Right Hand Rule (Physics): Direction Of Magnetic Forces

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:

\(F=qv\times B\)

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.

The general formula for a vector cross product is:

\(a \times b = |a| |b| \sin{\theta} n\)

•|​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​

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:

1. Point the index finger of your right hand along vector ​a​, the direction of the current or the charge's velocity.
2. Point the middle finger of your right hand along vector ​b​, into the direction of the magnetic field. 
3. 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?

\(F=I\times B \sin{\theta}=20\times 30\sin{15}=155.29\text{ N}\)

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)!