Maths

I will from time to time be changing the contents of this page.

But for now, here are copies of the understanding Maxwell’s electromagnetic equations.

Part 1:

For reasons too bizarre and convoluted to go into here, I needed to understand what Maxwell’s equations really felt like to help with some of my world building for Miranda. What do you do when you are faced with hieroglyphs like this:

Gauss’s law
Gauss’s law for magnetism
Maxwell–Faraday equation (Faraday’s law of induction)
Ampère’s circuital law (with Maxwell’s correction)

Scccrrrrreeeeeeeeeaaaaaaaaaaaammmmmmm…………..

Sorry, needed to get that long word out of my system, first.

The first equation [where E denotes electric field and ε (with subscript 0) is just a constant] is not much different in form to the law of gravity, only gravity relies on mass rather than electric charge density ρ. So like gravity the further away one electric charge was from another, the weaker the force between them. In fact the force is proportional to one over the distance between the charges squared. Yep, same as gravity that we all experience.

But there is one major difference. Gravity always attracts. Electric charges can be negative or positive. Opposite charges attract, like charges repel. So it is only truly like gravity if there are only two opposite charges involved. Doh! Not so simple is it?

Hold on a mo. We’ve all dealt with the north and south poles of magnets. Dealing with the positive and negative charges in an electric field is no different. So we have magnetism showing us the way for direction of force on electric charges, while we can have gravity giving us a sense of changing strength of force on a charge as it moves around. O.K. Got it.

What about the second equation? Well, it is the magnetic field, B, instead of the electric field. So the business about magnitude and direction of force is the same as for an electric field.

But hold on a sec. What is powering the magnetic field?

Well it is a moving electric charge. Here the emphasis is on MOVING.

This is where things like torque and rotation come into play… oh yuck… my head hurts… more in a later post… when I’ve figured out how to describe it…

Part 2:

To recap – we know the strength of an electric field (E) is weaker the further away from it another electric is – indeed it is proportional to one over the distance between them squared, just like gravity between masses. Unlike gravity, the direction is dependent on the polarity of the charges, which could be pushing as well as pulling at the charges. Like charges want to push each other away, unlike charges will pull at each other. Magnetic fields (B) have the same aspects.

What I now want to look at is the relationship between the electric fields and magnetic fields. This are described in two of Maxwell’s equations:

Maxwell–Faraday equation (Faraday’s law of induction)
Ampère’s circuital law (with Maxwell’s correction)

Anything with a subscript 0 is a constant. J is current density (i.e. moving charges). The first question I have to ask is what does that upside-down triangle followed by a multiplication sign actually mean. In mathematics, it is called a curl. And that is actually what it does to the thing it is operating on. It curls or rotates whatever it is operating on, the electric field in the first equation above and the magnetic field in the second equation.

Let us examine the first of the above equations in a little more depth. If the magnetic field does not change over time, then the right hand side of the equation would be 0. Which means there would be nothing to turn the direction of the electric field away from whatever direction it was working in.

If we increase the magnitude of the magnetic field over time, then the amount of twist in the electric field will also increase. For instance, if the magnetic field is pointing directly towards you, then the electric field increases its twistedness or curliness around that vector. If you draw contours of electric field strength around the magnetic field vector, you will see the contours becoming denser as the strength of the magnetic field increases.

Whichever point you look at, the electric field is at right angles to the change in the magnetic field over time.

In the second of the above equations and ignoring J for the moment (i.e. current density is assumed to be 0), we have a similar construct where if the direction of the change in electric field over time is pointing towards you, then the magnetic field twists around that vector.

Adding J into the picture distorts the magnitude of the electric field and the twist of the magnetic field. Well, it is a moving charge after all, so it makes sense, doesn’t it? But how does the current density, J, fit into the picture? Well the combination of the current density and the way the electric field changes is perpendicular to magnetic field. It is possible to have E, B and J all perpendicular to each other, and indeed this is the way most people think about the electric and magnetic fields, and current densities working. But we haven’t shown that the current density must be perpendicular to the electric field.

O.K. So how does impact my science fiction novel, Miranda? Well it gives room for a totally new experienceable phenomenon… oh, this is interesting… I’m off to write into Miranda…