Satisfying Maxwell's Equations

[MyName]

Homework Statement

Show whether or not the following functions satisfies Maxwell's Equations in free space. (That is, show whether or not they represent a valid electromagnetic wave).

$$E(x,y,t)=(0,0,E_0 sin(kx-ky+\omega t))$$
$$B(x,y,t)=B_0 (sin(kx-ky+\omega t),sin(kx-ky+\omega t),0)$$

Homework Equations

$$\nabla \times E=-\frac{\partial B}{\partial t}$$
$$\nabla \times B=\mu_0 \epsilon_0\frac{\partial E}{\partial t}$$
$$\nabla \cdot E=0$$
$$\nabla \cdot B=0$$

The Attempt at a Solution

Starting with the easy two, It is clear that these functions have zero divergence.

Then Ampere's Law gives us that $2B_0k=\mu_0 \epsilon_0 \omega E_0$
And Faraday tells us that $kE_0=\omega B_0$

We then elliminate $\frac{E_0}{B_0}$ from these two equations and note that $\frac{1}{\mu_0 \epsilon_0}=c^2$ to find that $\omega=\pm kc\sqrt{2}$

Up to this point i am fine. The field clearly satisfies maxwells equations aslong as these conditions are met. My problem is that it seems this implies the field pattern travels at $\frac{\omega}{k}=\pm c\sqrt{2}>c$ (the group velocity of the wave, correct?). This obviously can't be right. I do not know if I'm just interpreting the result wrong, or if somehow this field is invalid for some reason despite satisfying maxwells equations. I've been scouring my textbook and the internet for the last few hours and can't seem to find anything explaining my issue, and would really appreciate some help, thanks.

 

[Jonathan Scott]

Is there anything in the question that states that $k$ is the wavenumber in this case or are you just assuming it from the choice of letter?

 

[Jonathan Scott]

To put it another way, if the relationship between $k$ and $\omega$ is not defined in the question, then it is possible that $k$ is not actually the wave number, but rather some multiple of it, and that might help.

 

[MyName]

No, now i come to think of it there isn't anything stating that k is the wavenumber or that $\omega$ is the angular frequency. If i remember correctly though, i was taught that the coefficients of the spatial variables are always the components of k, and that the time coefficient is always the angular frequency, but this isn't necessarily the case?
If this is the case how would I go about figuring out the speed of this wave? (I've heard that Maxwell's Equations predict that any electromagnetic wave travels at c, so I assume this should be possible to do)

By the way, I also tried substituting the $E$ and $B$ fields into the wave equation itself;
$$(\nabla^2-\frac{1}{c^2}\frac{\partial}{\partial t})E=0$$
$$(\nabla^2-\frac{1}{c^2}\frac{\partial}{\partial t})B=0$$
And obtained the same results.

Also, is there any way two functions, say $E$ and $B$, can satisfy Maxwell's Equations without representing a valid electromagnetic wave (and is this perhaps the case?).
Thanks for your help.

 

[Jonathan Scott]

Sorry I don't have the time and patience to check everything out here, but it seems to me that in this case, the spatial term is $(x, y, z).(k, -k, 0)$ so the magnitude of the wavenumber seems to be $\sqrt{k^2+k^2+0} = k \sqrt{2}$, so this could be a valid electromagnetic wave if $\omega/(k\sqrt{2}) = c$.

 

[MyName]

Thats fair enough, Ireally appreciate your time and effort. That makes sense actually. I guess what I said is only true for waves with one spatial compnent! It definitely makes a bit more sense to me now.