Magnetic dipoles and their torque and energy

We studied the magnetic dipole in very much detail in one of my previous posts but, while we talked about an awful lot of stuff there, we actually managed to not talk about the torque on a it, when it’s placed in the magnetic field of other currents, or some other magnetic field tout court. Now, that’s what drives electric motors and generators, of course, and so we should talk about it, which is what I’ll do here. Let me first remind you of the concept of torque, and then we’ll apply it to a loop of current. 🙂

The concept of torque

The concept of torque is easy to grasp intuitively, but the math involved is not so easy. Let me sum up the basics (for the detail, I’ll refer you to my posts on spin and angular momentum). In essence, for rotations in space (i.e. rotational motion), the torque is what the force is for linear motion:

  1. It’s the torque (τ) that makes an object spin faster or slower around some axis, just like the force would accelerate or decelerate that very same object when it would be moving along some curve.
  2. There’s also a similar ‘law of Newton’ for torque: you’ll remember that the force equals the time rate of change of a vector quantity referred to as (linear) momentum: F = dp/dt = d(mv)/dt = ma (the mass times the acceleration). Likewise, we have a vector quantity that is referred to as angular momentum (L), and we can write: τ (i.e. the Greek tau) = dL/dt.
  3. Finally, instead of linear velocity, we’ll have an angular velocity ω (omega), which is the time rate of change of the angle θ that defines how far the object has gone around (as opposed to the distance in linear dynamics, describing how far the object has gone along). So we have ω = dθ/dt. This is actually easy to visualize because we know that θ, expressed in radians, is actually the length of the corresponding arc on the unit circle. Hence, the equivalence with the linear distance traveled is easily ascertained.

There are many more similarities, like an angular acceleration: α = dω/dt = d2θ/dt2, and we should also note that, just like the force, the torque is doing work – in its conventional definition as used in physics – as it turns an object instead of just moving it, so we can write:

ΔW = τ·Δθ

So it’s all the same-same but different once more 🙂 and so now we also need to point out some differences. The animation below does that very well, as it relates the ‘new’ concepts – i.e. torque and angular momentum – to the ‘old’ concepts – i.e. force and linear momentum. It does so using the vector cross product, which is really all you need to understand the math involved. Just look carefully at all of the vectors involved, which you can identify by their colors, i.e. red-brown (r), light-blue (τ), dark-blue (F), light-green (L), and dark-green (p).

Torque_animation

So what do we have here? We have vector quantities once again, denoted by symbols in bold-face. Having said that, I should note that τ, L and ω are ‘special’ vectors: they are referred to as axial vectors, as opposed to the polar vectors F, p and v. To put it simply: polar vectors represent something physical, and axial vectors are more like mathematical vectors, but that’s a very imprecise and, hence, essential non-correct definition. 🙂 Axial vectors are directed along the axis of spin – so that is, strangely enough, at right angles to the direction of spin, or perpendicular to the ‘plane of the twist’ as Feynman calls it – and the direction of the axial vector is determined by a convention which is referred to as the ‘right-hand screw rule’. 🙂

Now, I know it’s not so easy to visualize vector cross products, so it may help to first think of torque (also known, for some obscure reason, as the moment of the force) as a twist on an object or a plane. Indeed, the torque’s magnitude can be defined in another way: it’s equal to the tangential component of the force, i.e. F·sin(Δθ), times the distance between the object and the axis of rotation (we’ll denote this distance by r). This quantity is also equal to the product of the magnitude of the force itself and the length of the so-called lever arm, i.e. the perpendicular distance from the axis to the line of action of the force (this lever arm length is denoted by r0). So, we can define τ without the use of the vector cross-product, and in not less than three different ways actually. Indeed, the torque is equal to:

  1. The product of the tangential component of the force times the distance r: τ = r·Ft= r·F·sin(Δθ);
  2. The product of the length of the lever arm times the force: τ = r0·F;
  3. The work done per unit of distance traveled: τ = ΔW/Δθ or τ = dW/dθ in the limit.

Phew! Yeah. I know. It’s not so easy… However, I regret to have to inform you that you’ll need to go even further in your understanding of torque. More specifically, you really need to understand why and how we define the torque as a vector cross product, and so please do check out that post of mine on the fundamentals of ‘torque math’. If you don’t want to do that, then just try to remember the definition of torque as an axial vector, which is:

τ = (τyz, τzx, τxy) = (τx, τy, τz) with

τx = τyz = yFz – zFy (i.e. the torque about the x-axis, i.e. in the yz-plane),

τy = τzx = zFx – xFz (i.e. the torque about the y-axis, i.e. in the zx-plane), and

τz = τxy = xFy – yFx (i.e. the torque about the z-axis, i.e. in the xy-plane).

The angular momentum L is defined in the same way:

L = (Lyz, Lzx, Lxy) = (Lx, Ly, Lz) with

Lx = Lyz = ypz – zpy (i.e. the angular momentum about the x-axis),

Ly = Lzx = zpx – xpz (i.e. the angular momentum about the y-axis), and

Lz = Lxy = xpy – ypx (i.e. the angular momentum about the z-axis).

