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Relativity - Light At An Angle

  • Thread starter csnsc14320
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  • #1
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Homework Statement


A light beam is emitted at an angle [tex]\theta_o[/tex] with respect to the x' axis in S'

a) Find the angle [tex]\theta[/tex] the beam makes with respect to the x axis in S.
- Ans. : [tex] cos\theta = (cos\theta_o + \frac{v}{c})(1 + \frac{v}{c} cos\theta_o)[/tex]


The Attempt at a Solution


From an example problem in our book, we know that with a ruler at an angle in the same situation has angle in laboratory frame:

[tex] \theta = arctan(tan(\theta_o)\gamma)[/tex], where [tex]\gamma = \frac{1}{\sqrt{1-frac{v^2}{c^2}}}[/tex]

if you just take cos of both sides, then you have

[tex]cos\theta = cos(arctan((tan(\theta_o)\gamma)))[/tex]

Drawing a triangle with the right leg as [tex]tan(\theta_o)[/tex] and the bottom leg as [tex]\frac{1}{\gamma}[/tex] you get a hypotenuse of [tex]\sqrt{(\frac{1}{\gamma})^2 + tan(\theta_o)^2}[/tex]

from this i narrowed it down to

[tex]cos\theta = \frac{\frac{1}{\gamma}}{\sqrt{(\frac{1}{\gamma})^2 + tan(\theta_o)^2}}[/tex][tex] = \frac{1}{\sqrt{1 + frac{\gamma*2}{cos{\theta_o)^2} - \gamma^2}}[/tex]

I'm not seeing how this can reduce to my answer I'm suppose to get yet?
 

Answers and Replies

  • #2
Doc Al
Mentor
44,882
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Homework Statement


A light beam is emitted at an angle [tex]\theta_o[/tex] with respect to the x' axis in S'

a) Find the angle [tex]\theta[/tex] the beam makes with respect to the x axis in S.
- Ans. : [tex] cos\theta = (cos\theta_o + \frac{v}{c})(1 + \frac{v}{c} cos\theta_o)[/tex]
That answer doesn't look right. It should be:

[tex] cos\theta = \frac{(cos\theta_o + \frac{v}{c})}{(1 + \frac{v}{c} cos\theta_o)}[/tex]

The easy way to derive it is to use the velocity transformations.
 
  • #3
57
1
That answer doesn't look right. It should be:

[tex] cos\theta = \frac{(cos\theta_o + \frac{v}{c})}{(1 + \frac{v}{c} cos\theta_o)}[/tex]

The easy way to derive it is to use the velocity transformations.
oops I missed the divide sign in my book - your answer is the right one.

I'll give that a shot
 
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