Calculating Reflection Angle and Frequency for Moving Mirror?

In summary, To solve for the reflection angle and observed new frequency of a photon reflected off of a uniformly moving mirror, Lorentz transformations can be used. By considering the x and y components of momentum and dividing them, the ratio of the tangents of the incident and reflection angles can be obtained. The frequency of the photon will be different in the mirror frame compared to an external observer's frame due to the Doppler effect. The reflection angle can be expressed in terms of the incident angle and the mirror's velocity. Finally, the observed frequency of the reflected photon can be calculated using the first Lorentz transformation.
  • #1
xayon
3
0

Homework Statement


A mirror is moving uniformly in a direction normal to its plane with velocity v=βc. Given the angle of incidence and frequency of an incident photon (θ_e,nu_e in the figure), calculate the reflection angle and the observed new frequency (θ_i,nu_i in the figure).

Also prove that:
sin(θ_e)/(1+βcos(θ_e)) = sin(θ_i)/(1-βcos(θ_i))

Homework Equations


It has been suggested in a couple of threads to use Lorentz transformations, but i seem to have problems dealing with them.


The Attempt at a Solution


As said, my attempts to use them have been an utter failure. In the mirror's system the angles should be the same, but i get stuck at this point.
 

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  • #2
welcome to pf!

hi xayon! welcome to pf! :wink:
xayon said:
It has been suggested in a couple of threads to use Lorentz transformations, but i seem to have problems dealing with them.

yes, the Lorentz transformation should do it …

show us how far you've got, and where you're stuck :smile:
 
  • #3
Does it help to think about the momenta in the X and Y directions?
 
  • #4
Yes, I've thought of the momentum, h*nu. Is the frequency of the incident photon as measured in the mirror frame the same as the one from an external observer?
The other thing to take into account is that in the mirror frame the angles are the same isn't it? But again, by Doppler effect, the frequency of the photon would be different...
 
  • #5
xayon said:
Yes, I've thought of the momentum, h*nu.
[/QUOTE}
Ok, but have you thought about the x and y components of momentum? The ratio of these should give the tan of the angle. The answer will be different in the two frames.
Is the frequency of the incident photon as measured in the mirror frame the same as the one from an external observer?
No.
The other thing to take into account is that in the mirror frame the angles are the same isn't it? But again, by Doppler effect, the frequency of the photon would be different...
In the mirror frame, there'll be no change in frequency.
 
  • #6
Ok, having Lorentz transformations for the momentum:
[itex]E'=\gamma(E-\beta cp_{x})[/itex]
[itex]cp'_{x} =\gamma(cp_{x}-\beta E)[/itex]
[itex]cp'_y=cp_y[/itex]
[itex]cp'_z=cp_z[/itex]

Where the primed frame is the mirror's one.

As haruspex pointed, I have divided the x and y components of the momentum, getting two expressions for [itex]tan(\theta _{e})[/itex] proving the statement. From that i can get [itex]\theta_{i}[/itex] in terms of [itex]\theta _{e}[/itex] and v.

Finally getting [itex]\nu[/itex] from the 1st transformation.

Am I right?
 

What is reflection on a moving mirror?

Reflection on a moving mirror is a phenomenon where light rays are bounced off of a mirror that is in motion. This motion can be either linear or rotational, and it affects the angle at which the light rays are reflected.

How does the motion of a mirror affect reflection?

The motion of a mirror affects reflection by changing the angle at which light rays are reflected. If the mirror is moving linearly, the angle of reflection will change based on the mirror's velocity and the angle of incidence. If the mirror is moving rotationally, the angle of reflection will change as the mirror rotates, creating a different reflection angle for each position of the mirror.

What factors influence reflection on a moving mirror?

The main factors that influence reflection on a moving mirror are the mirror's motion (linear or rotational), the angle of incidence, and the mirror's velocity. Other factors such as the type of surface of the mirror and the wavelength of the light can also affect the reflection.

How is reflection on a moving mirror observed and measured?

Reflection on a moving mirror can be observed by shining a light source onto a moving mirror and observing the changes in the reflected light. The angle of reflection can be measured using a protractor or other measuring device. The velocity of the mirror can be measured using a sensor or by timing the motion.

What are the applications of reflection on a moving mirror?

Reflection on a moving mirror has several applications in scientific fields such as optics and astronomy. It is used in instruments like telescopes and laser scanners to capture images and measure distances. It also has practical applications in industries such as automotive and robotics for sensing and positioning purposes.

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