Deriving the wave equation for light from Maxwell's equations

[Schr0d1ng3r]

Homework Statement

Using the third and fourth of Maxwell’s equation in integral form, show that a plane polarized electromagnetic waves propagates in accordance with the generalized wave equation. Determine the velocity of light in terms of the permeability and permittivity constants.

Homework Equations

Faraday's Law

Maxwell-Ampere Equation

 

[Schr0d1ng3r]

I'm pretty sure that they are supposed to form some sort of differential equation, but I'm lost as to where I'm supposed to start.

 

[berkeman]

Homework Statement

Using the third and fourth of Maxwell’s equation in integral form, show that a plane polarized electromagnetic waves propagates in accordance with the generalized wave equation. Determine the velocity of light in terms of the permeability and permittivity constants.

Homework Equations

Faraday's Law

Maxwell-Ampere Equation

It's been a while since I did this problem, but start with the form of the Wave equation. What is the general form of the Wave equation? And can you post the two Maxwell's equations that they want you to use? What variables are involved?

 

[Schr0d1ng3r]

In order to satisfy the general wave equation, I'm pretty sure that means that it must fit the form x=(c1)cos(wt) + (c2)sin(wt), but I'm not sure

Faraday's Law:

$$\oint$$ E dot $$\partial$$L = -$$\partial$$$$\Phi$$_B/$$\partial$$t

Maxwell-Ampere Equation

$$\oint$$ B dot $$\partial$$L = $$\mu$$$$\epsilon$$$$\partial$$$$\Phi$$_E/$$\partial$$t

Sorry if the eqns are hard to read, I'm not great at putting them into script

 

[Schr0d1ng3r]

Never mind, I got it. I just had to evaluate over a rectangle Ldx and then do some differential manipulations and substitute the equations into each other. The general equation of the wave, by the way, was (d^2y/dx^2) = (1/(v^2))*(d^2y/dt^2), and the speed of light was, 1/(mu*epsilon)^(1/2), as expected.