Question

Derivation of speed of light by maxwell equation

Answer 1

Maxwell's equations describe electricity, magnetism, space, time and the relationships among them. They are simple and fundamental. As we saw in the introductory film clip, their simplicity, symmetry and beauty persuaded Einsten to develop a theory of relativity in which Maxwell's equations were invariant. Here we explain why, and derive an expression for the speed of light along the way.

Maxwell's equations: integral form,Maxwell's equations: differential formMaxwell's equations and the speed of lightMaxwell's equations: symmetric form

There are also properties of the medium, even if it is vacuum. ε is the dielectric permittivity. It is a property of a medium that determines the strength of the electric field produced by a given electric charge and geometry. Greater values of ε mean that more charge is required to produce the same electric field (for all materials in the liquid or solid phase, ε is greater than the vacuum value). μ is the magnetic permeability of a medium. Greater values of μ mean that a given value of electric current produces a stronger magnetic field (magnetic materials such as iron may have values thousands of times higher than the vacuum value).

For our purposes (dealing with light) we only need Maxwell's equations for the case of the vacuum, where ε and μ take their (uniform and constant) vacuum values of ε0 and μ0.

Maxwell's equations: integral form

Starting on the top left, Faraday's law is what makes electricity generation possible: relative motion of a coil and a magnet produces a voltage. Gauss' law is another way of writing Coulomb's law of electrostatic interaction. The other two laws together give the law of Biot-Savart. We have just seen the Coulomb and Biot-Savart laws in The electric and magnetic forces between moving charges. You can see that the Ampere-Maxwell law is something like a magnetic analogue of Faraday's law. As we shall see, it is also important in electromagnetic radiation: a magnetic field may arise not only from a current, but also from a varying electric field.

The symbol ∇ (pronounced 'del') expresses how strongly a quantity varies in space - it is a sort of spatial derivative. Loosely speaking, if you move in one direction, ∇.E(pronounced 'div E') expresses how much E varies spreads out or how much it changes in that direction. ∇xE (pronounced 'curl E') measures how much E curls around, or how much it changes in the perpendicular directions. This allows us to write:

Maxwell's equations: differential form

It is sometimes said that the beauty of these equations marks them as one of the great intellectual achievements of our species. You won't see them framed in a museum, or performed at a concert hall‚ but they do occasionally appear printed on the front of tee shirts worn by physics students. And in any case, they don't need a conventional exhibition space, for they are all around you in nearly all phenomena, and they are among the tools with which much of our technology has been created.

Below we write them in a more symmetric and beautiful form: a form that shows more clearly their various symmetries. However, for the moment, let's put them to use.

Maxwell's equations and the speed of light

Let's consider light travelling in a vacuum, ie in a region in which there are no electric charges, so that ρ = 0 and J = 0. Again for the vacuum, Maxwell's equations become:

To get a wave equation, we want second derivatives in both time and space. First, we take the curl of both sides of (1). (Informally, this is like differentiating in space, in the sense described above.) Then substitute in (3), and we have

A theorem in vector calculus is that, for any vector a, . Here, ∇.E = 0, so

This is the wave equation in three dimensions. In one dimension, we could write it

Let's look for a solution that is a sinusoidal wave, with speed v and wavelength λ. One such wave has the expression:

Differentiating, we get

    and    Substituting this back into the wave equation, we see that it is a solution, provided that

in other words, the equation has a wave solution and the speed of the waves is . When Hertz found this solution, he recognised that the value of this quantity was the speed already known for light, which strongly suggested that light was a wave, and part of the electromagnetic spectrum, along with radio. If you're interested, we show film clips of measurements of the speed of light and of (UHF) radio waves in a multimedia tutorial on The Nature of Light.

Maxwell's equations: symmetric form(ε0 = 1 and μ0 = 1)