- #1

roam

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## Homework Statement

560 nm light is collimated and passes as a parallel beam in a direction perpendicular to the ##x+y+z=0## plane. It is polarized parallel to the ##(y-z)## plane. Treating it as a plane wave, what are the real E and B fields?

Intensity of the beam is ##1 \ mW/cm^2##. Make your expressions for the fields as numerical as possible.

## Homework Equations

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

##B(r,t) = \frac{1}{c} E_0 \ cos (k.r - \omega t) (\hat{k} \times \hat{n})##

Where ##k## is the propagation/wave vector, and ##\hat{n}## is the direction of popularization.

##k=\frac{\omega}{\lambda f}=\frac{\omega}{c}= \frac{2 \pi}{\lambda}##

Using the following relationship, I used the given intensity to work out the amplitude:

##I=\frac{1}{2} c \epsilon_0 E_0^2 \implies E_0 = \sqrt{\frac{2I}{c \epsilon_0}} = 7.685 \times 10^{-10} \ V/m##

## The Attempt at a Solution

So, to find ##k.r## and ##k \times \hat{n}##, I need the equation for a plane perpendicular to the ##x+y+z=0## plane. But how can I find this equation when there are

**infinitely**many such vectors?

For instance the wave vector is ##k= \omega/c \ \hat{k}##, where ##\hat{k}## is a unit vector in the direction of the vector perpendicular to the ##x+y+z=0## plane. I also need that in order to work out the cross product ##\hat{k} \times \hat{n}## for the B field.

As for the polarization ##\hat{n}## (parallel to y-z plane), would the unit vector be ##\hat{n} = \frac{\hat{y}+\hat{z}}{\sqrt{2}}##? Or does the plus sign need to be changed to a minus? Any explanation would be greatly appreciated.