We started learning about electromagnetism in physics class, and the Right Hand Rule comes in handy as seems easy to use, but I'm curious as to how it actually works. I guess it's more of a math question since I think it just involves solving the cross product of the velocity of the charge and the magnetic field. I don't know anything about cross products, but I searched some things up and it seems that the matrix is has unit vectors in it which determine the directions, so would one have to solve the whole thing to determine the direction of the force on the charge? I know it has to be perpendicular to both of the vectors but that still leaves 2 directions.
[Physics] More about the right hand rule
conventionselectromagnetismvectors
Related Solutions
I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean?
Perhaps you would find the geometric interpretations of the dot and cross products more intuitive:
The dot product of A and B is the length of the projection of A onto B multiplied by the length of B (or the other way around--it's commutative).
The magnitude of the cross product is the area of the parallelogram with two sides A and B. The orientation of the cross product is orthogonal to the plane containing this parallelogram.
Why can't vectors be divided?
How would you define the inverse of a vector such that $\mathbf{v} \times \mathbf{v}^{-1} = \mathbf{1}$? What would be the "identity vector" $\mathbf{1}$?
In fact, the answer is sometimes you can. In particular, in two dimensions, you can make a correspondence between vectors and complex numbers, where the real and imaginary parts of the complex number give the (x,y) coordinates of the vector. Division is well-defined for the complex numbers.
The cross-product only exists in 3D.
Division is defined in some higher-dimensional spaces too (such as the quaternions), but only if you give up commutativity and/or associativity.
Here's an illustration of the geometric meanings of dot and cross product, from the wikipedia article for dot product and wikipedia article for cross product:
All of them in fact mean the same thing: given $\mathbf{A}$ and $\mathbf{B}$, you are finding (/deciding) the direction of $\mathbf{A}\times\mathbf{B}$.
For the situations in which Fleming's rules are of interest, we are interested in just one relation: $$\mathbf{F} = q\mathbf{v}\times\mathbf{B}$$ or its equivalent: $$\mathbf{F} = I\mathbf{l}\times\mathbf{B}$$ where $I\mathbf{l}$ is the current times the length of a conductor, in the direction of the current.
These different rules are just different ways of assigning quantities to fingers. All of them obey your general convention for the cross product, normally given by the "right hand screw rule".
For example, Fleming's right hand rule says that when a conductor is moving ($\mathbf{v}$) along your right hand thumb, and the field $\mathbf{B}$ is along the index finger, then the current (due to $\mathbf{F}$) is along your middle finger. But Fleming's left hand rule with $\mathbf{v}$ along your left hand middle finger and $\mathbf{B}$ along the left hand index finger gives the same direction (left hand thumb) for $\mathbf{F}$.
However, people seem to prefer always associating the thumb with the "Motion", and the middle finger with the "Current", giving rise to the two rules. Because of this, Fleming's right-hand rule is used when a moving conductor ($\mathbf{v}$) develops a current ($\mathbf{F}$ for positive $q$), as in a generator, and the left-hand rule is used when a current ($q\mathbf{v}$, more or less) generates motion (again $\mathbf{F}$), like in a motor.
The right hand palm rule is yet another (and as far as I can see, not very common) rule that says the same thing, which is used, given the current (right hand thumb) and the magnetic field (outstretched fingers), to find the force on the wire (upward normal from the palm i.e. on the side where you can fold fingers). It is completely equivalent to the other rules.
Best Answer
The formula for the force of a particle due to its magnetic field is $F = q \vec v \times \vec B$. The cross product has the property that its result is always perpendicular to both arguments.
Its direction is simply a result of how the cross product function is defined and the sign of electric charge (an electron is defined as negative).
It is important to note that the order of the arguments matters as the cross product function is not commutative; generally $ \vec A \times \vec B \neq \vec B \times \vec A$. For vectors the direction will be reversed; for matrices both sides can be wildly different.