# Maxwell and his ‘c’

If you ever type “Maxwell Equations” in ‘Google Images’ you get many images, like the one mentioned below. Have you ever thought why it is been shown like this.  What is so important about these equations.  If you want to know, read this article

If you ask me about these equations, I will say these are the greatest equations in the history of physics.  These equations may not look beautiful to a common man.  But truly, it looks beautiful for a physicist.  I am writing this article not to tell you the mathematics or the physics behind these equations.  But to tell you the beauty of nature, which is hidden in these equations!

Let me first tell you this,

• E= Electric Field
• B= Magnetic Field
•  ρ = Charge Density
•  ɛ = Permittivity of free space (Property of medium)
•  μ = Permeability of free space (Property of medium)
• J = Current Density
• Inverted triangle symbol is called ‘del’ It is just a differential operator

(read ɛ and μ as “epsilon not” and “mu not”)

That’s all about these equations.

Before Maxwell, Electricity and Magnetism are two separate theories in physics.  They were independent of each other.  But this person called “Maxwell” unified these two theories as a single theory, called “Electromagnetic Theory”.  He said, a moving charge produces a magnetic field, and a change in magnetic field produces current in a wire!  True, this looks obvious to us now.  But during Maxwell time (year 1865) this is a remarkable achievement in the field of Physics.  Later it said that, this is the second Greatest Unification Theory in the History of Physics since “Newton’s Universal Law of Gravitation”.  Einstein once remarked on Maxwell theory that “Most profound, most truthful that physics has experienced since the time of Newton”.  Believe it or not all his work on “Electromagnetic Theory” stands on the above four equations!  If you exploit each equation, hidden natures beauty will unfolds to you.  If you ask me what these equations means, in a crude language I will say, disturbance of Electric and Magnetic fields travel in space (medium/vacuum) as Electromagnetic Waves obeying above equations.

As I mentioned in the first paragraph, I want to show one such hidden beauty from these equations.  Before that let me introduce you to a “Wave Equations”.  It is just a mathematical equation which governs the propagation of wave in space.  And the equation is… Here, Ѱ is a wave, V= Velocity of the wave.  Don’t worry about the differentiation; just see the form of equation.  We will come back to this equation later on.

Now, What Maxwell did is, he took two of his last equation, that is… And his aim was to write down the final equation, which “looks like” wave equation, the one I showed you in the above paragraph.  He did some mathematical manipulation to his equation, and finally arrived what he wanted.  His final equation was this… One for Electric Field, another for Magnetic Field.

Here, if you compare these two equations to the wave equations.  You can identify E and B are Electric and Magnetic Wave respectively.  But notice that, these two equations don’t have the v term in it.  Instead, it has some other terms.  If you equate these two terms, you get…  So, you (he) finally got the velocity in which Electric and Magnetic Wave travel in space.  Alright, until now everything was fair enough.  But the beauty lies after this.  Let me tell you, we have… I said, in the first paragraph that  ɛ and  μ are the properties of medium in which wave is travelling.  And it’s a constant number for a given medium.

For instance, if you consider medium as vacuum. (Ignore the units)

If you substitute and calculate the velocity you get… This happens to be precisely the velocity of “Light”!

Before Maxwell people had did experiment and found velocity of light. But Maxwell did not even think of light, he took electricity and magnetic theory, combined these two theories to get “Electromagnetic Theory”.  And said the disturbance of electric and magnetic field will travel as “Electromagnetic Wave”.  And now he has calculated the velocity of “Electromagnetic Wave”, it turns out that it will travel at a velocity, precisely equal to the velocity of light.  This means, “Light is also an Electromagnetic Wave”!

Up to Maxwell time, Mankind did not know, what constitute the light, which was the most abounded form of the energy right through human civilization.  Maxwell unfolded this greatest mystery of nature by saying “Light is a wave made up of electric and magnetic field and they are called as Electromagnetic Wave”.  Truly a remarkable achievement by Maxwell!

It doesn’t stop here, there is another beautiful thing which is hidden in this formula.  Let me again tell you.  Now we can safely write, Velocity of light is… Again, as I said ɛ and μ are constant for a given medium, and they are the properties of medium itself.

Isn’t it strikes you that just a properties of a medium tells you the velocity of light, which is the ultimate speed in our universe!  And, what is also interesting is, experiment involving measurement of  ɛ and μ has nothing to do with ‘c’.  But yet their combination gives rise to ‘c’!  Velocity of light is controlled by the properties of medium!  Amazing result.  The same kind of argument can be seen in connection with velocity of sound. That is, you can write velocity of sound in terms of bulk modulus (property of medium) of the medium in which sound is travelling.  But in light, the bulk modulus turned out to be ɛ and μ of the medium.  Nature works same everywhere, and Maxwell showed us the beauty of nature through his equations.  I don’t have many words to say about these things, but this is truly amazing!  Let me remind you, Maxwell did not studied light, he just wrote down his equations on Electric and Magnetic field.  But all these things are just hidden in his equations; he himself did not know about these things.  But his results showed everything, about the hidden quantities.  He himself was surprised!  I have just showed you one example.  You can find many of them, if you exploit the equations.  Truly they are beautiful equations of Nature.

Now, I hope you got to know why people write Maxwell equations like the one which I showed you at the beginning.  Light is all about Maxwell’s equations.  I personally believe, existence of God is zero. But, if he is there, and if he controls the nature, then surely he must have said this!  Because that is how nature is working.  And this genius Maxwell unfolded the God’s secret to Mankind! James Clerk Maxwell (13 June 1831- 05 Nov 1879) was a Scottish man, and one of the greatest physicist during 19th century.  Apart from his “Electromagnetic Theory” he also worked on “Maxwell-Boltzmann Distribution” which is statistical way of describing “Kinetic Theory of Gases”.

Right: Maxwell Statue at Edinburgh. (Image from wikipedia)

-Viswa Keerthy S (29-06-2013)