Let’s now apply the concepts to a loop of current.

The forces on a current loop

The geometry of the situation is depicted below. I know it looks messy but let me help you identifying the moving parts, so to speak. 🙂 We’ve got a loop with current and so we’ve got a magnetic dipole with some moment μ. From my post on the magnetic dipole, you know that μ‘s magnitude is equal to |μ| = μ = (current)·(area of the loop) = I·a·b.

Geometry 2

Now look at the B vectors, i.e. the magnetic field. Please note that these vectors represent some external magnetic field! So it’s not like what we did in our post on the dipole: we’re not looking at the magnetic field caused by our loop, but at how it behaves in some external magnetic field. Now, because it’s kinda convenient to analyze, we assume that the direction of our external field B is the direction of the z-axis, so that’s what you see in this illustration: the B vectors all point north. Now look at the force vectors, remembering that the magnetic force is equal to:

Fmagnetic = qv×B

So that gives the F1F2F3, and F4 vectors (so that’s the force on the first, second, third and fourth leg of the loop respectively) the magnitude and direction they’re having. Now, it’s easy to see that the opposite forces, i.e. the F1F2 and F3Fpair respectively, create a torque. The torque because of Fand Fis a torque which will tend to rotate the loop about the y-axis, so that’s a torque in the xz-plane, while the torque because of Fand Fwill be some torque about the x-axis and/or the z-axis. As you can see, the torque is such that it will try to line up the moment vector μ with the magnetic field B. In fact, the geometry of the situation above is such that Fand Fhave already done their job, so to speak: the moment vector μ is already lined up with the xz-plane, so there’s not net torque in that plane. However, that’s just because of the specifics of the situation here: the more general situation is that we’d have some torque about all three axes, and so we need to find that vector τ.

If we’d be talking some electric dipole, the analysis would be very straightforward, because the electric force is just Felectric = qE, which we can also write as E = Felectric =/q, so the field is just the force on one unit of electric charge, and so it’s (relatively) easy to see that we’d get the following formula for the torque vector:

τ = p×E

Of course, the p is the electric dipole moment here, not some linear momentum. [And, yes, please do try to check this formula. Sorry I can’t elaborate on it, but the objective of this blog is not substitute for a textbook!]

Now, all of the analogies between the electric and magnetic dipole field, which we explored in the above-mentioned post of mine, would tend to make us think that we can write τ here as:

τ = μ×B

Well… Yes. It works. Now you may want to know why it works 🙂 and so let me give you the following hint. Each charge in a wire feels that Fmagnetic = qv×B force, so the total magnetic force on some volume ΔV, which I’ll denote by ΔF for a while, is the sum of the forces on all of the individual charges. So let’s assume we’ve got N charges per unit volume, then we’ve got N·ΔV charges in our little volume ΔV, so we write: ΔF = N·ΔV·q·v×B. You’re probably confused now: what’s the v here? It’s the (drift) velocity of the (free) electrons that make up our current I. Indeed, the protons don’t move. 🙂 So N·q·v is just the current density j, so we get: ΔF = j×BΔV, which implies that the force per unit volume is equal to j×B. But we need to relate it to the current in our wire, not the current density. Relax. We’re almost there. The ΔV in a wire is just its cross-sectional area A times some length, which I’ll denote by ΔL, so ΔF = j×BΔV becomes ΔF = j×BAΔL. Now, jA is the vector current I, so we get the simple result we need here: ΔF = I×BΔL, i.e.  the magnetic force per unit length on a wire is equal to ΔF/ΔL = I×B.

Let’s now get back to our magnetic dipole and calculate Fand F2. The length of ‘wire’ is the length of the leg of the loop, i.e. b, so we can write:

F= −F2 = b·I×B

So the magnitude of these forces is equal F= F2 = I·B·b. Now, The length of the moment or lever arm is, obviously, equal to a·sinθ, so the magnitude of the torque is equal to the force times the lever arm (cf. the τ = r0·F formula above) and so we can write:

τ = I·B·b·a·sinθ

But I·a·b is the magnitude of the magnetic moment μ, so we get:

τ = μ·B·sinθ

Now that’s consistent with the definition of the vector cross product:

τμ×= |μ|·|B|·sinθ·n = μ·B·sinθ·n

Done! Now, electric motors and generators are all about work and, therefore, we also need to briefly talk about energy here.

The energy of a magnetic dipole

Let me remind you that we could also write the torque as the work done per unit of distance traveled, i.e. as τ = ΔW/Δθ or τ = dW/dθ in the limit. Now, the torque tries to line up the moment with the field, and so the energy will be lowest when μ and B are parallel, so we need to throw in a minus sign when writing:

 τ = −dU/dθ ⇔ dU = −τ·dθ

We should now integrate over the [0, θ] interval to find U, also using our τ = μ·B·sinθ formula. That’s easy, because we know that d(cosθ)/dθ = −sinθ, so that integral yields:

U = 1 − μ·B·cosθ + a constant

If we choose the constant to be zero, and if we equate μ·B with 1, we get the blue graph below:

graph energy magnetic dipole 3

The μ·B in the U = 1 − μ·B·cosθ formula is just a scaling factor, obviously, so it determines the minimum and maximum energy. Now, you may want to limit the relevant range of θ to [0, π], but that’s not necessary: the energy of our loop of current does go up and down as shown in the graph. Just think about it: it all makes perfect sense!

