I do not even really know where to begin with this problem. Any help would be great. Q. Write an expression for a P-State (linearly polarized) lightwave of angular frequency [tex]\omega[/tex] and amplitude [tex]E_{0}[/tex] propogating along a line in the xy-plane at 45 degress to the x-axis and having its plane of vibration corresponding to the xy-plane. At t=0, y=0 and x=0 the field is zero. Like I said, I don't even know where to start. This prof is miserable, and the book is light on examples and explanations. I am figuring the equation will be of the form [tex] \vec{E}=(\tilde{i}E_{0x}+\tilde{j}E_{0y})cos(kz-\omega t) [/tex] This would be a wave traveling along the z-axis, so I expect that I have to change this term with a vector specifying the path 45 degrees from the x-axis. But how to do this? The prof did give a hint that we need to perform an operation of taking the dot product of two vectors, say k dot r. He was very vague, in fact down right confusing after that. Thanks in advance.
No takers on this question? I know I have shown no work, but I am completely stuck. Can anyone gander atleast a starting point?
A plane wave with wavenumber k travelling in an arbitrary direction given by the unit vector [itex]\hat k[/itex] can be represented as: [tex]A\cos(\vec k \cdot \vec r - \omega t - \varphi_0)[/tex] where [itex]|\vec k|=k[/itex]. Try to see why this is true. Draw a diagram or so. I think you'll learn the most by understanding this general case. Solving your problem is then easy.