The Genesis of Light

There's an old (at least, I assume it's fairly old) physics meme that goes:

“And God said…
\[\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_\circ}\\
\nabla\cdot\mathbf{B}=0\\
\nabla\times\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}\\
\nabla\times\mathbf{B}=\mu_\circ\mathbf{J}+\mu_\circ\epsilon_\circ\frac{\partial\mathbf{E}}{\partial t}\]

…and there was light.”

Here E is the electric field, \(\rho\) is the electric charge density, B is the magnetic field, J is the electric current density, and \(\mu_\circ\) and \(\epsilon_\circ\) are the permittivity and permeability of free space, respectively. Bold-face quantities are vector quantities. These four equations are Maxwell's equations which describe electricity and magnetism in terms of classical field theory, but more on that a little later.

Back in 2016 my parents took a trip to Israel and brought me back a shirt with this on it for Christmas, except in the original Hebrew. Pretty cool! Except, they weren't Maxwell's equations, and on a closer look they weren't actually equations at all, just collections of symbols that looked kinda like some equations from special relativity, and to top it off the shirt was just barely big enough for me so I never actually wore it.

But I loved the idea, and now (a little over a year later) I've created my own variation on the design and had a shirt printed with it:

This was a surprisingly difficult photo to take on my own.

Although known as Maxwell's equations (after the Scottish physicist James Clerk Maxwell, widely considered the third greatest physicist of all time after Newton and Einstein), these four equations are actually a reformulation of Maxwell's original twenty equations in twenty variables by Oliver Heaviside, who cast them into vector calculus form and condensed them down into four by use of the divergence (\(\nabla\cdot\) ) and curl (\(\nabla\times\) ) operators.

Being a classical description of electromagnetism Maxwell's equations have been superseded by quantum electrodynamics, but they are still very useful in a wide variety of situations that do not involve strong electromagnetic fields or individual photons, just as Newtonian gravitation is still a useful approximation to general relativity in areas of weak gravitational field.

These four equations, and what they represent, are a monumental achievement—second, at the time, only to Newton's work—and have a sublime beauty to the physicist. The seemingly-disparate forces of electricity and magnetism are revealed to be both aspects of a singular electromagnetic force. This is seen in the third and fourth equations, where curl (or rotation) of an electric field is seen to rely on a magnetic field, and vice versa. All four equations can be combined (in a vacuum, where J and \(\rho\) are both zero) to derive the electromagnetic wave equation which describes light as a series of correlated ripples in the electric and magnetic fields, or electromagnetic radiation. (In fact, it turns out that \(\frac{1}{\sqrt{\mu_\circ\epsilon_\circ}}=c\), the speed of light!)

The asymmetries between the electric and magnetic fields that at first glance might seem to mar the beauty of the whole only enhance it upon further inspection, as the minus sign in the third equation is crucial to forming the feedback loop in the electromagnetic wave equation that allows light to travel forever as self-contained photons. They helped motivate Einstein to develop special relativity (which Maxwell's equations are compatible with) to explain why the same phenomenon could be seen as an electric or a magnetic effect depending on the frame of reference chosen. And the second equation explains why you can't break a magnet in half and end up with two monopoles.

I used EqualX (which I wrote about last month) to typeset the equations in \(\LaTeX\), then exported them as SVG which I imported into Inkscape where I added the text (and did a lot of manual tweaking of the layout of the various elements to make it look nice). Switching my keyboard to Hebrew and figuring out the letters took quite a while, which is why there are no vowel pointings; Inkscape's support for right-to-left fonts is a bit fiddly (though I saw just the other day an update that supposedly improved it) and trying to figure out all the various pointings and getting them around the right letters was a nightmare, so I gave up after tortuously figuring out the first three. At least it's more authentic ancient Hebrew now…

While working on the design for this shirt (I uploaded five different versions to the printing site before I was satisfied) I thought sardonically to myself that I was making this shirt for my own enjoyment, and that of the perhaps five other people on the planet who understood both Biblical Hebrew and electromagnetism. Then lo and behold, the first day I wore it, while walking around Bunnings (basically Australian Home Depot) a gentleman stopped me, said he thought it was great, and asked where I'd gotten it. When I said I'd designed it and had it printed myself he then asked if I was selling it anywhere!

I ended up sending him the image file to use, but it got me thinking. I've gone ahead and uploaded the design to Spreadshirt.com, where I got the original shirt printed. You can find it (and a version in white for dark backgrounds) for sale on shirts here. It defaults to showing the men's styles, but there are women's styles as well and you can pick from a range of colors. (If any of you out there actually order one I'd love to hear about it!)