# 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.

Page: https://formulabro.yunusemrevurgun.com/electricity/magnetic-force-on-moving-charge/  
Category: Electricity  
Published: 2026-10-22  
Updated: 2026-10-22

## Formula

$$
F = |q|\, v\, B \sin\theta
$$

Read it out loud: "F equals the absolute value of q, times v, times B, times the sine of theta. The magnetic force F equals the size of the charge q, times its speed v, times the magnetic field strength B, times the sine of the angle theta between its motion and the field."

## Before you start

- **Magnetic field $B$ and the tesla (T)**: A **magnetic field** is the invisible influence around a magnet or a moving electric charge. It pushes on other magnets and on moving charges, but it does not push on a charge that is sitting still. Its strength is measured in **tesla** (T), where one tesla is one newton per ampere per meter. Earth's magnetic field at the surface is only tens of microteslas, so a tesla is a very strong field.
- **Electric charge and the elementary charge $e$**: Electric charge makes particles push or pull each other. Its unit is the **coulomb** (C). The **elementary charge**, written $e$, is the size of the charge on one electron or one proton: $e = 1.602176634 \times 10^{-19}$ C. An electron carries a negative charge of that size, written $-e$, and a proton carries $+e$.
- **Absolute value bars, $|q|$**: The vertical bars around a number mean **absolute value**, which is the size of the number with its sign removed. So $|-3| = 3$ and $|3| = 3$. The formula uses $|q|$ because the size of the push depends only on how much charge there is. The sign of the charge only sets the direction of the push.
- **Sine and degrees**: The **sine**, written $\sin$, turns an angle into a number from $-1$ to $1$. $\sin 0^\circ = 0$, $\sin 30^\circ = 0.5$ and $\sin 90^\circ = 1$. The small circle after a number means degrees, where a full turn is 360 degrees. A calculator in radian mode gives a different answer, so check its mode before you type an angle.
- **Powers of ten, like $10^{6}$ and $10^{-19}$**: A **power of ten** is ten multiplied by itself a set number of times. $10^{6}$ is a 1 followed by six zeros, which is 1,000,000. A negative power divides: $10^{-19}$ is 0.0000000000000000001, with the 1 in the nineteenth place after the decimal point. When two powers of ten are multiplied, their exponents add. A number such as $2.0 \times 10^{6}$ means 2 with the decimal point moved six places to the right.
- **Vectors and the cross product**: A **vector** has a size and a direction. Velocity is a vector: speed with a direction. An arrow over a letter, as in $\vec{v}$, marks a vector. The sign $\times$ between two vectors is the **cross product**. Its result points at right angles to both vectors, and the right-hand rule tells you which way. The vector form of the force uses this product.

More notation help: https://formulabro.yunusemrevurgun.com/math-basics/

## What each symbol means

- $F$: **Magnetic force**, the size of the push on the charge, measured in newtons (N). One newton is about the weight of a 100 gram apple, because Earth's gravity pulls on that mass with about 1 N. The example push is about $1.6 \times 10^{-13}$ N, far too small to feel.
- $|q|$: The **size of the charge**, measured in coulombs (C). For an electron, $|q| = e = 1.602176634 \times 10^{-19}$ C. Only the size goes into the formula, and the sign of the charge sets the direction of the push.
- $v$: The **speed** of the charge, measured in meters per second (m/s). The example electron moves at $2.0 \times 10^{6}$ m/s, which is 2,000,000 meters every second. That is far faster than a car, although it is still well below the speed of light.
- $B$: The **magnetic field strength**, measured in tesla (T). The example uses 0.5 T, which is thousands of times stronger than the magnetic field of the Earth at its surface, a field of tens of microteslas.
- $\theta$: The **angle** between the velocity and the magnetic field, measured in degrees. At 90 degrees the sine is 1, which gives the largest push. At 0 degrees the sine is 0, and there is no push. At 30 degrees the sine is 0.5, so the push is half the largest possible push.

