- #1
yungman
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Given a wave:
[tex] e(t)=\hat x E_{x0}\cos \omega t+\hat y E_{y0}\cos( \omega t+ \delta)[/tex](1)
The book claimed:
[tex]\sin^2\delta\;=\; \left[\frac {e_x(t)}{E_{x0}}\right]^2-2\left[\frac {e_x(t)}{E_{x0}}\right]\left[\frac {e_y(t)}{E_{y0}}\right]\cos\delta+ \left[\frac {e_y(t)}{E_{y0}}\right]^2[/tex]
I can not get this. Below is my work.
From (1)
[tex]e_x(t)=E_{x0}\cos\omega t\;\Rightarrow \;\cos \omega t=\frac{e_x(t)}{E_{x0}}[/tex]
[tex]\Rightarrow\; \sin\omega t\;=\;\sqrt{1-\left(\frac{e_x(t)}{E_{x0}}\right)^2}[/tex]
[tex]e_y(t)=E_{y0}\cos(\omega t+\delta) \;\Rightarrow \;\frac{e_y(t)}{E_{y0}}\;=\;\cos\omega t \cos\delta-\sin\omega t sin\delta\;=\; \frac{e_x(t)}{E_{x0}}\cos\delta-\sqrt{1-\left(\frac{e_x(t)}{E_{x0}}\right)^2}sin\delta[/tex]
[tex]\Rightarrow\;\sqrt{1-\left(\frac{e_x(t)}{E_{x0}}\right)^2}sin\delta \;=\; \frac{e_x(t)}{E_{x0}}\cos\delta-\frac{e_y(t)}{E_{y0}}[/tex]
[tex]\Rightarrow\;\left(1-(\frac{e_x(t)}{E_{x0}})^2\right) sin^2\delta \;=\;\left[\frac{e_x(t)}{E_{x0}}\right]^2 \cos^2\delta-2\frac{e_x(t)}{E_{x0}}\frac{e_y(t)}{E_{y0}}\cos \delta + \left[\frac{e_y(t)}{E_{y0}}\right]^2[/tex]
It is obvious that I am not going to get what the book gave:
[tex]\sin^2\delta\;=\; \left[\frac {e_x(t)}{E_{x0}}\right]^2-2\left[\frac {e_x(t)}{E_{x0}}\right]\left[\frac {e_y(t)}{E_{y0}}\right]\cos\delta+ \left[\frac {e_y(t)}{E_{y0}}\right]^2[/tex]
Please help my derivation.
Thanks
[tex] e(t)=\hat x E_{x0}\cos \omega t+\hat y E_{y0}\cos( \omega t+ \delta)[/tex](1)
The book claimed:
[tex]\sin^2\delta\;=\; \left[\frac {e_x(t)}{E_{x0}}\right]^2-2\left[\frac {e_x(t)}{E_{x0}}\right]\left[\frac {e_y(t)}{E_{y0}}\right]\cos\delta+ \left[\frac {e_y(t)}{E_{y0}}\right]^2[/tex]
I can not get this. Below is my work.
From (1)
[tex]e_x(t)=E_{x0}\cos\omega t\;\Rightarrow \;\cos \omega t=\frac{e_x(t)}{E_{x0}}[/tex]
[tex]\Rightarrow\; \sin\omega t\;=\;\sqrt{1-\left(\frac{e_x(t)}{E_{x0}}\right)^2}[/tex]
[tex]e_y(t)=E_{y0}\cos(\omega t+\delta) \;\Rightarrow \;\frac{e_y(t)}{E_{y0}}\;=\;\cos\omega t \cos\delta-\sin\omega t sin\delta\;=\; \frac{e_x(t)}{E_{x0}}\cos\delta-\sqrt{1-\left(\frac{e_x(t)}{E_{x0}}\right)^2}sin\delta[/tex]
[tex]\Rightarrow\;\sqrt{1-\left(\frac{e_x(t)}{E_{x0}}\right)^2}sin\delta \;=\; \frac{e_x(t)}{E_{x0}}\cos\delta-\frac{e_y(t)}{E_{y0}}[/tex]
[tex]\Rightarrow\;\left(1-(\frac{e_x(t)}{E_{x0}})^2\right) sin^2\delta \;=\;\left[\frac{e_x(t)}{E_{x0}}\right]^2 \cos^2\delta-2\frac{e_x(t)}{E_{x0}}\frac{e_y(t)}{E_{y0}}\cos \delta + \left[\frac{e_y(t)}{E_{y0}}\right]^2[/tex]
It is obvious that I am not going to get what the book gave:
[tex]\sin^2\delta\;=\; \left[\frac {e_x(t)}{E_{x0}}\right]^2-2\left[\frac {e_x(t)}{E_{x0}}\right]\left[\frac {e_y(t)}{E_{y0}}\right]\cos\delta+ \left[\frac {e_y(t)}{E_{y0}}\right]^2[/tex]
Please help my derivation.
Thanks