What does Maxwell's equations mean?

In summary, Maxwell's equations describe electricity and magnetism, and they predict the speed of light is constant.
  • #1
zafari
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I need some help figuring out what these formulas mean and what they relate to. All I know is the 'upside-down triangle' symbol is known as a "Del", and it's used in vector calculus.

Before the list of equations, it says "And God said...". After the list of equations, it says "...And there was light."

I'm guessing they might spell something in Latin which translates to "Let there be light", aka, "fiat lux".

Any help is appreciated!
 

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  • #2
These formulas are collectively called Maxwell's equations, and they describe electricity and magnetism. One solution to these equations describes the propagation of light. Hence, if God had "said" Maxwell's equations, then light would exist as a consequence of those equations.
 
  • #3
Ah, that makes sense. Can you describe what the variables mean? I am familiar with rho as density, and it seems like the -aB/at and aD/at seem to be derivatives taken with respect to t (time?), but that's about it.
 
  • #4
Yeah, it's a little hard to read the symbols; let me write out the equations here.

[tex]\nabla \cdot D = \rho \\
\nabla \cdot B = 0 \\
\nabla \times E = -\frac{\partial B}{\partial t} \\
\nabla \times H = i + \frac{\partial D}{\partial t}[/tex]

The symbol [itex]\partial[/itex] denotes partial differentiation--a derivative of a function of multiple variables that holds all others constant.

[itex]t[/itex] is time. [itex]\rho[/itex] here is the free charge density. We say "free" because charge can come from two sources; in particular, this charge density excludes charge that came from polarization of matter (where something that was neutrally charged gains positive charge in one place and negative in the other). Charge density that arises in this way is typically called "bound" charge.

[itex]i[/itex] is the free current density, and there can be bound current density in the same way. Together, [itex]\rho[/itex] and [itex]i[/itex] are the "sources" of electromagnetic fields, [itex]E, D, B, H[/itex]. [itex]E,D[/itex] have to do with electric fields. [itex]E[/itex] is usually called the electric field, while [itex]D[/itex] is sometimes called the "electric displacement field". Usually, [itex]B[/itex] is called the magnetic field nowadays, whereas [itex]H[/itex] has a few different names, but it's often just referred to by its symbol as "the H-field".

[itex]E,B,D,H[/itex] are vector fields, and the [itex]\nabla \cdot[/itex] and [itex]\nabla \times[/itex] describe how the derivatives of these fields must relate to the sources or to other time derivatives of fields.

This image appears to be in a specific set of units; often, you might see Maxwell's equations in SI units instead, and you'll see constant factors of [itex]\epsilon_0,\mu_0[/itex] in places. These just make the numbers work. Finally, you might see Maxwell's equations in vacuum, which simplify things considerably: without matter that can be polarized, D and H basically reduce to E and B respectively. There are also integral forms of Maxwell's equations, which are generated through a nifty piece of mathematics called Stokes' theorem, but they say the same thing.

That's the 30-second explanation of Maxwell's equations in a nutshell; wikipedia would do a far better job of explaining them than I ever could. The classical theory of electromagnetism is one of the most fascinating aspects of physics, and there's quite a bit of history to it that has helped shape our understanding not only of light but of space and time. Maxwell's equations predict that the speed of light is constant regardless of one's speed, and that helped motivate the theory of relativity. It is a robust, elegant theory and worthy of many hours of study, in my opinion.
 
  • #5
Whoa, that actually makes sense. It definitely seems like a very complex concept, which makes it all the more interesting. Thanks for the explanations, I would be very lost without them.
 

1. What are Maxwell's equations?

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields. They were developed by James Clerk Maxwell in the 19th century and are considered one of the cornerstones of classical electromagnetism.

2. What do Maxwell's equations mean?

Maxwell's equations describe the relationship between electric and magnetic fields, and how they interact with each other and with charged particles. They also explain how these fields can be generated and propagated through space.

3. Why are Maxwell's equations important?

Maxwell's equations are important because they provide a fundamental understanding of electromagnetism, which is essential in many fields such as physics, engineering, and technology. They have also been used to make groundbreaking discoveries, such as the prediction and confirmation of the existence of electromagnetic waves.

4. How are Maxwell's equations used in everyday life?

Maxwell's equations have many practical applications in everyday life. They are used in the design of electrical and electronic devices, such as computers, cell phones, and motors. They also play a crucial role in technologies like radio and television broadcasting, wireless communication, and medical imaging.

5. Are Maxwell's equations still relevant?

Yes, Maxwell's equations are still relevant today and continue to be a cornerstone of modern physics. They are used in many fields of research, such as quantum mechanics and general relativity, and have been confirmed to be accurate in a wide range of experiments and observations.

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