Now, there is, of course, more energy in the loop than this U energy because energy is needed to maintain the current in the loop, and so we didn’t talk about that here. Therefore, we’ll qualify this ‘energy’ and call it the mechanical energy, which we’ll abbreviate by Umech. In addition, we could, and will, choose some other constant of integration, so that amounts to choosing some other reference point for the lowest energy level. Why? Because it then allows us to write Umech as a vector dot product, so we get:

Umech = −μ·B·cosθ = −μ·B

The graph is pretty much the same, but it now goes from −μ·B to +μ·B, as shown by the red graph in the illustration above.

Finally, you should note that the Umech = −μ·B formula is similar to what you’ll usually see written for the energy of an electric dipole: U = −p·E. So that’s all nice and good! However, you should remember that the electrostatic energy of an electric dipole (i.e. two opposite charges separated by some distance d) is all of the energy, as we don’t need to maintain some current to create the dipole moment!

Now, Feynman does all kinds of things with these formulas in his Lectures on electromagnetism but I really think this is all you need to know about it—for the moment, at least. 🙂

Understanding gyroscopes

You know a gyroscope: it’s a spinning wheel or disk mounted in a frame that itself is free to alter in direction, so the axis of rotation is not affected as the mounting tilts or moves about. Therefore, gyroscopes are used to provide stability or maintain a reference direction in navigation systems. Understanding a gyroscope itself is simple enough: it only involves a good understanding of the so-called moment of inertia. Indeed, in the previous post, we introduced a lot of concepts related to rotational motion, notably the concepts of torque and angular momentum but, because that post was getting too long, I did not talk about the moment of inertia and gyroscopes. Let me do that now. However, I should warn you: you will not be able to understand this post if you haven’t read or didn’t understand the previous post. So, if you can’t follow, please go back: it’s probably because you didn’t get the other post.

The moment of inertia and angular momentum are related but not quite the same. Let’s first recapitulate angular momentum. Angular momentum is the equivalent of linear momentum for rotational motion:

  1. If we want to change the linear motion of an object, as measured by its momentum p = mv, we’ll need to apply a force. Changing the linear motion means changing either (a) the speed (v), i.e. the magnitude of the velocity vector v, (b) the direction, or (c) both. This is expressed in Newton’s Law, F = m(dv/dt), and so we note that the mass is just a factor of proportionality measuring the inertia to change.
  2. The same goes for angular momentum (denoted by L): if we want to change it, we’ll need to apply a force, or a torque as it’s referred to when talking rotational motion, and such torque can change either (a) L’s magnitude (L), (b) L’s direction or (c) both.

Just like linear momentum, angular momentum is also a product of two factors: the first factor is the angular velocity ω, and the second factor is the moment of inertia. The moment of inertia is denoted by I so we write L = Iω. But what is I? If we’re analyzing a rigid body (which is what we usually do), then it will be calculated as follows:

formula 1

This is easy enough to understand: the inertia for turning will depend not just on the masses of all of the particles that make up the object, but also on their distance from the axis of rotation–and note that we need to square these distances. The L = Iω formula, combined with the formula for I above, explains why a spinning skater doing a ‘scratch spin’ speeds up tremendously when drawing in his or her arms and legs. Indeed, the total angular momentum has to remain the same, but I becomes much smaller as a result of that r2 factor in the formula. Hence, if I becomes smaller, then ω has to go up significantly in order to conserve angular momentum.

Finally, we note that angular momentum and linear momentum can be easily related through the following equation:

Formula 2

That’s all kids stuff. To understand gyroscopes, we’ll have to go beyond that and do some vector analysis. In the previous post, we explained that rotational motion is usually analyzed in terms of torques than forces, and we detailed the relations between force and torque. More in particular, we introduced a torque vector τ with the following components:

τ = (τyz, τzx, τxy) = (τx, τy, τz) with

τx = τyz = yFz – zFy

τy = τzx = zFx – xFz

τz = τxy = xFy – yFx.

We also noted that this torque vector could be written as a cross product of a radius vector and the force: τ = F. Finally, we also pointed out the relation between the x-, y- and z-components of the torque vector and the plane of rotation:

(1) τx = τyz is rotational motion about the x-axis (i.e. motion in the yz-plane)

(2) τy = τzx is rotational motion about the y-axis (i.e. motion in the zx plane)

(3) τz = τxy is rotational motion about the z-axis (i.e. motion in the xy-plane)

The angular momentum vector L will have the same direction as the torque vector, but it’s the cross product of the radius vector and the momentum vector: L = p. For clarity, I reproduce the animation I used in my previous post once again.

