Maxwell’s Snowflakes

 Only one day, one tutorial and one lecture to go before Christmas holidays! We had our house dinner yesterday, which was awesome, but this blog is on something a bit different…

Basically, I was trying to listen to a lecture on matter reacting to magnetic fields, when I noticed a great tweet by symmetry mag showing their paper snowflakes in the shape of famous scientists. They look amazing, but are pretty intricate, and I don’t have a craft knife—also I was pretty sure I could make them even more nerdy, so I decided to come up with my own science inspired snowflakes.

So what part of science to choose?

As the lecture was on Magnetism, I decided on that Physicist’s firm favourite, Maxwell’s equations 😛

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If you’re up for a challenge try and match the snowflake to the equation without reading on…

These are four equations (though ACP at the moment is trying to convince me you can introduce some nightmarish notation to write them down as one) that describe how the Electric and Magnetic forces behave and relate to each other. Check out these snowflakes to find out more 🙂

Before we begin, here is a handy equation decoder for the tricky bits of the equations:

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Gauss’s (Christmas) Law

This can be written as:

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or:

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or:

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What the snowflake represents is a central positive charge with electric field lines radiating outwards from it (this is the ‘charge’ side of the equations). The circle round the edge represents the sphere you are integrating over, which is the other side of the red equation with the double integral sign on.

In practice, you can use Gauss’s equation to solve a situation where you know the charge (Q) inside a shape, to find an expression for what the electric field looks like. You try and pick a nice surface like a sphere which is easy to write down and in this case meets the field lines at 90 degrees so the maths all turns out easier 😛

Gauss’s (Santa Claus is coming to town) Law for magnetism

This is always the first Maxwell equation people say when trying to remember them all because it only has one non-zero side 😛

You might think that this makes it a useless equation, but actually it’s pretty cool because it is basically a statement that there are no magnetic monopoles. A magnetic monopole is like having the North end of a magnet without the South end (and it turns out you can’t just chop a magnet in half—that was the first thing they tried :P)

It is written as:

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or:

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or:

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The non-zero side is kind of the same as Gauss’s electrical law, which exposes a difference between electric and magnetic charges. You can obviously have a positive electric charge without a negative one right by it, which is why the electric equation is non-zero.

You can think of it in another way—that electric field lines start and end at the positive and negative charges as you can see from the first snowflake—the lines are starting right at the surface of the charge (well not really as the snowflake would have fallen apart, but in theory!) Magnetic field lines on the other hand, are always little loops. They never start or end anywhere, so when you try and make a nice surface to integrate over like before, you always get the same amount of lines coming into the surface as going out. This means the total ‘magnetic charge’ within any surface adds up to zero, because if you include a north pole (+1) you also have to take in another south pole (-1).

This is what this snowflake represents 🙂 The circles are the little loops of magnetic field lines and the wiggly line joining them is a desperate attempt to make some crazy path that takes in an uneven number of field lines, but of course, fails 😛

Faraday’s (Deck the halls with boughs of EM radiation) Law

This is a law you might have come across if you do A-level physics— it is the first equation on our list to contain both B and E (electric and magnetic fields) and relate the two together.

It is written as:

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or:

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or:

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It simply says that if you have a changing magnetic field, then you get an electric field generated. This is probably the most abstract snowflake—imagine time starts at zero at the centre and goes out towards the ends. The expanding circles are meant to be the increase in magnetic field, and the little flicks are meant to be the electric field lines generated 🙂

(Alex was least convinced by this one :P)

Ampere’s (Be merry and mathematically bright) Law

This is the final equation and I’ve decided to go the whole hog and use the full version that has a term to take into account ‘displacement current’—basically a changing electric field.

It can be written as:

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or:

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or:

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The bendy lines on the snowflakes are wires carrying current (as demonstrated by my signature flicks for the electric field lines of course), and the circle demonstrates the magnetic field that has been generated by the current flow. To factor in the fact we need to be wary of changing electric fields, I’ve put in two little dials  with arows which are meant to demonstrate that the electric field is changing…:P

And that’s all of them!

You might be confused as to why I have written out two forms of all the equations… Surely one or the other version would have done? Well, the main reason is that I really like the differential (non-integral) versions, and the two relations you need to change one type to the other 😛

These two relations are called ‘Stokes’s Theorem’ and the ‘Divergence Theorem’ and I’m pretty sure I’ve mentioned them in blogs many times before because they are cool <3

So, in the spirit of Christmas (?) I have decided to make snowflakes to explain these guys too, because maths relations need holidays as well… 🙂

Stokes’ (Oh I wish it could be Christmas everyday [but that’s not mathematically possible]) Theorem

Stokes’ theorem says that if you find the curl (the triangle with the x next to it) of a field—here an electric or magnetic field—over an area, you can also get the same result by integrating around any curve that encloses that surface.

It can be written as:

or:

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You can see the curl on the snowflake (the curls in the middle) and also the nice round curve that encloses them. 🙂

This theorem is pretty crazy because you can have a legit mental surface like a plastic bag, all crumpled up and folded over and be like omg this is going to take ages to write down as an equation… but you can instead just integrate around a nice circle at the opening of the bag that bounds it. Phew.

Also, even more weirdly, that means that that crazy surface must end up with the same result as a flat surface across the curve.

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It’s like a Delia Smith cheat’s recipe for Physics. If it ain’t a straight line or a square you don’t want to know. Even with ellipses (squished circle) you usually perform a co-ordinate transform so you can get it back into a circle and breathe easy.

The (Rudolph the 620–750 nm-wavelength-light-nosed reindeer) Divergence Theorem

This is my most ambitious snowflake, no doubt. To demonstrate that this one is an integral over a volume, I thought I’d try and make it 3D so I stacked three snowflakes on top of each other. 🙂

 It can be written as:

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or:

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What you can see in the snowflake is a little sun-shaped thing in a box that is putting out rays that are meant to look like they are escaping through the surface of the box.

The divergence theorem says that if you take the divergence (the triangle with the dot next to it) of the electric or magnetic field over the volume, then you get the same result as if you integrate the amount of field lines coming through the surface of the box.

(As you might have guessed you can try this for the magnetic field but you will always get zero [one line going in (-ve) for every line going out (+ve)] so give it up now ;))

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Oh, and I also made this Rosette and Philae one, because #2014

Merry Christmas!

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