How do you derive Fleming's left hand rule? What is the theoretical explanation for the directions of the magnetic field, current and the force on the current for being oriented in that way relative to one another?
[Physics] How to derive Fleming’s left hand rule
conventionselectromagnetism
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It is unfortunate that the physics of magnetism got saddled with several different *-hand rules, and that they use different hands. Let's pull them apart:
Fleming's left-hand rule
gives you the direction of the force that acts on a current if you know the magnetic field.
This rule applies to motors, i.e. devices which use currents in a magnetic field to generate motion. It derives its validity from the Lorentz force, $$ \mathbf F=q\mathbf v\times\mathbf B, $$ in which the current goes with the charge's velocity and the induced motion is along the direction of the force. This is why this rule coincides with the left-hand rule used in cross-products in general.
Fleming's right-hand rule
is much less used in physics (though I can't speak for how engineers do things). It applies to generators, i.e. devices which use motion in a magnetic field to generate currents. This again relies on the cross product in the Lorentz force, except that now the charge's velocity is given by the object's motion, and the force along the wire is what establishes the current. This means you've swapped the middle finger with the thumb with respect to Fleming's left-hand rule, which you can do by keeping the (vague) assignments to 'motion' and 'current' and switching hands.
I dislike this convention very much and I would encourage you to forget all about it except the fact that it exists and should be avoided. In any situation where you need it, you can simply use the Lorentz force to figure out which way the current will go.
Ampère's right-hand rule
is quite different, and it gives you the magnetic field generated by a straight wire.
It derives its validity from the Biot-Savart law, which gives the magnetic field at position $\mathbf r$ generated by an infinitesimal current element of current $I$ and directed length $\mathrm d\mathbf l$ at position $\mathbf r'$, as $$ \mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\frac{I\mathrm d\mathbf l\times(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3} $$ Again, it is the cross product which dictates the direction of the field, and you should check by yourself that it works out as indicated in the picture.
As you can see, the rules are quite different. It is therefore crucial that, if you want to use them as mnemonics, you learn correctly which one applies where, and that you apply them correctly. (It is no use to learn which hand to use if you e.g. swap the assignments for the index and middle finger.)
The most important thing to learn, though, is the Lorentz force law, which is based on a left-hand rule (charge-times-current on your middle finger, field on the index, force on the thumb) indicated by the cross product. This is essentially failsafe if you apply it correctly and is less subject to confusion with other rules.
[Physics] Is it possible to express Fleming’s Left Hand Rule and Right Hand Rule in terms of vectors
You could just calculate the vector components. These hand rules are used whenever vector quantities are related by a cross product. Let's take as an example the force on a moving charge due to a magnetic field: $$\vec{F}=q \vec{v} \times \vec{B}$$
Let's write this out in terms of its components (assuming we're working in 3D):
$$\left( \begin{array}{} F_x \\ F_y \\ F_z \end{array} \right) = q \left( \begin{array}{} v_x \\ v_y \\ v_z \end{array} \right) \times \left( \begin{array}{} B_x \\ B_y \\ B_z \end{array} \right) = q \left( \begin{array}{} v_y B_z - v_z B_y \\ v_z B_x - v_x B_z \\ v_x B_y - v_y B_x \end{array} \right)$$
In principle, this gives you the direction of the force... but I doubt it will be faster than using a hand rule. I generally only use one hand rule, namely the right hand rule for cross products:
Given two vectors $\vec{A}$ and $\vec{B}$, point the fingers of your right hand in the direction of $\vec{A}$, then curl them towards $\vec{B}$. Your thumb then points in the direction of the outer product $\vec{A}\times\vec{B}$.
If you know the equation for the cross product quantity you're trying to calculate, you can always use this rule and don't have to think about whether you need to use a left-hand or right-hand rule.
Best Answer
The fact that Fleming's is a left-hand rule is an artifact of the completely arbitrary choice of the right-hand rule to define the direction of the magnetic field. If electromagnetic induction had been discovered by people who put South at the tops of their maps, we might well define the direction of a cross product using the left hand instead of the right. Since every prediction of an acceleration in electrodynamics involves an even number of right-hand rules, and complete and consistent switch to the left hand would be mathematically identical.
That said, Fleming's rule is a consequence of the Lorentz force between the fields and currents in the motor (or generator).