A Bit on Work: Part III of III

Hold a permanent magnet above a paper clip and the paper clip "jumps" up to the magnet. How does this happen given that the magnetic field is not doing any work on the paper clip?

First off, the paper clip is made of steel, which contains a large amount of iron. Iron is a ferromagnetic material, meaning it can become magnetized when placed in a magnetic field and remain magnetized when removed from that field. Non ferromagnetic materials, on the other hand, would lose their magnetization upon removal of the external magnetic field. (In this case, we're not removing the magnetic field, but it's still nice that we're working with a ferromagnetic material. You'll see why later.) When we place a permanent magnet above a paper clip, the magnetic field produced by the magnet induces magnetism in the paper clip by applying a torque to the magnetic dipoles in the iron, lining them up.

What's a magnetic dipole? A small current loop (say, electrons flowing around a tiny loop of wire) is, more or less, a magnetic dipole. We call the small current loop a magnetic dipole because it produces a magnetic field, at some distance, that is strictly "dipolar" in nature. Not all systems produce a magnetic field that is dipolar in nature. Some systems produce a field that is not dipolar at all, but perhaps "quadrupolar" or "octopolar." Other systems might produce fields that are largely dipolar but a little quadrupolar, too. What does it mean for the field to be strictly "dipolar" in nature? It means that, as you move away from the system, the strength of the magnetic field drops off as one over the distance cubed. There's no component of the field that drops off as one over the distance or one over the distance to the fourth power, etc. Not too many systems can actually produce a field that is strictly dipolar. It's easy to produce one that is largely dipolar, but quite difficult to produce one that is strictly dipolar. A tiny current loop does the trick, however. But it has to be really tiny, as in infinitesimally small. How do you make such a thing in the lab? You don't. Luckily, I suppose, they already exist in nature as electrons whizzing about nuclei inside of atoms. (No wires necessary.) Previously, I said that the magnetic field of a permanent magnet applies a torque to the magnetic dipoles in a sample of iron, lining them up. Now it should be fairly clear that it is atomic electrons (acting in their capacity as magnetic dipoles) that are doing the lining up. Note: not every electron in a sample of iron experiences this torque. Only the unpaired electrons do. (Each iron atom has 4 unpaired electrons.)

OK, so how can an electron point in a particular direction? Really, it can't. What I mean is that this whirrling electron, this magnetic dipole, points its magnetic dipole moment in a particular direction. This so-called dipole moment is a property of the dipole and can act to represent the physical dipole. It's a vector (with, of course, a magnitude and a direction) that quantifies the contribution of a system's internal magnetism to the external dipolar magnetic field produced by the system. That is, a measure of how much what's going on inside the system is effecting the magnetic field observed outside the system. The moment may be non-physical, but it often proves useful to picture an electron as a little vector when doing calculations or thinking through problems like the one we're addressing here. Therefore, to say that dipoles are lined up is to say that their dipole moments are lined up (or parallel to one another).

Magnets come in different strengths, which we quantify through the concept of magnetization. Something with a large magnetization is both strongly affected by external magnetic fields and the source of its own strong magnetic field. We define magnetization as the amount of magnetic dipole moment per unit volume. Therefore, given a unit volume, we perform a vector sum of all the little moments (vectors) in that volume, and we see how strong our magnet is. Two vectors of equal magnitude pointing in opposite directions sum to zero. Likewise, a large number of arbitrarily directed vectors also sums to zero. This explains why the paper clip, before being magnetized by the permanent magnet, isn't magnetic. It has plenty of little moments (or vectors) inside, but they are arbitrarily directed (well, sorta) and so the net sum of these moments, per unit volume, is pretty close to zero. Once the permanent magnet acts on the dipole moments in the paper clip and lines them up, the vector sum no longer equals zero. Rather, it is now rather large, and the paper clip now has a large magnetization and acts outwardly like a magnet.

