Subtitle: Basic derivation of Ampere’s Law from the Biot-Savart equation.
Know your meme.
It’s been a while since this became a thing, but I think it’s actually a really good question. If you stop to think about it, magnets are one of those things where the structure goes deep down and the pieces which drive the phenomenon become quite confusing and mind bending. Truly, the original meme exploded from an unlikely source who wanted to relish in appreciating those things that seem magical without really appreciating how mind-bending and thought-expanding the explanation to this seemingly earnest question actually is.
As I got on in this writing, I realized that the scope of the topic is bigger than can be tackled in a single post. What is presented here will only be the first part. The succeeding posts may end up being as mathematical as this, but perhaps less so. Moveover, as I got to writing, I realized that I haven’t posted a good bit of math here in a while: what good is the the mathematical poetry of physics if nobody sees it?
Magnets do not get less magical when you understand how they work: they get more compelling.
This image, taken from a website that sells quackery, highlights the intriguing properties of magnets. A solid object with apparently no moving parts has this manner of influencing the world around it. How can that not be magical? Lodestones have been magic forever and they do not get less magical with the explanation.
Truthfully, I’ve been thinking about the question of how they work for a couple days now. When I started out, I realized that I couldn’t just answer this out of hand, even though I would like to think that I’ve got a working understanding of magnetic fields. How the details fit together gets deep in a hurry. What makes a bar magnet like the one in the picture above special? You don’t put batteries in it. You don’t flick a switch. It just works.
For most every person, that pattern above is the depth of how it works. How does it work? Well, it has a magnetic field. When a piece of a certain kind of metal is in a magnetic field, it feels a force due to the magnet and this causes the magnet to pull on it or maybe to stick to it. If you have two magnets together, you can orient them in a certain way and they push each other apart.
In this picture from penguin labs, these magnets are exerting sufficient force on one another that many of them apparently defy gravity. Here, the rod simply keeps the magnets confined so that they can’t change orientations with respect to one another and they exert sufficient repulsive force to climb up the rod as if they have no weight.
It’s definitely cool, no denying.
But, is it better knowing how they work, or just blindly appreciating them because it’s too hard to fill in the blank?
Maybe we can answer that.
The central feature of how magnets work is quite effortlessly explained by the physics of Electromagnetism. Or, maybe it’s better to say that the details are laboriously and completely explained. People rebel against how hard it is to understand the details, but no true explanation is required to be easily explicable.
The forces which hold those little pieces of metal apart are relatively understandable.
Here’s the Lorentz force law. It says that the force (F) on an object with a charge is equal to sum of the electric force on the object (qE) plus the magnetic force (qvB). Magnets interact solely by magnetic force, the second term.
In this picture from Wikipedia, if a charge (q) moving with speed (v) passes into a region containing this thing we call a “magnetic field,” it will tend to curve in its trajectory depending on whether the charge is negative or positive. We can ‘see’ this magnetic field thing in the image above with the bar magnet and iron filings. What is it, how is it produced?
The fundamental observation of magnetic fields is tied up into a phenomenological equation called the Biot-Savart law.
This equation is immediately intimidating. I’ve written it in all of it’s horrifying Jacksonian glory. You can read this equation like a sentence. It says that all the magnetic field (B) you can find at a location in space (r) is proportional to a sum of all the electric currents (J) at all possible locations where you can find any current (r’) and inversely proportional to the square of the distance between where you’re looking for the magnetic field and where all the electrical currents are –it may say ‘inverse cube’ in the equation, but it’s actually an inverse square since there’s a full power of length in the numerator. Yikes, what a sentence! Additionally, the equation says that the direction of the magnetic field is at right angles to both the direction that the current is traveling and the direction given by the line between where you’re looking for magnetic field and where the current is located. These directions are all wrapped up in the arrow scripts on every quantity in the equation and are determined by the cross-product as denoted by the ‘x’. The difference between the two ‘r’ vectors in the numerator creates a pure direction between the location of a particular current element and where you’re looking for magnetic field. The ‘d’ at the end is the differential volume that confines the electric currents and simply means that you’re adding up locations in 3D space. The scaling constants outside the integral sign are geometrical and control strength; the 4 and Pi relate to the dimensionality of the field source radiated out into a full solid angle (it covers a singularity in the field due to the location of the field source) and the ‘μ’ essentially tells how space broadcasts magnetic field… where the constant ‘μ’ is closely tied to the speed of light. This equation has the structure of a propagator: it takes an electric current located at r’ and propagates it into a field at r.
