Infinitesimal, Snell's law, and ray tracing

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Discussion Overview

The discussion revolves around the application of Snell's law in the context of light refraction between two fluids, specifically focusing on the relationship between the angle of incidence and the velocity of light in the first medium. Participants explore how to analytically derive the differentiation of the angle with respect to the velocity of light.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant seeks to calculate d(alpha1)/d(v1) analytically, noting that v2 is constant and that changes in v1 affect both alpha1 and alpha2.
  • Another participant suggests drawing a diagram to visualize the relationship between alpha1 and alpha2.
  • A participant emphasizes that they are not looking for a relationship between alpha1 and alpha2, but rather the differentiation of alpha1 with respect to v1.
  • Some participants propose using a relationship between alpha1 and alpha2 to eliminate alpha2 from Snell's law to find the desired differentiation.
  • One participant expresses frustration at the lack of clarity in the responses and requests specific equations to assist in their calculations.
  • Another participant encourages the original poster to provide coordinates for the source and receiver to help derive a second expression relating alpha1 and alpha2.
  • The original poster later indicates they have resolved their issue using the diagram they created.

Areas of Agreement / Disagreement

Participants generally agree on the need for a diagram and the use of relationships between angles to derive the necessary differentiation. However, there is no consensus on the specific analytical method to achieve this differentiation, as some participants provide different approaches and levels of detail.

Contextual Notes

Limitations include the dependence on the accuracy of the diagram and the assumptions made about the fixed positions of the source and receiver. The discussion does not resolve the analytical method for differentiation, leaving it open to interpretation.

ytht100
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With FIXED SOURCE AND RECEIVER, I have a light incident from fluid 1 with velocity v1 into fluid 2 with velocity v2. Obviously, according to Snell's law, v1/v2=sin(alpha1)/sin(alpha2), where alpha1 and alpha2 are the angles with regard to the vertical line.

My question is: how to calculate d(alpha1)/d(v1)?

Obviously, I can obtain the result from numerical calculation, but I am seeking analytical method. I guess maybe I need some approximation to calculate that?

Please note v2 is always constant, please also note when v1 changes, alpha1 and alpha2 change. Please note direct differentiation of Snell's law, I can obtain d(alpha1)/d(alpha2). I am clueless and have already tried several methods but obtain no result, please help!
 
Last edited:
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Have you drawn a diagram? If you do so, you ought to be able to see a relationship between ##\alpha_1## and ##\alpha_2##.
 
Thank you very much for your kind answer. Yes, I have drawn it. I am NOT seeking a relationship between alpha1 and alpha2. I am seeking relationship between d(alpha1) and d(v1), in which d represents differentiation.
 
And you can get that by using a relationship between ##\alpha_1## and ##\alpha_2## to eliminate ##\alpha_2## from your Snell's Law expression.
 
Last edited:
Ibix said:
And you can get that by using a relationship between ##\alpha_1## and ##\alpha_2## to eliminate ##\alpha_2## from your Snell's Law expression.

Sorry you answer is NOT clear. I have tried everything that I can think of for several days, can you write down specific equations? Thank you very much!
 
I can help you find the answer, but I can't give it to you. The aim of this site is to help you learn, not to do things for you.

You say the source and receiver are fixed. Where are they? Write down the coordinates. Then, work out where the ray traveling from source to receiver crosses from one medium into the other (this is where a diagram will help you). That will give you a second expression relating ##\alpha_1## and ##\alpha_2##.
 
ytht100 said:
Thank you very much for your kind answer. Yes, I have drawn it.

show us your diag. so we can confirm you are on the right path
 
Thank you all for your kind answer. I have figured it out by using the diagram. Kind regards!
 

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