Electromagnetism Basics Part 1 : The Cross Product

Electromagnetism is a very interesting phenomenon and can be a little tough to understand when starting to develop applications that include the concept. The post will cover some of the basic concepts like the “Right Hand Rule”, Magnetic Permeability, Faraday’s Law, Lenz’s Law, and other related topics.

Right Hand Rule for Vectors

The first detail I should cover is the right hand rule for vectors. Some of the mathematic formulas include vectors. Vectors are quantities with magnitude and direction (quantities can be anything from length, force, magnetic field, etc…). The rule applies to an operation called the cross product. Here is the format shown in a formula:

\vec{A}\times\vec{B}=|\vec{A}|*|\vec{B}|*sin(\theta)*\hat{n}

The formula is read as vector A cross vector B equals the magnitude of vector A times the magnitude of vector B times the sine of the angle between the vectors times the unit vector n. The unit vector n has a magnitude of one and has a direction based on the right hand rule. The order of multiplication matters in the cross product. A X B will not equal B X A because the result vector will be in the opposite, orthagonal direction. Let’s take a look at an example of the cross product:
VectorsExample
Let’s assume we take the cross product of AXB which equals 5x7.2xsin(98º) times the unit vector n. The value for the magnitude of vector C would be 35.6, however, we need to determine if the vector points towards us or points away from us because it must be orthagonal to both vectors A and B. This is where the right hand rule is used.

Three Finger Method

RHRVectorDiagram
The pointer finger represents the first vector used (A in this case), the middle finger represents the second vector, and the thumb represents the result vector. Start with the fingers extended as seen in the photo, keep them looking like a 3D axis. Next, you will have to twist your hand in a way where the pointer and middle fingers line up with the direction of the other two vectors (can be difficult to line up depending on the angle). Here is how I oriented my hand to match the vectors:


Of course I can’t really draw vector C straight out of the page, so technically the visual is more of an “perspective” view of what is occuring. When drawn on a 2D diagram, a circle with a dot (⦿) in the center stands for coming out of the plane while a circle with a cross(⦻) stands for going into the plane.
VectorCrossAXBResult
Now if you cross vector B with A, you will got the opposite direction because the order of which vector is multiplied first, therefor making the right hand rule very useful to determine the direction of the result vector. Here is how I oriented my hand:

This time B is the first vector, so the pointer finger represents B, thus changing C to point into the page/plane. Here is the graphical representation of B cross A with C pointed into the page/plane.
VectorsCrossBXA
Personally, trying to twist the right hand wrist into the right orientation can be a bit confusing or painful and the thumb doesn’t exactly look like it is on an exact right angle at times. However, it is easier to set the angle between the pointer finger and the middle finger for visualizing.

Palm Method

The palm method is a second way to use the right hand rule:
RHRSlapRule
The first vector follows the thumb, the second vector follows the direction of the four fingers when making a flat palm, and the result vector is in the direction that the palm is facing. The results are the same as before because if the thumb points in the direction that vector A does, the palm has to face towards you when the other fingers point in the direction that vector B does. However, if you point the thumb in the direction that Vector B does, the palm faces away from you. This is harder to form the angles, but is much more comfortable and somewhat easier to see that the thumb and four fingers are in the same plane and the palm faces completely in an orthagonal direction.