- #1

roam

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## Homework Statement

A monochromatic, electromagnetic plane wave is traveling in a non-conducting, transparent medium. Its real electric field is described by:

##E=100 \ cos (7.62 \times 10^6 (x+y+z) \ - \ 2.98 \times 10^{15} t) \ \frac{1}{\sqrt{5}} (\hat{y} + 2 \hat{z}) \ V/m##

**(a)**What would be the vacuum wavelength of the wave?

**(b)**What is the refractive index of the medium?

**(c)**In what direction is the wave travelling?

## Homework Equations

##\lambda = 2\pi/k, \ \ v=\omega/k, \ \ n=ck/\omega##

Real electric field general expression:

##E(r,t)=E_0 \ cos(k.r-\omega t) \hat{n}##

Where ##\hat{n}## represents the polarization.

## The Attempt at a Solution

**(a)**I have tried to do this part using the two different equations: ##\lambda = 2\pi/k## and ##\lambda=c/\nu##. But each time I get a different answer:

##\lambda = \frac{2 \pi}{k} = \frac{2 \pi}{7.62 \times 10^6}= 824.5 \ nm##

And since ##\nu = \omega / 2\pi## we have:

##\lambda = \frac{c}{\nu} = \frac{3 \times 10^8}{(2.98\times 10^{15})/2\pi} = 632.5 \ nm##

So which method is correct?

**(b)**For refractive index I got:

##n= \frac{ck}{\omega} = \frac{(3 \times 10^8)(7.62 \times 10^6)}{2.98 \times 10^{15}} = 0.76##

But this doesn't look right. Shouldn't n be greater than or equal to 1?

**(c)**Since the expression for electric field has the general form ##cos(kz-\omega t)##, I believe it is traveling to the right. But how do I explain the "x+y+z" part? What plane or line is the direction of travel parallel/perpendicular to?

Any explanation would be greatly appreciated.