Maxwell's Equation: Proving the Speed of Light

In summary, Maxwell proved the speed of light to be equal to 1/√με by using two equations from classical electrodynamics and transforming them into a wave equation. He also suggested that light was a form of electromagnetic radiation. His equations were later verified experimentally and it was shown that any wave, regardless of its shape, would satisfy the 3D wave equation with a propagation speed of 1/√με. However, it should be noted that Maxwell's original method of determining the speed of electromagnetic waves would have involved using the integral-differential forms of his equations instead of the vector operators introduced by Oliver Heaviside.
  • #1
Ebolamonk3y
180
0
How exactly did Maxwell prove speed of light is [tex]\frac{1}{\sqrt{\mu\epsilon}}[/tex]
 
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  • #2
Well, that is mathematically rather sophisticated. To prove it, the following two equations are needed (2 of the 4 famous equations that explain classical electrodynamics completely:)

[tex]\oint\vec{B}d\vec{s}=\mu_0 I+\mu_0\varepsilon_0\frac{d}{dt}\oint E_n dA[/tex]

[tex]\oint\vec{E}d\vec{s}=-\frac{d}{dt}\int B_n dA[/tex]

Now, there are two possibilties to get the speed of light:

1) The equations can be transformed into a wave equation (difficult) and you can get the speed of the wave.

2) A special and simple case can be constructed where it is rather easy to get the speed. Though, this way isn't really sufficient.
 
  • #3
The speed of light was already closely approximated at the time (I believe). What Maxwell really did is to suggest that light was a form of electromagnetic radiation through the close numerical agreement of c ~ 1/√με.
 
  • #4
Hehe. I like the first treatment better... Its more "convincing" as of now. :)
 
  • #5
In a vacuum, Maxwell's Equation (differential form) reduce to

[Tex] \nabla * E = 0 [/Tex]

[Tex]
\nabla * B = 0[/Tex]

[Tex]
\nabla \times B = \epsilon_0 \mu_0 \frac{\delta E} {\delta t}
[/Tex]

[Tex]
\nabla \times E = -\frac{\delta B}{\delta t}
[/tex]

These are obtainable from the integral forms Sitewinder mentioned, which themselves can be experimentally verified with batteries and wire loops and compasses and such.

Take the curl of both sides of the bottom two equations. For example, the last one is
[Tex]

\nabla \times \nabla \times E = -\frac{\delta \nabla \times B}{\delta t}
[/Tex]

Use a vector calculus identity (think of it as a special case of the chain rule, if you want):
[Tex]
\nabla \times \nabla \times A = \nabla(\nabla * A) - \nabla^2 A
[/Tex]
(I fudged the first term; I know that it's zero in a vacuum because div E is 0.)

Substitute in -mu0 epsilon0 dE/dt for curl of B, and you've got
[tex]
\nabla^2 E = \mu_0 * \epsilon_0 * -\frac{\delta^2 E}{ \delta t^2}
[/tex]
This is the 3D version of the wave equation; you can show that any wave (spherical, cylindrical, plane) satisfies this equation, with a propagation speed of
[tex] \frac{1}{(\mu_0 \epsilon_0 )^(0.5)}[/tex]

P
 
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  • #6
Ebolamonk3y,
Just a short note: Rocketcity showed a valid method of determining the speed of an electromagnetic waves equations using the vector operators.However, vector operators were introduced by Oliver Heaviside several years later after Maxwell published his original findings. So, the method that Rocket city posted is not the "exact" method that James Maxwell would have used to prove what the speed of electromagnetic waves was. He would have started usinfg the integral-differential forms of his equations instead.
Regards,
Roland
 
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1. How did James Clerk Maxwell prove the speed of light?

James Clerk Maxwell proved the speed of light by using his famous set of equations, now known as Maxwell's Equations. These equations describe the behavior of electric and magnetic fields and their relationship to each other. By solving these equations, Maxwell was able to calculate the speed of light to be approximately 299,792,458 meters per second.

2. What is the significance of Maxwell's Equations in physics?

Maxwell's Equations are considered one of the most important discoveries in the field of physics. They unified the theories of electricity and magnetism and laid the foundation for modern electromagnetic theory. These equations have been crucial in understanding the behavior of light, electricity, and magnetism, and have led to the development of many technologies we use today.

3. How do Maxwell's Equations relate to the speed of light?

One of the four equations in Maxwell's set is known as the wave equation, which describes the propagation of electromagnetic waves. By solving this equation, Maxwell was able to calculate the speed of light, which is a constant in free space and is known as the speed of electromagnetic waves.

4. Can Maxwell's Equations be used to prove the speed of light in other mediums?

Yes, Maxwell's Equations can be used to calculate the speed of light in any medium. The speed of light in a medium is determined by the properties of that medium, such as its refractive index. By using Maxwell's Equations and taking into account the properties of the medium, the speed of light in that medium can be calculated.

5. How has the discovery of Maxwell's Equations impacted our understanding of the universe?

The discovery of Maxwell's Equations has had a significant impact on our understanding of the universe. It has helped us understand the fundamental principles of electricity and magnetism, and their relationship to each other. These equations have also led to the development of many technologies, such as radio, television, and wireless communication, which have greatly influenced our daily lives. Furthermore, Maxwell's Equations have also played a crucial role in the development of Einstein's theory of relativity, which has revolutionized our understanding of space and time.

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