Parametric representation of a vector electric field

[jjand]

Homework Statement

See Attachment

Homework Equations

None I can think of

The Attempt at a Solution

I'm fairly certain that phi_yx is zero
Also I tried factoring out the cos and splitting up the equation into it's respective components, but to no avail. Am I even going about this correctly?

 

[vela]

The last bit is a typo. It should be $\phi_{xy} = \phi_x$.

I'm not sure what you mean by "splitting up the equation into its respective components" since the equation you're supposed to show is true is a scalar equation.

 

[jjand]

Thanks for the reply Vela! Actually I think I figured it out. Maybe you or someone else can check my work? I assumed that it the phi's were not typos, but they could very well be. I went ahead and attached my work.

 

[vela]

Your expressions for E_x and E_y are slightly wrong. For example,
$$E_x = A_x\cos(kz-\omega t+\phi_x) = A_x\cos(-r+\phi_x) = A_x\cos(r-\phi_x)$$since cosine is an even function. You have an extra negative signs floating around.

You must have made a mistake somewhere in the middle because the cross term has the wrong sign. The sign doesn't flip the way you did it because, again, $\cos (-\theta)=\cos\theta$.

You actually made it more complicated by keeping the E's and A's around. You know that
\begin{align*}
\frac{E_x}{A_x} &= \cos(\phi_x-r) \\
\frac{E_y}{A_y} &= \cos(r) \\
\end{align*}so you could have just written the lefthand side in terms of cosines and then shown it simplifies down to the righthand side.

 

[jjand]

Ah that I forgot about that cosine property and I found a convenient sign mistake half way through the problem. Thanks Vela!