## The formula piece by piece

- $|q|$: Start with the size of the charge, with its sign removed. It is how much charge the particle carries, in coulombs. The more charge there is, the bigger the push, in direct proportion.
- $v\, B$: Multiply the speed by the field strength. A faster particle pushes harder in the same field, and a stronger field pushes harder on the same particle. Doubling either one doubles the push, because the force grows in step with each of them.
- $\sin\theta$: The angle factor, a number from 0 to 1. It says how much of the motion is across the field, which is the only part that gets pushed. At right angles it is 1, and along the field it is 0.
- $F = |q|\, v\, B \sin\theta$: The whole formula in one sentence: the size of the magnetic force on a moving charge equals the size of the charge times its speed times the field strength times the sine of the angle between its motion and the field.
- $\vec{F} = q\,\vec{v} \times \vec{B}$: The same law in vector form, which also gives the direction. The cross product $\times$ makes a result at right angles to both the velocity and the field. The sign of $q$ decides whether the push points along that direction or against it.

## What the formula is doing, and why

A magnetic field $B$, measured in tesla (T), exerts a force only on a charge that is moving. A charge at rest feels no magnetic force, although it may still feel an electric force. The charge $|q|$ is measured in coulombs (C), and the speed $v$ is in meters per second (m/s). The bars around $q$ mean that we use only the size of the charge here.

**Why multiply?** Each factor is a separate reason for a bigger push. More charge means more charge to push. A faster charge is pushed harder by the same field. A stronger field pushes harder. Measurements show the force is proportional to each of the three, so the formula multiplies them. Double any one of the three and the force doubles.

**Why the sine?** Only the part of the velocity that is across the field produces the push. Split the velocity into two parts, one along the field and one across it. The part along the field produces no push. The sine is 1 at right angles and 0 when the motion lines up with the field, so it picks out the part that matters. That is why a charge moving along the field lines feels nothing.

**Why at right angles, and why circles?** The force is always at right angles to both the velocity and the field. A force at right angles to the motion does no work, so it changes the direction of motion but not the speed. A charge moving across a uniform field, meaning a field with the same strength and direction everywhere, therefore travels in a circle. This is the principle behind cyclotrons and mass spectrometers.

**Which way?** The direction follows the **right-hand rule**. In vector form the force is $\vec{F} = q\,\vec{v} \times \vec{B}$. Point the fingers of your right hand along the velocity, curl them toward the field, and your thumb points where a positive charge is pushed. An electron has a negative charge, so it is pushed the opposite way.

## An everyday picture

Think of a ball on a string swung in a circle. The string pulls sideways, always at right angles to the motion, so the ball keeps its speed while its path bends into a circle. A magnetic field acts on a moving charge in the same way. It pushes sideways, at right angles to the motion, and bends the path without changing the speed.

## Worked example: Electron crossing a magnetic field

An electron moves at $v = 2.0 \times 10^{6}$ m/s through a uniform magnetic field of $B = 0.5$ T. The size of its charge is $|q| = e = 1.602176634 \times 10^{-19}$ C. Find the magnetic force when the electron moves at right angles to the field, then when it moves at 30 degrees to the field.