It may also be confusing to you that I’m calling current ‘J’ when nearly every basic physics class calls it ‘I’… well, get used to it. ‘Current vector’ is a subtle variation of current.
I looked for some diagrams to help depict Biot-Savart’s components, but I wasn’t satisfied with what Google coughed up. Here’s a rendering of my own with all the important vectors labeled.
Now, I showed the crazy Biot-Savart equation, but I can tell you right now that it is a pain in the ass to work with. Very few people wake up in the morning and say “Boy oh boy, Biot-Savart for me today!” For most physics students this equation comes with a note of dread. Directly using it to analytically calculate magnetic fields is not easy. That cross product and all the crazy vectors pointing in every which direction make this equation a monster. There are some basic feature here which are common to many fields, particularly the inverse square, which you can find in the Newtonian gravity formula or Coulomb’s law for electrostatics, and the field being proportional to some source, in this case an electric current, where gravity has mass and electrostatics have charge.
Magnetic field becomes extraordinary because of that flipping (God damned, effing…) cross product, which means that it points in counter-intuitive directions. With electrostatics and gravity, the field is usually going toward or away from the source, while magnetism has the field seems to be going ‘around’ the source. Moreover, unlike electrostatics and gravity, the source isn’t exactly a something, like a charge or a mass, it’s dynamic… as in a change in state; electric charges are present in a current, but if you have those charges sitting stationary, even though they are still present, they can’t produce a magnetic field. Moreover, if you neutralize the charge, a magnetic field can still be present if those now invisible charges are moving to produce a current: current flowing in a copper wire is electric charges that are moving along the wire and this produces a magnetic field around the wire, but the presence of positive charges fixed to the metal atoms of the wire neutralizes the negative charges of the moving electrons, resulting in a state of otherwise net neutral charge. So, no electrostatic field, even though you have a magnetic field. It might surprise you to know that neutron stars have powerful magnetic fields, even though there are no electrons or protons present in order give any actual electric currents at all. The requirement for moving charges to produce a magnetic field is not inconsistent with the moving charge required to feel force from a magnetic field as well. Admittedly, there’s more to it than just ‘currents’ but I’ll get to that in another post.
With a little bit of algebraic shenanigans, Biot-Savart can be twisted around into a slightly more tractable form called Ampere’s Law, which is one of the four Maxwell’s equations that define electromagnetism. I had originally not intended to show this derivation, but I had a change of heart when I realized that I’d forgotten the details myself. So, I worked through them again just to see that I could. Keep in mind that this is really just a speed bump along the direction toward learning how magnets work.
For your viewing pleasure, the derivation of the Maxwell-Ampere law from the Biot-Savart equation.
In starting to set up for this, there are a couple fairly useful vector identities.
This trio contains several basic differential identities which can be very useful in this particular derivation. Here, the variables r are actually vectors in three dimensions. For those of you who don’t know these things, all it means is this:
These can be diagrammed like this:
This little diagram just treats the origin like the corner of a 3D box and each distance is a length along one of the three edges emanating from the corner.
I’ll try not to get too far afield with this quick vector tutorial, but it helps to understand that this is just a way to wrap up a 3D representation inside a simple symbol. The hatted symbols of x,y and z are all unit vectors that point in the relevant three dimensional directions where the un-hatted symbols just mean a variable distance along x or y or z. The prime (r’) means that the coordinate is used to tell where the electric current is located while the unprime (r) means that this is the coordinate for the magnetic field. The upside down triangle is an operator called ‘del’… you may know it from my hydrogen wave function post. What I’m doing here is quite similar to what I did over there before. For the uninitiated, here are gradient, divergence and curl:
Gradient works on a scalar function to produce a vector, divergence works on a vector to produce a scalar function and curl works on a vector to produce a vector. I will assume that the reader can take derivatives and not go any further back than this. The operations on the right of the equal sign are wrapped up inside the symbols on the left.
One final useful bit of notation here is the length operation. Length operation just finds the length of a vector and is denoted by flat braces as an absolute value. Everywhere I’ve used it, I’ve been applying it to a vector obtained by finding the distance between where two different vectors point:
As you can see, notation is all about compressing operations away until they are very compact. The equations I’ve used to this point all contain a great deal of math lying underneath what is written, but you can muddle through by the examples here.