Electrons, bound to atoms, move in two ways. This leads to two magnetic dipoles or, better put, two contributions to a single magnetic dipole. (It's a simple vector sum.) Firstly, an electron "orbits" the nucleus. Even though it's not accurate, people often picture this motion as being like a planet orbiting the sun. That's a good way to think of it at the present. Secondly, an electron "spins." You might think of this as an electron spinning about its own axis, just as the Earth spins about its axis once every 24 hours. But this is a rather horrible and misleading analogy, because this "spin" is not really a physical rotation about an axis. The electron is a point particle with no physical size, so there's really no way it could have a component off-axis that could move around some central point. For this reason, physicists say the electron has an intrinsic magnetic dipole moment that originates with its spin. It just exists.

Let's step back now and look at what we have. A permanent magnet (which itself is comprised of magnetic dipoles, with the moments all pointing in the same direction, hence its large magnetization) produces a magnetic field that exists in the space around the magnet. This space includes the paper clip, sitting on a table. The magnetic field interacts with the electrons in the iron/steel paper clip, changing the magnetic dipole moments of these electrons, and inducing magnetism in the paper clip. How exactly?

The magnetic field produced by the permanent magnet has the property, determined through experimentation, that it can exert a force on a moving charged particle, like our atomic electrons. (This force, called the Lorentz force, was defined in the previous blog entry.) Acting on each magnetic dipole, this force (actually, torque) acts to twist the dipole moments such that they line up parallel to the field. The result: countless magnetic dipole moments in our paper clip are now pointing in the direction of the field. (It is at this point we appreciate the paper clip being made of iron, a ferromagnetic material. If it was not, the magnetic force would have a more difficult time turning all of the dipole moments and would ultimately manage to turn only some of them, diminishing the strength of the magnetization induced in the paper clip.) The paper clip is now a magnet.

How else does the magnetic field interact with the atomic electrons? Surprisingly, it can change the speed with which the electrons orbit their nuclei! Before the magnetic field enters the picture, the electrons are held in their orbits by electrical forces alone. (Unlike charges, i.e. protons and electrons, attract.) When the magnetic field shows up, it produces a force that acts in the opposite direction as the electrical force, at the location of each orbiting electron, and serves to weaken the pull on the electron towards the center of the orbit. The electron no longer needs to travel so quickly to maintain its orbital radius and it slows down. Now, think back to the previous blog entry and the example of you holding on to a string, at the other end of which is attached a ball. You're whirling this ball about your head. In this case, there is a centripetal force pulling the ball towards the center of its circular path, thereby acting, at all times, perpendicular to the direction of the ball's motion. (This is like the electric force holding the electron in its orbit about the atom's nucleus.) Likewise, the magnetic force, once introduced, acts perpendicular to the circling electron. The magnetic force, however, is acting not towards the center of the circle but radially outward, away from the center. As stated before, this diminished centripetal force causes the electron to slow down. We'll soon see that this slowing of the electrons in the atoms of the paper clip is key to the lifting of the paper clip by the permanent magnet.

Now is the time to point out that the magnetic field produced by the permanent magnet is non-uniform. It generally is pointing downwards, assuming the north pole of the magnet is nearer the paper clip than the south pole, but it also flares out. It's the vertical component of the field which acts to slow down the electrons but the horizontal component, existing in the plane of the orbiting electrons, that provides an upward force. Adding together these two components we end up with a force (represented by a vector) that is pointing up and out (away from the center of the electron's circular orbit). We know this force must be perpendicular to the motion of the electron, and indeed it is, as the electron begins to move upwards along a helical path.

The motion of an electron in its orbit constitutes stored kinetic energy. It's this energy that is tapped to lift the paper clip off the table. As the paper clip (acting as a magnet) rises, the unpaired electrons inside slow down and the stored kinetic energy decreases. The magnetic force redirects this energy into lifting the paper clip/magnet against the force of gravity. The net magnetic force, like the normal force mentioned in the previous blog entry, is responsible for the vertical motion of the object (previously a box and now an electron) despite the fact that it doesn't do work on that object. Both of these forces (the magnetic and the normal) are redirecting work done by another agent. In the case of the box being pushed up the incline, the other agent is a person. And in this case, it's ... Well, who or what is this agent?

Uhhhhhh. Well, the agent is whatever got all those electrons circling around all those nuclei to begin with. Whatever it was, it did work and imparted kinetic energy to each little electron. Trying to trace the formation of these iron atoms back to the original source of energy would lead us back to the Big Bang. So it was God, I guess. "God" lifted the paper clip.