1. **Write down the size of the charge.** The electron's charge has size $|q| = e = 1.602176634 \times 10^{-19}$ C. The $10^{-19}$ means divide by ten nineteen times, so one electron carries a very small fraction of a coulomb. Electrons are the usual charge carriers in metal wires and in electron beams.
2. **Set up the formula.** We use $F = |q|\, v\, B \sin\theta$ with $v = 2.0 \times 10^{6}$ m/s and $B = 0.5$ T. The $2.0 \times 10^{6}$ means 2 with the decimal point moved six places to the right, which is 2,000,000 meters per second. That is a very fast speed compared with a car.
3. **Use 90 degrees.** The sine of 90 degrees is 1. That means all of the velocity is across the field, so the sine does not reduce the push at all. This is the largest force the electron can feel at this speed and field.
4. **Multiply the numbers.** The ordinary numbers multiply to $1.602176634 \times 2.0 \times 0.5 = 1.602176634$. The powers of ten multiply to $10^{-19} \times 10^{6} = 10^{-13}$, because the exponents add: $-19 + 6 = -13$. So $F = 1.602 \times 10^{-13}$ N. In full, that is 0.0000000000001602 newtons.
5. **Check the unit.** A tesla is one newton per ampere per meter, and an ampere is one coulomb per second. Multiply coulombs by meters per second and by teslas, and the coulombs and seconds cancel, which leaves newtons. So the answer really is a force.
6. **Use 30 degrees.** The sine of 30 degrees is 0.5, so only half of the velocity is across the field. The force is half as big: $F = 1.602 \times 10^{-13}\ \text{N} \times 0.5 = 0.801 \times 10^{-13}$ N. Moving the decimal point one place right makes 0.801 into 8.01, and the exponent drops by one, so $F = 8.01 \times 10^{-14}$ N.
7. **Find what the force does to the electron.** Force divided by mass gives acceleration. The electron's mass is $9.1093837 \times 10^{-31}$ kg. Dividing $1.602 \times 10^{-13}$ N by that mass gives about $1.76 \times 10^{17}$ m/s$^2$. The exponents subtract, $-13 - (-31) = 18$, and $1.602 / 9.109$ is about 0.176, so the result is $0.176 \times 10^{18}$, which is $1.76 \times 10^{17}$ m/s$^2$. The force is tiny, but the electron's mass is so small that this force bends its path sharply.

**Answer:** At right angles the magnetic force on the electron is about **$1.60 \times 10^{-13}$ N**. At 30 degrees it drops to about **$8.01 \times 10^{-14}$ N**. The force is tiny in everyday terms, but the electron's mass is so small that the force gives it an acceleration of about $1.76 \times 10^{17}$ m/s$^2$.

## Common mistakes

- Entering the angle in degrees into a calculator set to radians. Typing 90 with the calculator in radian mode gives a sine of about 0.89, not 1. Set the mode to degrees, or convert the angle to radians by multiplying by $\pi/180$.
- Expecting a charge at rest to feel a magnetic force. Only moving charges feel this push. A charge sitting still in a magnetic field feels nothing from it.
- Forgetting the sine. The force is zero when the charge moves along the field lines, and a formula without the sine would wrongly give a push in that case.
- Mixing up the direction. The right-hand rule gives the direction for a positive charge. An electron is pushed the opposite way, because its charge is negative.

## Where you will meet it

- Cyclotrons and other particle accelerators, which bend charged particles into circles with magnetic fields.
- Mass spectrometers, which sort ions by how much their paths bend in a magnetic field.
- Older cathode ray tube televisions and monitors, which steer the electron beam with magnetic fields.
- Auroras, where charged particles from the Sun are guided by Earth's magnetic field toward the polar regions.

## Questions people ask

### Why does a magnetic field not speed up a charge?

The force is always at right angles to the motion, so it does no work on the charge. It can change the direction of travel, but it cannot change the kinetic energy, which is the energy of motion. The speed stays the same.

### What is a tesla?

The SI unit of magnetic field strength: one newton per ampere per meter. Earth's magnetic field at the surface is only tens of microteslas, so a tesla is a very strong field.

### Why does the force vanish when the charge moves along the field?

The sine of 0 degrees is zero. When the motion is parallel to the field, there is no sideways part of the velocity to push, so the force is zero.

## Related formulas

- [Force on a Current-Carrying Wire](https://formulabro.yunusemrevurgun.com/electricity/force-on-current-carrying-wire/)
- [Magnetic Field of a Straight Wire](https://formulabro.yunusemrevurgun.com/electricity/magnetic-field-of-straight-wire/)
- [Faraday's Law of Induction](https://formulabro.yunusemrevurgun.com/electricity/faradays-law/)
- [Electric Field of a Point Charge](https://formulabro.yunusemrevurgun.com/electricity/electric-field-of-point-charge/)

## Sources

- [Lorentz force, Wikipedia](https://en.wikipedia.org/wiki/Lorentz_force)
- [Right-hand rule, Wikipedia](https://en.wikipedia.org/wiki/Right-hand_rule)
- [Elementary charge, Wikipedia](https://en.wikipedia.org/wiki/Elementary_charge)

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