Getting back to my identity trio:
The first identity here (I1) takes the vector object written on the left and produces a gradient from it… the thing in the quotient of that function is the length of the difference between those two vectors, which is simply a scalar number without a direction as shown in the length operation as written above.
The second identity (I2) here takes the divergence of the gradient and reveals that it’s the same thing as a Dirac delta (incredibly easy way to kill an integral!). I’ve not written the operation as divergence on a gradient, but instead wrapped it up in the ‘square’ on the del… you can know it’s a divergence of a gradient because the function inside the parenthesis is a scalar, meaning that the first operation has to be a gradient, which produces a vector, which automatically necessitates the second operation to be a divergence, since that only works on vectors to produce scalars.
The third identity (I3) shows that the gradient with respect to the unprimed vector coordinate system is actually equal to a negative sign times the primed coordinate system… which is a very easy way to switch from a derivative with respect to the first r and the same form of derivative with respect to the second r’.
To be clear, these identities are tailor-made to this problem (and similar electrodynamics problems) and you probably will never ever see them anywhere but the *cough cough* Jackson book. The first identity can be proven by working the gradient operation and taking derivatives. The second identity can be proven by using the vector divergence theorem in a spherical polar coordinate system and is the source of the 4*Pi that you see everywhere in electromagnetism. The third identity can also be proven by the same method as the first.
There are two additional helpful vector identities that I used which I produced in the process of working this derivation. I will create them here because, why not! If the math scares you, you’re on the wrong blog. To produce these identities, I used the component decomposition of the cross product and a useful Levi-Civita kroenecker delta identity –I’m really bad at remembering vector identities, so I put a great deal of effort into learning how to construct them myself: my Levi-Civita is ghetto, but it works well enough. For those of you who don’t know the ol’ Levi-Civita symbol, it’s a pretty nice tool for constructing things in a component-wise fashion: εijk . To make this work, you just have to remember it as I just wrote it… if any indices are equal, the symbol is zero, if they are all different, they are 1 or -1. If you take it as ijk, with the indices all different as I wrote, it equals 1 and becomes -1 if you reverse two of the indices: ijk=1, jik=-1, jki=1, kji=-1 and so on and so forth. Here are the useful Levi-Civita identities as they relate to cross product:
Using these small tools, the first vector identity that I need is a curl of a curl. I derive it here:
Let’s see how this works. I’ve used colors to show the major substitutions and tried to draw arrows where they belong. If you follow the math, you’ll note that the Kroenecker deltas have the intriguing property of trading out indices in these sums. Kroenecker delta works on a finite sum the same way a Dirac delta works on an integral, which is nothing more than an infinite sum. Also, the index convention says that if you see duplicated indices, but without a sum on that index, you associate a sum with that index… this is how I located the divergences in that last step. This identity is a soft stopping point for the double curl: I could have used the derivative produce rule to expand it further, but that isn’t needed (if you want to see it get really complex, go ahead and try it! It’s do-able.) One will note that I have double del applied on a vector here… I said that it only applies on scalars above… in this form, it would only act on the scalar portion of each vector component, meaning that you would end up with a sum of three terms multiplied by unit vectors! Double del only ever acts on scalars, but you actually don’t need to know that in the derivation below.
This first vector identity I’ve produced I’ll call I4:
Here’s a second useful identity that I’ll need to develop:
This identity I’ll call I5:
*Pant Pant* I’ve collected all the identities I need to make this work. If you don’t immediately know something off the top of your head, you can develop the pieces you need. I will use I1, I2, I3, I4 and I5 together to derive the Maxwell-Ampere Law from Biot-Savart. Most of the following derivation comes from Jackson Electrodynamics, with a few small embellishments of my own.
In this first line of the derivation, I’ve rewritten Biot-Savart with the constants outside the integral and everything variable inside. Inside the integral, I’ve split the meat so that the different vector and scalar elements are clear. In what follows, it’s very important to remember that unprimed del operators are in a different space from the primed del operators: a value (like J) that is dependent on the primed position variable is essentially a constant with respect to the unprimed operator and will render a zero in a derivative by the unprimed del. Moreover, unprimed del can be moved into or out of the integral, which is with respect to the primed position coordinates. This observation is profoundly important to this derivation.