Torque_animation

How do we get that cross vector product for L? We noted that τ (i.e. the Greek tau) = dL/dt. So we need to take the time derivative of all three components of L. What are the components of L? They look very similar to those of τ:

L = (Lyz, Lzx, Lxy) = (Lx, Ly, Lz) with

Lx = Lyz = ypz – zpy

Ly = Lzx = zpx – xpz

Lz = Lxy = xpy – ypx.

Now, just check the time derivatives of Lx, Ly, and Lz and you’ll find the components of the torque vector τ. Together with the formulas above, that should be sufficient to convince you that L is, indeed, a vector cross product of r and p: L = p.

Again, if you feel this is too difficult, please read or re-read my previous post. But if you do understand everything, then you are ready for a much more difficult analysis, and that’s an explanation of why a spinning top does not fall as it rotates about.

In order to understand that explanation, we’ll first analyze the situation below. It resembles the experiment with the swivel chair that’s often described on ‘easy physics’ websites: the man below holds a spinning wheel with its axis horizontal, and then turns this axis into the vertical. As a result, the man starts to turn himself in the opposite direction.

Rotating angular momentum

Let’s now look at the forces and torques involved. These are shown below.

Angular vectors in gyroscope

This looks very complicated–you’ll say! You’re right: it is quite complicated–but not impossible to understand. First note the vectors involved in the starting position: we have an angular momentum vector L0 and an angular velocity vector ω0. These are both axial vectors, as I explained in my previous post: their direction is perpendicular to the plane of motion, i.e. they are arrows along the axis of rotation. This is in line with what we wrote above: if an object is rotating in the zx-plane (which is the case here), then the angular momentum vector will have a y-component only, and so it will be directed along the y-axis. Which side? That’s determined by the right-hand screw rule. [Again, please do read my previous post for more details if you’d need them.]

So now we have explained L0 and ω0. What about all the other vectors? First note that there would be no torque if the man would not try to turn the axis. In that case, the angular momentum would just remain what it is, i.e. dL/dt = 0, and there would be no torque. Indeed, remember that τ = dL/dt, just like F = dp/dt, so dL/dt = 0, then τ = 0. But so the man is turning the axis of rotation and, hence, τ = dL/dt ≠ 0. What’s changing here is not the magnitude of the angular momentum but its direction. As usual, the analysis is in terms of differentials.

As the man turns the spinning wheel, the directional change of the angular momentum is defined by the angle Δθ, and we get a new angular momentum vector L1. The difference between L1 and L0 is given by the vector ΔL. This ΔL vector is a tiny vector in the L0L1 plane and, because we’re looking at a differential displacement only, we can say that, for all practical purposes, this ΔL is orthogonal to L0 (as we move from L0 to L1, we’re actually moving along an arc and, hence, ΔL is a tangential vector). Therefore, simple trigonometry allows us to say that its magnitude ΔL will be equal to L0Δθ. [We should actually write sin(Δθ) but, because we’re talking differentials and measuring angles in radians (so the value reflects arc lengths), we can equate sin(Δθ) with Δθ).]

Now, the torque vector τ has the same direction as the ΔL vector (that’s obvious from their definitions), but what is its magnitude? That’s an easy question to answer: τ = ΔL/Δt = L0Δθ/Δt = L0 (Δθ/Δt). Now, this result induces us to define another axial vector which we’ll denote using the same Greek letter omega, but written as a capital letter instead of in lowercase: Ω. The direction of Ω is determined by using that right-hand screw rule which we’ve always been using, and Ω‘s magnitude is equal to Ω = Δθ/Δt. So, in short, Ω is an angular velocity vector just like ω: its magnitude is the speed with which the man is turning the axis of rotation of the spinning wheel, and its direction is determined using the same rules. If we do that, we get the rather remarkable result that we can write the torque vector τ as the cross product of Ω and L0:

τ = Ω×L0

Now, this is not an obvious result, so you should check it yourself. When doing that, you’ll note that the two vectors are orthogonal and so we have τ = Ω×L0 = Ω×L0 =|Ω||L0|sin(π/2)n = ΩL0n with n the normal unit vector given, once again, by the right-hand screw rule. [Note how the order of the two factors in a cross product matters: b = –a.]

You’re probably tired of this already, and so you’ll say: so what?

Well… We have a torque. A torque is produced by forces, and a torque vector along the z-axis is associated with rotation about the z-axis, i.e. rotation in the xy-plane. Such rotation is caused by the forces F and –F that produce the torque, as shown in the illustration. [Again, their direction is determined by the right-hand screw rule – but I’ll stop repeating that from now on.] But… Wait a minute. First, the direction is wrong, isn’t it? The man turns the other way in reality. And, second, where do these forces come from? Well… The man produces them, and the direction of the forces is not wrong: as the man applies these forces, with his hands, as he holds the spinning wheel and turns it into the vertical direction, equal and opposite forces act on him (cf. the action-reaction principle), and so he starts to turn in the opposite direction.

