#vector calculus

(will add to this list as I find/remember more)

Faraday showed that electricity moved because it was pushed by a field.

But the study of fields required a new branch of mathematics, which was eventually codified by Cambridge mathematician James Clerk Maxwell and called vector calculus. 

So in the same way that Kepler and Galileo laid the foundation for Newtonian physics, Faraday paved the way for Maxwell’s equations.  Maxwell was a virtuoso in mathematics who made astonishing breakthroughs in physics. He realized that the behavior of electricity and magnetism, as discovered by Faraday and others, could be summarized in precise mathematical language.

One law stated that a moving magnetic field could create an electric field. 

Another law stated the opposite, that a moving electric field could create a magnetic field.

 Then Maxwell had an idea for the ages. What if a changing electric field created a magnetic one that then created another electric field that then created another magnetic field, etc.?

He had the brilliant insight that the end product of this rapid back-and-forth motion would be a moving wave, where electric and magnetic fields were constantly turning into each other.

This infinite sequence of transformations has a life of its own, creating a moving wave of vibrating electric and magnetic fields.

 Using vector calculus, he calculated the speed of this moving wave, and he found it to be 310,740 kilometers per second. He was shocked beyond belief. To within experimental error, this speed was remarkably close to the speed of light (which is now known to be 299,792 kilometers per second).

He then made the next bold step to claim that this was light! Light is an electromagnetic wave.

 Maxwell then wrote prophetically, “We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

 Today, every physics student and electrical engineer has to memorize Maxwell’s equations. They are the basis for TVs, lasers, dynamos, generators, and more.