What is the Relationship Between EM Wave Velocity and Electric Field in Space?

[kasse]

Homework Statement

Find the velocity of EM waves as a function of [tex]\epsilon_{0}[/tex] and [tex]\mu_{0}[/tex]

2. The attempt at a solution

[tex]E = E_{0}cos(kx-\omega t)[/tex]

Using [tex]v= \frac{\omega}{k}[/tex]

 

[gabbagabbahey]

When you used Maxwell's equations to derive the wave equation, you should have ended up with an answer to this

 

[kasse]

No, I didn't. But I can substitute my expression for E into the wave equation. What is [tex]\vec{\nabla}^{2}E[/tex]?

[tex]\frac{\partial^{2}E}{\partial x^{2}} + \frac{\partial^{2}E}{\partial t^{2}}[/tex]?

 

[gabbagabbahey]

No, I didn't.

Last time I checked, Maxwell's equations were in terms of [itex]\epsilon_0[/tex] and [itex]\mu_0[/itex] not [itex]c[/itex]; so you should have ended up with a wave equation where the propagation speed is in terms of [itex]\epsilon_0[/tex] and [itex]\mu_0[/itex]...if you didn't, then you did something wrong...I think you should go back to that problem and show me your work.

 

[kasse]

double

 

[kasse]

Of course...

So [tex]\vec{\nabla}^{2}E[/tex] = [tex]\frac{\partial^{2}E}{\partial x^{2}} + \frac{\partial^{2}E}{\partial y^{2}} + \frac{\partial^{2}E}{\partial z^{2}}[/tex] (only spatial dimension, not time)?

 

[gabbagabbahey]

Did you even read my last post?

 

[kasse]

Yes.[tex]\frac{1}{v^{2}} = \mu_{0}\epsilon_{0}[/tex], so [tex]\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}} = v[/tex]. That's what you meant, right?

That would mean that (if I substitute my expression for E into the wave equation) [tex]\vec{\nabla}^{2}E = \frac{\partial^{2}E}{\partial x^{2}} + \frac{\partial^{2}E}{\partial y^{2}} + \frac{\partial^{2}E}{\partial z^{2}}[/tex].

Can I also write [tex]\vec{\nabla}^{2}E = \frac{\partial^{2}E}{\partial \vec{r}^{2}}[/tex]?

 

[gabbagabbahey]

Yes.[tex]\frac{1}{v^{2}} = \mu_{0}\epsilon_{0}[/tex], so [tex]\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}} = v[/tex]. That's what you meant, right?

Yes.

That would mean that (if I substitute my expression for E into the wave equation) [tex]\vec{\nabla}^{2}E = \frac{\partial^{2}E}{\partial x^{2}} + \frac{\partial^{2}E}{\partial y^{2}} + \frac{\partial^{2}E}{\partial z^{2}}[/tex].

Can I also write [tex]\vec{\nabla}^{2}E = \frac{\partial^{2}E}{\partial \vec{r}^{2}}[/tex]?

First, the electric field is vector, not a scalar so this relation is incorrect...second what does this have to do with finding v...or anything else for that matter?