Determining the direction in which magnetic forces act can be tricky. Understanding the right-hand rule makes this easier.
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:
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| 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
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.)
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?
And the direction is 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)!