- #1
maverick280857
- 1,774
- 5
Hi everyone
A few weeks ago, I had worked out the relations between the angle of reflection and angle of incidence in case of reflection from a plane mirror,
(1) moving toward the incident ray
(2) moving normal to the incident ray
(PS -- This is not homework.)
The way I did it was (for the first case, as an example)
1. Assume the angle of incidence is \theta in the lab frame.
2. The velocity of the mirror is
\mathbf{v}_{m} = -v_{m}\sin\theta\hat{x} + v_{m}\cos\theta\hat{y}
3. The velocity of the light ray in the lab frame is,
\mathbf{v}_{l} = c\sin\theta\hat{x} - c\cos\theta\hat{y}
4. Find the velocity of the incident light ray in the mirror frame,
\mathbf{v}_{i;m} = \frac{\mathbf{v}_l + (\gamma-1)\frac{\mathbf{v_l}\cdot\mathbf{V_m}}{V_m^2}\mathbf{V}_m -\gamma\mathbf{V}_m}{\gamma\left(1-\frac{\mathbf{v_l}\cdot\mathbf{V_m}}{c^2}\right)} = c\cos i \hat{x} - c\sin i \hat{y}
where i is the angle of incidence in the frame of the mirror.
5. Use the law of reflection in the frame of the mirror, to write the velocity of the reflected ray as
\mathbf{v}_{r;m} = c\cos i \hat{x} + c\sin i \hat{y}
6. Transform this velocity back to the lab frame, using the inverse of formula 4 (with i --> r, for the reflected ray of course).
7. From this transformed velocity, find the angle of reflection in the lab frame.
The original question was to find the modification to Snell's Laws for a moving mirror. I thought the above procedure is a correct way of doing it. I was able to show that for low velocities, Snell's law does also hold in the laboratory frame (we have assumed in step 5 that it always holds in the frame of the mirror -- which seems reasonable to me).
But I came across this short article: http://home.c2i.net/pb_andersen/pdf/aberration.pdf , and it seems to suggest that I made some mistake.
What is going wrong? Is the procedure I outlined above correct?
Does Snell's Law hold even in the laboratory frame for a moving mirror?
Would appreciate inputs/insights.
Thanks in advance,
Vivek
A few weeks ago, I had worked out the relations between the angle of reflection and angle of incidence in case of reflection from a plane mirror,
(1) moving toward the incident ray
(2) moving normal to the incident ray
(PS -- This is not homework.)
The way I did it was (for the first case, as an example)
1. Assume the angle of incidence is \theta in the lab frame.
2. The velocity of the mirror is
\mathbf{v}_{m} = -v_{m}\sin\theta\hat{x} + v_{m}\cos\theta\hat{y}
3. The velocity of the light ray in the lab frame is,
\mathbf{v}_{l} = c\sin\theta\hat{x} - c\cos\theta\hat{y}
4. Find the velocity of the incident light ray in the mirror frame,
\mathbf{v}_{i;m} = \frac{\mathbf{v}_l + (\gamma-1)\frac{\mathbf{v_l}\cdot\mathbf{V_m}}{V_m^2}\mathbf{V}_m -\gamma\mathbf{V}_m}{\gamma\left(1-\frac{\mathbf{v_l}\cdot\mathbf{V_m}}{c^2}\right)} = c\cos i \hat{x} - c\sin i \hat{y}
where i is the angle of incidence in the frame of the mirror.
5. Use the law of reflection in the frame of the mirror, to write the velocity of the reflected ray as
\mathbf{v}_{r;m} = c\cos i \hat{x} + c\sin i \hat{y}
6. Transform this velocity back to the lab frame, using the inverse of formula 4 (with i --> r, for the reflected ray of course).
7. From this transformed velocity, find the angle of reflection in the lab frame.
The original question was to find the modification to Snell's Laws for a moving mirror. I thought the above procedure is a correct way of doing it. I was able to show that for low velocities, Snell's law does also hold in the laboratory frame (we have assumed in step 5 that it always holds in the frame of the mirror -- which seems reasonable to me).
But I came across this short article: http://home.c2i.net/pb_andersen/pdf/aberration.pdf , and it seems to suggest that I made some mistake.
What is going wrong? Is the procedure I outlined above correct?
Does Snell's Law hold even in the laboratory frame for a moving mirror?
Would appreciate inputs/insights.
Thanks in advance,
Vivek
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