Determining the direction in which magnetic forces act can be tricky. Understanding the right-hand rule makes this easier.

## Magnetic Forces

The Lorentz force law relates a magnetic field to the force felt by a moving electric charge or current that encounters it. This law can be expressed as a vector cross product:

F = qv × B

for a charge *q* (in coulombs, C) moving with velocity *v* (in meters per second, m/s) in a magnetic field *B* (measured in teslas, T). The SI unit of force is the newton (N).

For a collection of moving charges, a current, this can be expressed instead as F = I × B, where current *I* is measured in amperes (A).

The direction of the force acting on either the charge or the current in a magnetic field is determined by the right-hand rule. Additionally, because force is a vector, if the terms in the law are not at right angles to one another, its magnitude and direction are a component of the given vectors. In this case, some trigonometry is needed.

## Vector Cross Products and the Right-Hand Rule

The general formula for a vector cross product is:

**a** × **b** = |**a**| |**b**| sin(θ) **n**

- |
**a**| is the magnitude (length) of vector**a** - |
**b**| is the magnitude (length) of vector**b** - θ is the angle between
**a**and**b** **n**is the unit vector at right angles to both**a**and**b**

[insert supporting diagram]

If vector **a** and vector **b** are in a plane, the resulting direction of the cross product (vector **c**) could be perpendicular in two ways: pointing up or down from that plane (pointing into or out of it). In a Cartesian coordinate system, this is another way to describe the z-direction when vectors **a** and **b** are in the x-y plane.

In the case of the Lorentz force law, vector **a** is either the charge's velocity *v* or the current *I*, vector **b** is the magnetic field *B* and vector **c** is the force *F.*

So how can a physicist tell if the resulting force vector is pointing up or down, into or out of the plane, or in the positive or negative z-direction, depending on the vocabulary she wants to use? Easy: She uses the right-hand rule:

- Point the index finger of your right hand along vector
**a**, the direction of the current or the charge's velocity. - Point the middle finger of your right hand along vector
**b**, into the direction of the magnetic field. - Look where the thumb points. This is the direction of vector
**c**, the cross product and the resulting force.

Note that this works for a positive charge only. If the charge or current is *negative*, the force will actually be in the *opposite* direction of where the thumb ends up pointing. However, the *magnitude* of the cross product doesn't change. (Alternatively, using the left hand with a negative charge or current will result in the thumb pointing in the correct direction of the magnetic force.)

## Examples

A 20-A conventional current flows in a straight wire at a 15-degree angle through a 30-T magnetic field. What force does it experience?

F = I × B sin(θ)

F = (20 A)(30 T)sin(15) = 155.29 N outwards (positive z-direction).

Note that the direction of the magnetic force remains perpendicular to the plane containing both the current and the magnetic field; the angle between those two differing from 90 degrees only changes the *magnitude* of the force.

This also explains why the sine term can be dropped when the vector cross product is for perpendicular vectors (since sin(90) = 1) and also why a charge or current moving **parallel to a magnetic field** experiences **no force** (since sin(0) = 0)!