## Conventions

I woke this morning thinking about something I learned in physics at school. A wire carrying a current in a magnetic field is acted on by a force (if it is not parallel to the field): in what direction does the force act? Dually, a wire in a closed circuit which is moved in a magnetic field has a current induced in it; which way does the current flow?

These questions can be answered, but the answers seem to depend on five somewhat arbitrary conventions.

• First, we have to distinguish between left and right. Now certainly most people can do this; but this is the only point where we make contact with “reality”. Also, the variety of words for “left” and “right” in European languages suggests that the ability (and necessity) to distinguish does not go back to the roots of language. Admittedly, “right” seems a bit more uniform than “left”, and so is maybe older, perhaps because of its association with a different meaning of “right”, cf. the French droit. As has often been remarked, the words for “left” in French and Latin, “gauche” and “sinister”, do suggest some prejudice against left-handed people; maybe one day we will become so tolerant that these layers of meaning will disappear. (In my youth it was not uncommon for children to be “converted” to right-handedness, not always successfully.)
• Next, we have to remember the association “left” and “motor”, and between “right” and “dynamo” (the two problems mentioned in the opening paragraph). I am sure we learned a mnemonic for this at school, though I can’t clearly remember what it was. Perhaps it was an association with the idea that the right hand is more dynamic, another instance of the prejudice mentioned above.
• Next, we have to remember the bijection between the thumb, index, and middle fingers and force (or motion), field, and current. The mnemonic was thuMb, First, and seCond finger. (This sounds like six choices, but there are really only two, since an even permutation doesn’t change the convention).
• Next, we have to remember the direction of a magnetic field line. I remember diagrams showing the field lines leaving the north pole of a magnet and entering the south pole, which of course were brought to life by experiments with iron filings. But added confusion comes from the fact that the north magnetic pole of the earth is actually a south pole and vice versa. Easily explained: the north pole of a compass magnet is the one that points north, since opposites attract.
• Finally, we have to remember which way current flows. It flows in the opposite direction to the way the electrons flow. This convention, of course, was established before the discovery of electrons, and involved an arbitrary choice of which terminal of a battery is positive and which is negative.

Given all these conventions, to solve the first problem, hold the left hand so that the first finger points in the direction of the magnetic field and the second in the direction of the current; the thumb will indicate the direction of the force. For the second problem, use the right hand, with thumb in direction of motion and first finger in the direction of field, the second finger will indicate the direction of current flow.

So you have to remember 5 bits of information, or (at least) get an even number of them wrong.

There are various connections here. The right-hand rule is related to the right hand rule for the vector product (cross product) of two three-dimensional vectors: if the first and second fingers of the right hand are in the direction of the two vectors, the thumb will be in the direction of the vector product. This rule is usually formulated as a screw rule: if we turn a right-hand screw in the direction from the first vector to the second, the screw will move forward in the direction of the product. This also seems to connect with reality. It is more natural to turn a screwdriver in one direction than the other: this is presumably because we use different muscles for the two actions. The more natural direction tightens a right-handed screw if done with the right hand. (Some people transfer the screwdriver to their left hand to undo a right-hand screw.)

Also, the left-right distinction connects with the direction of the magnetic field. In the northern hemisphere, if you stand facing the midday sun, the sun will rise on your left and set on your right, and the earth’s magnetic field will come from in front of you. (These things reverse in the southern hemisphere, and the tropics require special care; also, in the region within the polar circles, it may not be clear where the midday sun is.) Of course, if the earth’s magnetic field were to reverse, the force acting on a current-carrying wire in a magnetic field would not change!

To a mathematician, of course, the cross product (or exterior product) of two vectors from the real 3-dimensional vector space V lives in a different 3-dimensional space, the exterior square VV. I believe that physicists do recognise the distinction, by calling the vectors of V axial vectors and those of the exterior square polar vectors. (I think that is right, but this is another of those things which you presumably just have to remember.) They distinguish them by the fact that they behave differently under transformations of the underlying space. So the convention here is actually a choice of identification between V and its exterior square.

In Oxford undergraduate courses they use $\wedge$ for vector product, instead of $\times$. I tried to complain about it, to no avail. 🙂