So there we are: we have explained this complex situation fully in terms of torques and forces now. So that’s good. [If you don’t believe the thing about those forces, just get one of your wheels out of your mountainbike, let it spin, and try to change the plane in which it is spinning: you’ll see you’ll need a bit of force. Not much, but enough, and it’s exactly the kind of force that the man in the illustration is experiencing.]

Now, what if we would not be holding the spinning wheel? What if we would let it pivot, for example? Well… It would just pivot, as shown below.

 gyroscope_diagram

But… Why doesn’t it fall? Hah! There we are! Now we are finally ready for the analysis we really want to do, i.e. explaining why these spinning tops (or gyros as they’re referred to in physics) don’t fall.

Such spinning top is shown in the illustration below. It’s similar to the spinning wheel: there’s a rotational axis, and we have the force of gravity trying to change the direction of that axis, so it’s like the man turning that spinning wheel indeed, but so now it’s gravity exerting the force that’s needed to change the angular momentum. Let’s associate the vertical direction with the z-axis, and the horizontal place with the xy-axis, and let’s go step-by-step:

  1. The gravitational force wants to pull that spinning top down. So the ΔL vector points downward this time, not upward. Hence, the torque vector will point downward too. But so it’s a torque pointing along the z-axis.
  2. Such torque along the z-axis is associated with a rotation in the xy-plane, so that’s why the spinning top will slowly revolve about the z-axis, parallel to the xy-plane. This process is referred to as precession, and so there’s a precession torque and a precession angular velocity.

spinning top

So that explains precession and so that’s all there is to it. Now you’ll complain, and rightly so: what I write above, does not explain why the spinning top does not actually fall. I only explained that precession movement. So what’s going on? That spinning top should fall as it precesses, shouldn’t it?

It actually does fall. The point to note, however, is that the precession movement itself changes the direction of the angular momentum vector as well. So we have a new ΔL vector pointing sideways, i.e. a vector in the horizontal plane–so not along the z axis. Hence, we should have a torque in the horizontal plane, and so that implies that we should have two equal and opposite forces acting along the z-axis.

In fact, the right-hand screw rule gives us the direction of those forces: if these forces were effectively applied to the spinning top, it would fall even faster! However, the point to note is that there are no such forces. Indeed, it is not like the man with the spinning wheel: no one (or nothing) is pushing or applying the forces that should produce the torque associated with this change in angular momentum. Hence, because these forces are absent, the spinning top begins to ‘fall’ in the opposite direction of the lacking force, thereby counteracting the gravitational force in such a way that the spinning top just spins about the z-axis without actually falling.

Now, this is, most probably, very difficult to understand in the way you would like to understand it, so just let it sink in and think about it for a while. In this regard, and to help the understanding, it’s probably worth noting that the actual process of reaching equilibrium is somewhat messy. It is illustrated below: if we hold a spinning gyro for a while and then, suddenly, we let it fall (yes, just let it go), it will actually fall. However, as it’s falling, it also starts turning and then, because it starts turning, it also starts ‘falling’ upwards, as explained in that story of the ‘missing force’ above. Initially, the upward movement will overshoot the equilibrium position, thereby slowing the gyro’s speed in the horizontal plane. And so then, because its horizontal speed becomes smaller, it stops ‘falling upward’, and so that means it’s falling down again. But then it starts turning again, and so on and so on. I hope you grasp this–more or less at least. Note that frictional effects will cause the up-and-down movement to dampen out, and so we get a so-called cycloidal motion dampening down to the steady motion we associate with spinning tops and gyros.

Actual gyroscope motion

That, then, is the ‘miracle’ of a spinning top explained. Is it less of a ‘miracle’ now that we have explained it in terms of torques and missing forces? That’s an appreciation which each of us has to make for him- or herself. I actually find it all even more wonderful now that I can explain it more or less using the kind of math I used above–but then you may have a different opinion.

In any case, let us – to wrap it all up – ask some simple questions about some other spinning objects. What about the Earth for example? It has an axis of rotation too, and it revolves around the Sun. Is there anything like precession going on?

The first answer is: no, not really. The axis of rotation of the Earth changes little with respect to the stars. Indeed, why would it change? Changing it would require a torque, and where would the required force for such torque come from? The Earth is not like a gyro on a pivot being pulled down by some force we cannot see. The Sun attracts the Earth as a whole indeed. It does not change its axis of rotation. That’s why we have a fairly regular day and night cycle.

The more precise answer is: yes, there actually is a very slow axial precession. The whole precessional cycle takes approximately 26,000 years, and it causes the position of stars – as perceived by us, earthlings, that is – to slowly change. Over this cycle, the Earth’s north axial pole moves from where it is now, in a circle with an angular radius of about 23.5 degrees, as illustrated below.

