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:
- The index finger points in the direction of the momentum vector qv.
- The middle finger points in the direction of the magnetic field vector B.
- 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:
- The thumb points in the direction of current I.
- The index finger points in the direction of the observation vector r.
- 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:
- The index finger points in the direction of the current I.
- The middle finger points in the direction of the magnetic field vector B.
- 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.