What are Maxwell's Equations?

Maxwell's Equations (Home)

I've discussed the equations of Maxwell individually up until now. On this page I'd like to give the overview of them again - with no math - and explain what they mean as a whole.

1. Gauss' Law

Gauss' Law is equivalent to the Force Equation for Electric Charges: like charges repel each other and opposite charges (i.e. positive and negative charge) attract.

Gauss' Law also says that Electric Field lines diverge away from Electric Charges. This means that positive charge acts as a source of Electric Fields (like the way a faucet is a source of water). Gauss' Law means that negative charges acts as a sink for Electric Fields (the way water drains or exits a region via a sink hole). This means Electric Field lines start and stop on Electric Charge.

2. Gauss' Law for Magnetism

Maxwell's Second Equation says that magnetic monopoles do not exist. While we have Electric Charges (Electric Monopoles), we have never found the magnetic equivalent - magnetic charge or a magnetic monopoles. This equation states that the magnetic field tends to wrap around things - since the divergence is zero the fields tend to form closed loops.

3. Faraday's Law

Faraday's Law tells us that a magnetic field that is changing in time will give rise to a circulating E-field. This means we have two ways of generating E-fields - from Electric Charges (or flowing electric charge, current) or from a magnetic field that is changing.

4. Ampere's Law

Ampere's Law tells us that a flowing electric current gives rise to a magnetic field that circles the wire. In addition to this, it also says that an Electric Field that is changing in time gives rise to a magnetic field that encircles the E-field - this is the Displacement Current term that Maxwell himself introduced.

This means there are 2 ways to generate a solenoidal (circulating) H-field - a flowing electric current or a changing Electric Field. Both give rise to the same phenomenon.

Maxwell's Equations as a Whole

As a whole, what do Maxwell's Equations mean?

The first two are mainly used at D.C. - i.e. when all the voltages and currents are constant and nothing is changing with time. These are somewhat less important in my opinion, and in fact, they can be derived from the second set of equations. Therefore, these are somewhat of a weaker copy of the second two.

The 3rd and 4th equations really dictate the rules for Electric and Magnetic Fields. For instance, in a popular Electromagnetics Solver known as the Finite-Difference Time Domain (FDTD) method, only the second two equations are used to numerically solve for the E- and H- fields. The first two are not needed.

Let's look at the 3rd and 4th equations, two succinct statements that govern all of E- and H- field propagation:

the main maxwell's equations

Farday's Law says that a changing Magnetic Field gives rise to a circling Electric Field. Now, things in the universe do not continually grow or continually shrink - they oscillate (move up and down about an average). This means the Magnetic Field increases and then decreases - which means the Electric Field wraps back and forth around a changing Magnetic Field. And this means that the Electric Field is changing in time as well.

Look now at Ampere's Law. This says that a changing Electric Field gives rise to a circling Magnetic Field. And in the same way, the Electric Field will be oscillating in time, the encircling Magnetic Field will be wrapping around it and changing in time as well.

Let's think about the last two paragraphs. A changing magnetic field gives rise to a changing Electric Field. And a changing Electric Field gives rise to a changing Magnetic Field - which itself will produce a changing Electric Field which will give rise to .....

What is this? This phenomenon - is known as propagation. This is what gives rise to the propagation of Electromagnetic Waves. This is the eternal wheel of motion that enables sunlight to travel through a vaccuum without any medium. And from these two equations we can determine that all propagating waves travel at a single speed, the speed of light. And this speed can be directly determined from the constants found within Maxwell's Equations.

Every electromagnetic form or radiation - visible light, x-rays, sunlight that heats the earth, radio waves, television waves, wifi signals, bluetooth signals, cell phone transmission, and GPS all consist solely of Electric and Magnetic Fields. And everything you need to know about how they propagate and interact with materials is completely determined by the above 2 equations - which were unified and explained originally by James Clerk Maxwell in the 1860s. And that is that.

Maxwell's Equations

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