640px-Earth_precession

What is this precession caused by? There must be some torque. There is. The Earth is not perfectly spherical: it bulges outward at the equator, and the gravitational tidal forces of the Moon and Sun apply some torque here, attempting to pull the equatorial bulge into the plane of the ecliptic, but instead causing it to precess. So it’s a quite subtle motion, but it’s there, and it’s got also something to do with the gravitational force. However, it’s got nothing to do with the way gravitation makes a spinning top do what it does. [The most amazing thing about this, in my opinion, is that, despite the fact that the precessional movement is so tiny, the Greeks had already discovered it: indeed, the Greek astronomer and mathematician Hipparchus of Nicaea gave a pretty precise figure for this so-called ‘precession of the equinoxes’ in 127 BC.]

What about electrons? Are they like gyros rotating around some pivot? Here the answer is very simple and very straightforward: No, not at all! First, there are no pivots in an atom. Second, the current understanding of an electron – i.e. the quantum-mechanical understanding of a electron – is not compatible with the classical notion of spin. Let me just copy an explanation from Georgia State University’s HyperPhyics website. It basically says it all:

“Experimental evidence like the hydrogen fine structure and the Stern-Gerlach experiment suggest that an electron has an intrinsic angular momentum, independent of its orbital angular momentum. These experiments suggest just two possible states for this angular momentum, and following the pattern of quantized angular momentum, this requires an angular momentum quantum number of 1/2. With this evidence, we say that the electron has spin 1/2. An angular momentum and a magnetic moment could indeed arise from a spinning sphere of charge, but this classical picture cannot fit the size or quantized nature of the electron spin. The property called electron spin must be considered to be a quantum concept without detailed classical analogy.

So… I guess this should conclude my exposé on rotational motion. I am not sure what I am going to write about next, but I’ll see. 🙂

Post scriptum:

The above treatment is largely based on Feynman’s Lectures.(Vol. I, Chapter 18, 19 and 20). The subject could also be discussed using the concept of a force couple, aka pure moment. A force couple is a system of forces with a resultant moment but no resultant force. Hence, it causes rotation without translation or, more generally, without any acceleration of the centre of mass. In such analysis, we can say that gravity produces a force couple on the spinning top. The two forces of this couple are equal and opposite, and they pull at opposite ends. However, because one end of the top is fixed (friction forces keep the tip fixed to the ground), the force at the other end makes the top go about the vertical axis.

The situation we have is that gravity causes such force couple to appear, just like the man tilting the spinning wheel causes such force couple to appear. Now, the analysis above shows that the direction of the new force is perpendicular to the plane in which the axis of rotation changes, or wants to change in the case of our spinning top. So gravity wants to pull the top down and causes it to move sideways. This horizontal movement will, in turn, create another force couple. The direction of the resultant force, at the free end of the axis of rotation of the top, will, once again, be vertical, but it will oppose the gravity force. So, in a very simplified explanation of things, we could say:

  1. Gravity pulls the top downwards, and causes a force that will make the top move sideways. So the new force, which causes the precession movement, is orthogonal to the gravitation force, i.e. it’s a horizontal force.
  2. That horizontal force will, in turn, cause another force to appear. That force will also be orthogonal to the horizontal force. As we made two 90 degrees turns, so to say, i.e. 180 degrees in total, it means that this third force will be opposite to the gravitational force.
  3. In equilibrium, we have three forces: gravity, the force causing the precession and, finally, a force neutralizing gravity as the spinning top precesses about the vertical axis.

This approach allows for a treatment that is somewhat more intuitive than Feynman’s concept of the ‘missing force.’

Spinning: the essentials

When introducing mirror symmetry (P-symmetry) in one of my older posts (time reversal and CPT-symmetry), I also introduced the concept of axial and polar vectors in physics. Axial vectors have to do with rotations, or spinning objects. Because spin – i.e. turning motion – is such an important concept in physics, I’d suggest we re-visit the topic here.

Of course, I should be clear from the outset that the discussion below is entirely classical. Indeed, as Wikipedia puts it: “The intrinsic spin of elementary particles (such as electrons) is quantum-mechanical phenomenon that does not have a counterpart in classical mechanics, despite the term spin being reminiscent of classical phenomena such as a planet spinning on its axis.” Nevertheless, if we don’t understand what spin is in the classical world – i.e. our world for all practical purposes – then we won’t get even near to appreciating what it might be in the quantum-mechanical world. Besides, it’s just plain fun: I am sure you have played, as a kid of as an adult even, with one of those magical spinning tops or toy gyroscopes and so you probably wonder how it really works in physics. So that’s what this post is all about.

The essential concept is the concept of torque. For rotations in space (i.e. rotational motion), the torque is what the force is for linear motion:

  • It’s the torque (τ) that makes an object spin faster or slower, just like the force would accelerate or decelerate that very same object when it would be moving along some curve (as opposed to spinning around some axis).
  • There’s also a similar ‘law of Newton’ for torque: you’ll remember that the force equals the time rate-of-change of a vector quantity referred to as (linear) momentum: F = dp/dt = d(mv)/dt = ma (the mass times the acceleration). Likewise, we have a vector quantity that is referred to as angular momentum (L), and we can write: τ (i.e. the Greek tau) = dL/dt.
  • Finally, instead of linear velocity, we’ll have an angular velocity ω (omega), which is the time rate-of-change of the angle θ defining how far the object has gone around (as opposed to the distance in linear dynamics, describing how far the object has gone along). So we have ω = dθ/dt. This is actually easy to visualize because we know that θ is the length of the corresponding arc on the unit circle. Hence, the equivalence with the linear distance traveled is easily ascertained.

