Before jumping into the equations, famous as they are, I believe it’s important to explain how to approach scientific principles in general. There is a tendency for text books to present equations and then try to explain the science through the equations.  I recall often struggling to understand a new concept by staring at the equation and hoping that the fundamental principle would suddenly appear in my mind; this is somewhat backwards. There are times when this approach works, but typically it is not the right approach.

In my experience, the best textbooks can be discovered in 15 seconds by this method:  take the book and slowly thumb the pages while glancing in as they go by.  If you see more text than equations, it is probably a well written book.  If it has more equations than text, it may be that the author is trying to impress his or her colleagues.  This may sound critical, but this site is about inspiring STEM education.  Good authors will take the time to explain concepts before describing them mathematically.

One example of a well written book on the subject of physics is The Feynman Lectures on Physics, by Richard Feynman, Robert Leighton and Matthew Sands, (Addison-Wesley publishing company).

Here is an excerpt from The Feynman Lectures on Physics, Volume II, page 2-1:  “A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist. Ordinarily, a course like this is given by developing gradually the physical ideas-by starting with simple situations and going on to more and more complicated situations. This requires that you continuously forget things you previously learned-things that are true in certain situations, but which are not true in general.”  Case in point: electrons do not “orbit” around an atom’s nucleus like planets around the sun. The orbit concept helps introduce the scientific principles of electrons in matter, but has no semblance of reality.

Maxwell’s Equations quite elegantly describe electromagnetism and the propagation of electromagnetic waves. However, the mathematics requires an understanding of differential and integral calculus. I had considered not even showing the equations and just provide a description of the science. Sometimes students panic at first and completely dismiss a course because of the math. I am trying to assuage these fears and concerns – sometimes the concepts are really not difficult to understand.

So, to that end, let’s present the electric field E, and magnetic field B.  Electric fields arise in the presence of charge, like electrons. The electric field can be described by lines that extend away from charge. So if you had a bunch of electrons between your finger tips, the E field would be strongest between your fingers and would reduce in intensity at distances away from your fingers in all directions. This is the concept of divergence. Divergence is like light from a lantern that reduces in intensity as you move away from the source in all directions. The first of Maxwell’s equations below describes the divergence of the electric field, where ρ is the charge density.

Next, there is no equivalent magnetic “charge” that you can hold between your fingers. A magnetic field B always encloses in complete end-to-end loops like rubber bands that never cross over each other. Magnetic fields have no beginning or end point and always enclose upon themselves. Therefore, the concept of a diverging B field makes no sense and we call it zero, (meaning it doesn’t exist).  This is the second of Maxwell’s equations below which again tells us that nature permits no magnetic monopoles.

Mathematically, divergence is expressed by the del operator ∇, as in ∇ • E. This operator applies differentiation across each axis: x, y & z.

When electric charge moves, for example in a copper wire, we get a current. In fact, current is defined as the time-rate-of-change of charge, and the density of current is referred to as J.  When charge moves, or piles up like on the plates of a capacitor, there is an induced rotation in the magnetic field. Field rotations (electric or magnetic) can be thought of as similar to water draining (circulating) out of a bathtub. However, the rotating field doesn’t “drain” away as water would when you pull the plug in the tub.

Mathematically, we use the same ∇ operator but in a cross-product as in ∇ x B or ∇ x E to describe a rotating field.  These rotations are also referred to as the “curl” of a field.  So, ∇ x B is called the “curl of the magnetic field”.