Right-Hand Rule

Edited by Darye Ji Fall 2018 (25 November 2018)

The Main Idea

The Right-Hand Rule is an easy way to find the direction of a cross product interaction before doing the math. For any equation involving a cross product, the right hand rule is a valuable tool for finding the direction. There are two primary ways of using the right hand rule.

1) The first method is to use your entire right hand. In the example below, the velocity is pointing north up (+y) and the magnetic field is pointing to the left (-x). We place our hand with our thumb sticking up along the velocity because that is the first variable in F= qv X B. Next, we curl our fingers towards the magnetic field. Our thumb is pointing outwards toward us; therefore, the direction of magnetic force is out of the page (+z). It is important to note that this only applies to positive charges. For negative charges, this direction is opposite.

2) The second method also uses your entire right hand to find the direction of the conventional current from your magnetic field. In this method, you use your thumb to point in the direction of the magnetic field, and the direction your other fingers point is the direction of the conventional current. In this specific example, the direction of the magnetic field is out of the page (+z), meaning that the conventional current goes in the counter-clockwise direction. Vice versa, if the magnetic field is into the page (-z), then the conventional current would be in the clockwise direction.

Right-Hand Rule for Force

A Mathematical Model

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

[math]\displaystyle{ \mathbf{u\times v}=(u_2v_3sin\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]

The cross product can also be solved in the following form (using u and v):

[math]\displaystyle{ \mathbf{u} \times \mathbf{v} = \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| \sin (\theta) }[/math]

Cross Product: Alternate Method 1

If you are like me and for some particular reason it was hard to grasp the traditional method of carrying out a cross product, fear not! There is an alternate method that will lead you to the same answer in a more an intuitive manner.

This is a method I picked up a while back and has been proving quite useful since.1

It involves structuring the three components in a circle and assigning a clockwise direction around this circle.

Every time you cross one component with another component that can be reached clockwise on on the next move around the circle, you know that you will have a positive direction in the direction of the third direction.

This can be seen in the picture below.

In order to quickly solve a cross product using this alternate method, see the following example.

1 NOTE: I did not devise this method, but simply learned it.

Cross Product: Alternate Method 2

Cross Product: Alternate Method 3

If you're personally more of a math person, and you have taken linear algebra or have done determinants before, cross products are essentially taking determinants of a 3x3 matrix.

1. For the first row, write down i, j, k. These are your directional components with "i" representing the +x direction, "j" the +y direction, and "k" the +z direction.

2. If you are crossing a x b, write down the a components in the second row and b components in the third row corresponding to the x,y,z direction. If for example, ax is in the -x direction, simply make the value of ax negative.

3. Then take the determinant of the 3x3 matrix which is shown below. Don't forget that the second term (j) must be negative.

Another Alternate Right Hand Rule

If you didn't understand the first three right hand rules, hopefully this fourth one will be the charm.

Let us take a simple physics problem as an example. We know that [math]\displaystyle{ \mathbf{F} = q\mathbf{v} \times \mathbf{B} }[/math]

Instead of using our right hand, we can just use the method pictured below to find the direction of the magnetic force given the direction of the moving particle and the magnetic field.

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)	Right-hand (alternative 2)
A	First or index	Thumb	Palm
B	Second finger or palm	First or index	First finger
C	Thumb	Second finger or palm	Thumb

Another method for determining the direction of the product orthogonal vector is to place the fingers of your right hand in the direction of the first vector(A). Curl your fingers in the direction of the second vector(B), effectively making the "thumbs up" sign in whichever direction the thumb happens to be pointing. The resulting vector(C) is in the direction in which your thumb is now pointing.

Examples

Magnetic Field for a Point Charge (Bio-Savart Law)

The equation written in pink in the picture below is the Bio-Savart Rule for a single charged particle which will be later explained in further chapters. As you can see, there is a cross product within the equation, and it can be a little tricky finding the direction of the magnetic field.

Using the Right-Hand rule (alternative 2):

    1. place your palm in the direction of velocity since it is velocity crossed rhat
          2. curl your finger to the direction of rhat
                3. Point your thumb up( perpendicular from your fingers) and this direction is the direction of your magnetic field, assuming that the charge is positive.
                    If your charge is negative, then simply flip the direction of your thumb.

In the picture below, the direction of the magnetic field would be out of the page (+z) if the particle is positively charged. If the charge was an electron (negatively charged), then the direction of the magnetic field would be into the page (-z).

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.

Direction of Magnetic Field made by a Current

The direction of curl of the magnetic field can be found using a modified right hand rule:

1. The thumb points in the direction of conventional current.
2. The curl of the fingers indicates the direction of magnetic field at each location surrounding the flow of current.

For example:

1. In the image below, conventional current flows to the left (-x). By pointing your thumb in the -x direction and curling your fingers, the magnetic field curls in the y,z-plane, as shown.

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.

Another Force on a Current from a Magnetic Field

In the situation shown below, we have a current pointing downward or in the negative y direction and we have a magnetic field into the page or in the negative z direction. Instead of using the typical right hand rule, we can use easy cross product method below to find the direction of the force.

Hall Effect Example

In the above picture, you see that all the positive charges accumulated at the top and all the negative charges accumulated to the bottom. You know the direction the particles velocity and magnetic field. Can you find out the charge of the particle using the right hand rule?

In the picture, the particles are coming out of the negative terminal, so they are electrons. This can be verified by using the right hand rule. You curl your fingers from velocity vector to magnetic field vector to find the direction of magnetic force perpendicular. Your thumb is pointing up, but since these are negative charges, its opposite and you flip your hand and you find that the direction of the magnetic force is actually pointing down. Therefore it makes sense that the electrons would accumulate at the bottom since its magnetic force is pushing them towards there.

Connectedness

1. How is the topic connected to something you are interested in?

I am interested in being able to find how induced currents are generated from the presence of other particles or electric fields. Using the Right-Hand-Rule is actually a simple application of complex Physics principles. A lot of what we learn is Physics 2 is abstract knowledge, so it is interesting to think of these concepts in a more tangible form.

2. How is it connected to your major?

As an Industrial Engineering major, knowing how to calculate cross products is not directly related to my major, nor my Supply Chain concentration. However, being able to understand how magnetic fields, currents, charges of particles, force, and electric fields all connect to each other is an important concept to understand both in college and in the real world. Physics principles, specifically cross product, are used quite frequently in shortest path problems.

3. Is there an industrial application?

When creating wheels, knowledge of the cross product controls how the wheel is built and fixed when it is broken. Whenever two objects work in tandem or against each other, maximum impact is produced when the two vectors are perpendicular (because [math]\displaystyle{ \sin (90)=1 }[/math]).

History

John Ambrose Fleming is credited with devising the right hand rule. He was a professor at the University College, London where he was liked by many of his students. He taught them how to easily determine the direction of a current. He made directional relationships easier between current, its magnetic field, and the electromotive force.

See Also

External Links

1. https://www.khanacademy.org/test-prep/mcat/physical-processes/magnetism-mcat/a/using-the-right-hand-rule
2. https://ocw.mit.edu/courses/physics/8-02t-electricity-and-magnetism-spring-2005/lecture-notes/prs_w06d1.pdf
3. http://physics.bu.edu/~duffy/semester2/d12_RHR_practice.html

References

1. https://en.wikipedia.org/wiki/Right-hand_rule
2. https://en.wikipedia.org/wiki/Magnetic_field
3. https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/john-ambrose-fleming

--Cjacobson7 (talk) 13:45, 10 November 2015 (EST)