There are numerous other equivalences. For example, we also have an angular acceleration: α = dω/dt = d2θ/dt2; and we should also note that, just like the force, the torque is doing work – in its conventional definition as used in physics – as it turns an object:

ΔW = τ·Δθ

However, we also need to point out the differences. The animation below does that very well, as it relates the ‘new’ concepts – i.e. torque and angular momentum – to the ‘old’ concepts – i.e. force and linear momentum.

Torque_animation

So what do we have here? We have vector quantities once again, denoted by symbols in bold-face. However, τ, L and ω are special vectors: axial vectors indeed, as opposed to the polar vectors F, p and v. Axial vectors are directed along the axis of spin – so that is, strangely enough, at right angles to the direction of spin, or perpendicular to the ‘plane of the twist’ as Feynman calls it – and the direction of the axial vector is determined by the direction of spin through one of two conventions: the ‘right-hand screw rule’ or the ‘left-hand screw rule’. Physicists have settled on the former.

If you feel very confused now (I did when I first looked at it), just step back and go through the full argument as I develop it here. It helps to think of torque (also known, for some obscure reason, as the moment of the force) as a twist on an object or a plane indeed: the torque’s magnitude is equal to the tangential component of the force, i.e. F·sin(Δθ), times the distance between the object and the axis of rotation (we’ll denote this distance by r). This quantity is also equal to the product of the magnitude of the force itself and the length of the so-called lever arm, i.e. the perpendicular distance from the axis to the line of action of the force (this lever arm length is denoted by r0). So we can write τ as:

  1. The product of the tangential component of the force times the distance r: τ = r·Ft = r·F·sin(Δθ)
  2. The product of the length of the lever arm times the force: τ = r0·F
  3. The torque is the work done per unit of distance traveled: τ = ΔW/Δθ or τ = dW/dθ in the limit.

So… These are actually only the basics, which you should remember from your high-school physics course. If not, have another look at it. We now need to go from scalar quantities to vector quantities to understand that animation above. Torque is not a vector like force or velocity, not a priori at least. However, we can associate torque with a vector of a special type, an axial vector. Feynman calls vectors such as force or (linear) velocity ‘honest’ or ‘real’ vectors. The mathematically correct term for such ‘honest’ or ‘real’ vectors is polar vector. Hence, axial vectors are not ‘honest’ or ‘real’ in some sense: we derive them from the polar vectors. They are, in effect, a so-called cross product of two ‘honest’ vectors. Here we need to explain the difference between a dot and a cross product between two vectors once again:

(1) A dot product, which we denoted by a little dot (·), yields a scalar quantity: b = |a||b|cosα = a·b·cosα with α the angle between the two vectors a and b. Note that the dot product of two orthogonal vectors is equal to zero, so take care:  τ = r·Ft = r·F·sin(Δθ) is not a dot product of two vectors. It’s a simple product of two scalar quantities: we only use the dot as a mark of separation, which may be quite confusing. In fact, some authors use ∗ for a product of scalars to avoid confusion: that’s not a bad idea, but it’s not a convention as yet. Omitting the dot when multiplying scalars (as I do when I write |a||b|cosα) is also possible, but it makes it a bit difficult to read formulas I find. Also note, once again, how important the difference between bold-face and normal type is in formulas like this: it distinguishes vectors from scalars – and these are two very different things indeed.

(2) A cross product, which we denote by using a cross (×), yields another vector: τ = r×F =|r|·|F|·sinα·n = r·F·sinα·n with n the normal unit vector given by the right-hand rule. Note how a cross product involves a sine, not a cosine – as opposed to a dot product. Hence, if r and F are orthogonal vectors (which is not unlikely), then this sine term will be equal to 1. If the two vectors are not perpendicular to each other, then the sine function will assure that we use the tangential component of the force.

But, again, how do we go from torque as a scalar quantity (τ = r·Ft) to the vector τ = r×F? Well… Let’s suppose, first, that, in our (inertial) frame of reference, we have some object spinning around the z-axis only. In other words, it spins in the xy-plane only. So we have a torque around (or about) the z-axis, i.e. in the xy-plane. The work that will be done by this torque can be written as:

ΔW = FxΔx + FyΔy = (xFy – yFx)Δθ

Huh? Yes. This results from a simple two-dimensional analysis of what’s going on in the xy-plane: the force has an x- and a y-component, and the distance traveled in the x- and y-direction is Δx = –yΔθ and Δy = xΔθ respectively. I won’t go into the details of this (you can easily find these elsewhere) but just note the minus sign for Δx and the way the x and y get switched in the expressions.