The usage of the first two identities here manages to extract the cross product from the midst of the function and puts it into a manipulable position where the del is unprimed while the integral is primed, letting me move it out of the integrand if I want.
This intermediate contains another very important magnetic quantity in the form of the vector potential (A) –“A” here not to be confused with the alphabetical placeholder I used while deriving my vector identities. I may come back to vector potential later, but this is simply an interesting stop-over for now. From here, we press on toward the Maxwell-Ampere law by acting in from the left with a curl onto the magnetic field…
The Dirac delta I end with in the final term allows me to collapse r’ into r at the expense of that last integral. At this point, I’ve actually produced the magnetostatic Ampere’s law if I feel like claiming that the current has no divergence, but I will talk about this later…
This substitution switches del from being unprimed to primed, putting it in the same terms as the current vector J. I use integration by parts next to switch which element of the first term the primed del is acting on.
Were I being really careful about how I depicted the integration by parts, there would be a unit vector dotted into the J in order to turn it into a scalar sum in that first term ahead of the integral… this is a little sloppy on my part, but nobody ever cares about that term anyway because it’s presupposed to vanish at the limits where it’s being evaluated. This is a physicist trick similar to pulling a rug over a mess on the floor –I’ve seen it performed in many contexts.
This substitution is not one of the mathematical identities I created above, this is purely physics. In this case, I’ve used conservation of charge to connect the divergence of the current vector to the change in charge density over time. If you don’t recognize the epic nature of this particular substitution, take my word for it… I’ve essentially inverted magnetostatics into electrodynamics, assuring that a ‘current’ is actually a form of moving charge.
In this line, I’ve switched the order of the derivatives again. Nothing in the integral is dependent on time except the charge density, so almost everything can pass through the derivative with respect to time. On the other hand, only the distance is dependent on the unprimed r, meaning that the unprimed del can pass inward through everything in the opposite direction.
At this point something amazing has emerged from the math. Pardon the pun; I’m feeling punchy. The quantity I’ve highlighted blue is a form of Coulomb’s law! If that name doesn’t tickle you at the base of your spine, what you’re looking at is the electrostatic version of the Biot-Savart law, which makes electric fields from electric charges. This is one of the reasons I like this derivation and why I decided to go ahead and detail the whole thing. This shows explicitly a connection between magnetism and electrostatics where such connection was not previously clear.
And thus ends the derivation. In this casting, the curl of the magnetic field is dependent both on the electric field and on currents. If there is no time varying electric field, that first term vanishes and you get the plain old magnetostatic Ampere’s law:
This says simply that the curl of the magnetic field is equal to the current. There are some interesting qualities to this equation because of how the derivation leaves only a single positional dependence. As you can see, there is no separate position coordinate to describe magnetic field independently from its source. And, really, it isn’t describing the magnetic field as ‘generated’ by the current, but rather that a deformation to the linearity of the magnetic field is due to the presence of a current at that location… which is an interesting way to relate the two.
This relationship tends to cause magnetic lines to orbit around the current vector.
This image from hyperphysics sums up the whole situation –I realize I’ve been saying something similar from way up, but this equation is proof. If you have current passing along a wire, magnetic field will tend to wrap around the wire in a right handed sense. For all intents and purposes, this is all the Ampere’s law says, neglecting that you can manipulate the geometry of the situation to make the field do some interesting things. But, this is all.
Well, so what? I did a lot of math. What, if anything, have I gained from it? How does this help me along the path to understanding magnets?
The Ampere Law is useful in generating very simple magnetic field configurations that can be used in the Lorentz force law, ultimately showing a direct dynamical connection between moving currents and magnetic fields. I have it in mind to show a freshman level example of how this is done in the next part of this series. Given the length of this post, I will do more math in a different post.
This is a big step in the direction of learning how magnets work, but it should leave you feeling a little unsatisfied. How exactly do the forces work? In physics, it is widely known that magnetic fields do no work, so why is it that bar magnets can drag each other across the counter? That sure looks like work to me! And if electric currents are necessary to drive magnets, why is it that bar magnets and horseshoe magnets don’t require batteries? Where are the electric currents that animate a bar magnet and how is it that they seem to be unlimited or unpowered? These questions remain to be addressed.
Until the next post…