Right-Hand Rule

Short Description of Topic

The Main Idea

The Right-Hand Rule is an easy way to find the direction of a cross product interaction before doing the math.

A Mathematical Model

The Right-Hand Rule is mathamatically modeled by the cross product:

[math]\displaystyle{ \mathbf{u\times v}=(u_2v_3\mathbf{i}+u_3v_1\mathbf{j}+u_1v_2\mathbf{k}) -(u_3v_2\mathbf{i}+u_1v_3\mathbf{j}+u_2v_1\mathbf{k}) }[/math]

A Computational Model

The cross product is used to describe many magnetic interactions, for example, magnetic field created by a moving charge or a current and magnetic force on a particle by a magnetic field. Because of this, using the right hand rule, to determine the direction of a cross product, can be a useful to check behind the math for sign errors.

Follow the chart bellow to find which fingers correspond to which vectors.

[math]\displaystyle{ \mathbf{A\times B}=\mathbf{C} }[/math]

Vector	Right-hand	Right-hand (alternative)
A	First or index	Thumb
B	Second finger or palm	First or index
C	Thumb	Second finger or palm

Examples

Magnetic Force on a Moving Particle

[math]\displaystyle{ \mathbf{F} = q\mathbf{v} \times \mathbf{B} }[/math]

The direction of the cross product may be found by application of the right hand rule as follows:

1. The index finger points in the direction of the momentum vector qv.
2. The middle finger points in the direction of the magnetic field vector B.
3. The thumb points in the direction of magnetic force F.

For example, for a positively charged particle moving to the right, in a region where the magnetic field points up, the resultant force points out of the page.

Magnetic Field made by a Current

[math]\displaystyle{ \mathbf{B} = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2}, }[/math]

The direction of the cross product may be found by application of the right hand rule as follows:

1. The thumb points in the direction of current I.
2. The index finger points in the direction of the observation vector r.
3. The middle finger points in the direction of the magnetic field vector B.

For example, for a current moving out of the page, the magnetic field points up, when the observation location is to the right of the current.

Force on a Current from a Magnetic Field

[math]\displaystyle{ \mathbf{F} = mathbf{I} \times \mathbf{B} }[/math]

The direction of the cross product may be found by application of the right hand rule as follows:

1. The index finger points in the direction of the current I.
2. The middle finger points in the direction of the magnetic field vector B.
3. The thumb points in the direction of magnetic force F.

For example, for a current moving into the page, in a region where the magnetic field points up, then the force is to the right of the current.

References

1. https://en.wikipedia.org/wiki/Right-hand_rule
2. https://en.wikipedia.org/wiki/Magnetic_field