So the torque in the xy-plane is given by τxy = ΔW/Δθ = xFy – yFx. Likewise, if the object would be spinning about the x-axis – or, what amounts to the same, in the yz-plane – we’d get τyz = yFz – zFy. Finally, for some object spinning about the y-axis (i.e. in the zx-plane – and please note I write zx, not xz, so as to be consistent as we switch the order of the x, y and z coordinates in the formulas), then we’d get τzx = zFx – xFz. Now we can appreciate the fact that a torque in some other plane, at some angle with our Cartesian planes, would be some combination of these three torques, so we’d write:

(1)    τxy = xFy – yFx

(2)    τyz = yFz – zFy and

(3)    τzx = zFx – xFz.

Another observer with his Cartesian x’, y’ and z’ axes in some other direction (we’re not talking some observer moving away from us but, quite simply, a reference frame that’s being rotated itself around some axis not necessarily coinciding with any of the x-, y- z- or x’-, y’- and z’-axes mentioned above) would find other values as he calculates these torques, but the formulas would look the same:

(1’) τx’y’ = x’Fy’ – y’Fx’

(2’) τy’z’ = y’Fz’ – z’Fy’ and

(3’) τz’x’ = z’Fx’ – x’Fz’.

Now, of course, there must be some ‘nice’ relationship that expresses the τx’y’, τy’z’ and τz’x’ values in terms of τxy, τyz, just like there was some ‘nice’ relationship between the x’, y’ and z’ components of a vector in one coordination system (the x’, y’ and z’ coordinate system) and the x, y, z components of that same vector in the x, y and z coordinate system. Now, I won’t go into the details but that ‘nice’ relationship is, in fact, given by transformation expressions involving a rotation matrix. I won’t write that one down here, because it looks pretty formidable, but just google ‘axis-angle representation of a rotation’ and you’ll get all the details you want.

The point to note is that, in both sets of equations above, we have an x-, y- and z-component of some mathematical vector that transform just like a ‘real’ vector. Now, if it behaves like a vector, we’ll just call it a vector, and that’s how, in essence, we define torque, angular momentum (and angular velocity too) as axial vectors. We should note how it works exactly though:

(1) τxy and τx’y’ will transform like the z-component of a vector (note that we were talking rotational motion about the z-axis when introducing this quantity);

(2) τyz and τy’z’ will transform like the x-component of a vector (note that we were talking rotational motion about the x-axis when introducing this quantity);

(3) τzx and τz’x’ will transform like the y-component of a vector (note that we were talking rotation motion when introducing this quantity). So we have

τ = (τyz, τzx, τxy) = (τx, τy, τz) with

τx = τyz = yFz – zFy

τy = τzx = zFx – xFz

τz = τxy = xFy – yFx.

[This may look very difficult to remember but just look at the order: all we do is respect the clockwise order x, y, z, x, y, z, x, etc. when jotting down the x, y and z subscripts.]

Now we are, finally, well equipped to once again look at that vector representation of rotation. I reproduce it once again below so you don’t have to scroll back to that animation:

Torque_animation

We have rotation in the zx-plane here (i.e. rotation about the y-axis) driven by an oscillating force F, and so, yes, we can see that the torque vector oscillates along the y-axis only: its x- and z-components are zero. We also have L here, the angular momentum. That’s a vector quantity as well. We can write it as

L = (Lyz, Lzx, Lxy) = (Lx, Ly, Lz) with

Lx = Lyz = ypz – zpy (i.e. the angular momentum about the x-axis)

Ly = Lzx = zpx – xpz (i.e. the angular momentum about the y-axis)

Lz = Lxy = xpy – ypx (i.e. the angular momentum about the z-axis),

And we note, once again, that only the y-component is non-zero in this case, because the rotation is about the y-axis.

We should now remember the rules for a cross product. Above, we wrote that τ = r´F =|r|×|F|×sina×n = = r×F×sina×n with n the normal unit vector given by the right-hand rule. However, a vector product can also be written in terms of its components: c = a´b if and only

cx = aybz – azby,

cy = azbx – axbz, and

cz = axby – aybx.

Again, if this looks difficult, remember the trick above: respect the clockwise order when jotting down the x, y and z subscripts. I’ll leave it to you to work out r´F and r´p in terms of components but, when you write it all out, you’ll see it corresponds to the formulas above. In addition, I will also leave it to you to show that the velocity of some particle in a rotating body can be given by a similar vector product: v = ω´r, with ω being defined as another axial vector (aka pseudovector) pointing along the direction of the axis of rotation, i.e. not in the direction of motion. [Is that strange? No. As it’s rotational motion, there is no ‘direction of motion’ really: the object, or any particle in that object, goes round and round and round indeed and, hence, defining some normal vector using the right-hand rule to denote angular velocity makes a lot of sense.]

I could continue to write and write and write, but I need to stop here. Indeed, I actually wanted to tell you how gyroscopes work, but I notice that this introduction has already taken several pages. Hence, I’ll leave the gyroscope for a separate post. So, be warned, you’ll need to read and understand this one before reading my next one.