$ m. The formula needs meters, so this conversion comes before any other step.
Use 90 degrees first. When the wire is at right angles to the field, $\sin 90^\circ = 1$ . The whole length of the wire feels its full push, so the sine does not reduce anything here.
Multiply the numbers. $F = B I L \sin\theta = (0.2\ \text{T})(5\ \text{A})(0.30\ \text{m})(1)$ . First $0.2 \times 5 = 1.0$ , then $1.0 \times 0.30 = 0.30$ . So $F = 0.30$ N, which is a push of 0.30 newtons on the wire.
Check the unit. A tesla is one newton per ampere per meter. Multiplying by amperes and meters cancels both of them, which leaves newtons. So the product is a force, as it should be.
Use 30 degrees. When the wire is tilted to 30 degrees from the field, $\sin 30^\circ = 0.5$ , so only half of the wire’s length lies across the field. The force is half as big: $F = 0.30\ \text{N} \times 0.5 = 0.15$ N.
Say the force in everyday terms. A force of 0.30 N is about the weight of 31 grams. Weight is the force of gravity, about 9.8 N for each kilogram, so 0.30 divided by 9.8 is about 0.0306 kg, which is about 31 grams.

Answer: At right angles the wire feels a force of 0.30 N, and at 30 degrees it feels 0.15 N. The force pushes the wire sideways, at right angles to its length. The 0.30 N force is about the weight of 31 grams.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using the full length of the wire when only part of it is in the field. Only the length inside the field counts for $L$ . A part of the wire outside the field gives no push from the field.
Forgetting the sine. A wire lying along the field feels no force, even though it carries current. A formula without the sine would give a push where there is none.
Assuming the force points along the wire. The force is at right angles to both the wire and the field, so it pushes the wire sideways.
Using a field in gauss without converting. One tesla equals $10^{4}$ gauss, so a field of 2000 gauss is 0.2 T. Divide the gauss value by 10,000 before you use it.

Where you will meet it

Electric motors in fans, drills, electric cars and many appliances, where forces on a coil turn the shaft.
Loudspeakers, where a voice coil sits in a magnet and is pushed back and forth by the current that carries the sound signal.
Moving-coil meters, which swing a needle in proportion to the current in a coil.
Hard disk drive actuators, where a coil in a magnet swings the read head across the disk.

Questions people ask

Why is the wire pushed sideways?

Each moving charge in the wire feels a force at right angles to its motion and to the field. Added together, these forces push the whole wire at right angles to its length.

How does this relate to the force on a single moving charge?

The force on the whole wire is the sum of the forces on all the charges moving in it. The current $I$ counts how many charges pass each second, so it turns the single-charge force $qvB\sin\theta$ into $ILB\sin\theta$ for the whole wire.

What happens if I reverse the current?

The force reverses direction. Reversing the field also reverses it, but reversing both leaves the push in the same direction as before.

Magnetic Force on a Moving Charge: A magnetic field pushes a moving charge sideways, and the push is strongest when the charge moves straight across the field.
Magnetic Field of a Straight Wire: A current in a straight wire makes a magnetic field that wraps around the wire, and the field gets weaker the farther away you measure it.
Magnetic Field of a Solenoid: A long coil of wire makes a strong, nearly even magnetic field along its middle, and the field gets stronger when more loops are packed into each meter of coil.
Faraday's Law of Induction: A changing magnetic field through a loop of wire pushes a voltage into the wire, and that is how generators make electricity.