Determining Electric and Magnetic field given certain conditions

[guyvsdcsniper]

Homework Statement

Refer to attached image

Relevant Equations

comlplex wave equation,

I am unsure of my solutions and am looking for some guidance. a.)The real part of the wave in complex notation can be written as $\widetilde{A} = A^{i\delta}$. Writing the Complex Wave equation, we have $\vec E(t) = \widetilde{A}e^{(-kz-\Omega t)} \hat x$. Therefore the real part is $\vec E(t) =Ae^{(-kz-\Omega t+\delta)} \hat x$. The negative in front of kz indicates it is a left traveling wave.

b.) The unit vector of $\hat B = \frac{(\hat x - 2\hat z)}{\sqrt{5}}$. I know that $\hat E$ must be perpendicular to $\hat B$, so simply,
$\hat E = \frac{(\hat x + 2\hat z)}{\sqrt{5}}$

c.) I am not so sure about this problem. I know that $\vec E = \widetilde{E}_oe^{i(ky-wt)}\hat x$
Griffiths states $\widetilde{B}_o = \widetilde{E}_o/c$ and $v=c/n$.

So $\vec B = \frac{c\widetilde{E}_o}{1.7}e^{i(ky-wt)}\hat z$

 

[vela]

a.)The real part of the wave in complex notation can be written as $\widetilde{A} = A^{i\delta}$. Writing the Complex Wave equation, we have $\vec E(t) = \widetilde{A}e^{(-kz-\Omega t)} \hat x$. Therefore the real part is $\vec E(t) =Ae^{(-kz-\Omega t+\delta)} \hat x$. The negative in front of kz indicates it is a left traveling wave.

The problem statement said the wave travels in the $-x$ direction. Your answer isn't sinusoidal. It decays with time. What is $\delta$?

b.) The unit vector of $\hat B = \frac{(\hat x - 2\hat z)}{\sqrt{5}}$. I know that $\hat E$ must be perpendicular to $\hat B$, so simply,
$\hat E = \frac{(\hat x + 2\hat z)}{\sqrt{5}}$

It doesn't look like $\hat E \cdot \hat B=0$.

 

[guyvsdcsniper]

The problem statement said the wave travels in the $-x$ direction. Your answer isn't sinusoidal. It decays with time. What is $\delta$?

You're right. I should have $\vec E(t) =Acos{(-kz-\Omega t+\delta)} \hat x$
$\delta$ is the phase
So I need to make $\hat x$ be $\hat - x$ as well as account for it is a wave traveling left by the $-kz$ in the $cos$?

I assumed the $-kz$ in the $cos$ accounted for the negative direction.

 

[vela]

I assumed the $-kz$ in the $cos$ accounted for the negative direction.

In the $-z$ direction, not the $-x$ direction.

 

[vanhees71]

Hint: The wave travels in the negative $x$ direction and the $\vec{B}$-field (sic!) is polarized in $z$-direction. So the complex ansatz for $\vec{B}$ is (using the HEP physicists' convention concerning the signs in the exponential)
$$\vec{B}=A \vec{e}_z \exp(-\mathrm{i} \Omega t-\mathrm{i} k x), \quad \Omega,k>0.$$
Now just use the source-free Maxwell equations to get the dispersion relation $\Omega=\Omega(k)$ and the $\vec{E}$-field!

 

[guyvsdcsniper]

I figured it out. I had a big misunderstanding on the equations I was using but took the time to read through my book and was able to come to the correct answer. Thanks all for the help!