## Maxwell's Idea of the Field: Maxwell's Equations
The development of "Maxwell's Equations" is a complicated subject and its historical roots are no longer relevant as you'll maybe see. What Maxwell faced was the complete works of Faraday in electricity and magnetism which it's worth summarizing, so that we face the same challenge as Maxwell.
### The Four Electricity and Magnetism Challenges to Maxwell
Remember that Faraday studied quantitatively or outright discovered four phenomena:
1. **Electric charges create forces.** Electrically charged objects (remember, at that time, macroscopically sized "chunks" of material) come in two forms arbitrarily designated as positive and negative.
* Attractive forces occur between oppositely electrically charged objects and repellant forces occur between same-charged objects.
* Faraday pictured the space between charges as consisting of material "lines of force," actual material disturbances in space. He coined the term "fields" to this idea. This is the **Electric Field**.
* Prior to Faraday, Coulomb showed that the amount of force between two charged objects reduces by the square of the distance from them; separate them by twice as much, and the force will decrease by a factor of 4. This is the "**inverse square" force** relationship that gravity also respects.
1. **Magnetic poles create forces.** Magnetized objects also come in two forms, historically dubbed as north and south.
* Forces are exerted between magnetic poles in a similar sense as for electrical charges; attraction for N-S poles and repulsion for N-N or S-S poles.
* Unlike electrical charges, magnetic poles appear to always exist in pairs. Bar magnets cannot be separated into isolated magnetic charges (a phenomenon still with us today).
* Although we didn't consider this, Coulomb also showed an inverse square force relationship for magnetic poles.
* Faraday came to his lines of force picture originally from his observation of the patterns of iron filings when in the presence of a magnet. It's hard to not believe that "something is there." This is Faraday's **Magnetic Field**.
1. **Changing magnetic fields create currents.** Faraday's discovery of induction: a magnetic field that changes intensity in time will create a current in a nearby wire. When the magnetic intensity is constant, the current stops. This too could be restated as: "**Changing magnetic fields create changing electric fields**."
1. **Currents create magnetic fields.** Oersted's discovery: electrical currents create a magnetic field (Faraday's interpretation) that can exert a force on magnetic poles.
Stay tuned to this last one.
The connection between electricity and magnetism was too intimate to be unrelated. Ampere and others had created mathematical models that ignored the field aspect, which was roundly dismissed by almost all of the community of physicists. For example, Ampere could calculate the force on a current-carrying wire by another current-carrying wire without reference to any field---just "action at a distance," which was still as unsatisfying to 19th century people as in Newton's time.
While Maxwell was still a student another genius student, himself only a few years older, William Thomson (the future 1st Baron Kelvin, OM, GCVO, PC, PRS, FRSE...whew) had worked out the mathematics of continuous heat and fluid flow.
```{admonition} Wait. Buried in that long list of titles, the Kelvin name sounds familiar
:class: warning
**Glad you asked:** Lord Kelvin, was one of the greats of late classical physics and one of only three people to have their own temperature scale named for them. There was Daniel Gabriel Fahrenheit, Anders Celsius, and Baron Lord Kelvin. The Kelvin scale has the same incremental size as Celsius (or centigrade) but its zero is at the absolute zero of temperature. You'll find light bulbs' color gauged on packages in “K"...5000K would be a bulb that has the color of bright daylight.
```
Thomson suggested, and Maxwell wrestled to the ground, the analogy that the equations that described the continuous flow of heat were similar to those that might describe the "flow" of electrical and magnetic fields...now proposed by Maxwell to be a continuous "flux" that the rate and direction of the direction of heat flow is analogous to the strength and direction of the "flow" of electric field. Just like heat can have a *source* (something hot) and a *sink* (something cold), the sources and sinks of electricity could be thought of as opposite electrical charges. You can sort of see this, right?
He took Faraday's field idea and make it the actual "substance" of electricity and magnetism. He could account for both #1 and #2 above–static electric and magnetic fields with this direct analogy with heat or fluids. The time changing magnet and currents arrangements of #3 and #4 eluded him. But he thought he was onto something. He published his work in 1855, *On Faraday’s Lines of Force*.
In some ways, he'd not expanded on Faraday's original "thought-pictures" but he did give them a respectable mathematical basis. Mind you: he was only a student, on his way to Cambridge! He was surely thrilled by the note he received from Faraday:
>I received your paper, and thank you very much for it. I do not venture to thank you for what you have said about 'Lines of Force', because I know you have done it for the interests of philosophical truth; but you must suppose it is work grateful to me, and gives me much encouragement to think on. I was at first almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well.
The subsequent six years saw him complete his degree; receive (and lose) his job at Marishal College;[^1]
[^1]: Marishal College was merged with another university and Maxwell was deemed "redundant" because there is only one "professor" in any subject.
become appointed as the Professor of Natural Philosophy at King's College, London; receive the Rumford Medal of the Royal Society (for his color vision work); and election to the Royal Society (for color and for his Saturn rings work). It was time then to take up electricity and magnetism again, with an eye on those pesky #3 and #4 Faraday discoveries.