Snell's Law Derivaiton: Solving the Minima in 8:53

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Discussion Overview

The discussion revolves around the derivation of Snell's Law, specifically focusing on the analysis of minima in the total time taken for light to travel through different media. Participants explore the implications of derivatives in this context, addressing the relationship between speed and the conditions under which the derivative is set to zero.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question the interpretation of the derivative dT/dx being set to zero, suggesting that it should reflect a non-zero value related to speed (1/v).
  • Others clarify that the derivative itself is not zero, but rather the rate of change of total time with respect to distance is analyzed at the minima.
  • A participant points out that the dimensions of dT/dv are similar to 1/speed, raising the question of which speed is relevant in this context.
  • Some argue that the minimum of the expression T(x) is not directly related to any physical speed, including the speed of light.
  • Another participant introduces the concept of phase velocity, suggesting that it can be infinite under certain conditions without violating known physics, as information transmission remains limited to the speed of light.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the derivative in the context of Snell's Law. There is no consensus on the relationship between the derivative, speed, and the implications for the physical interpretation of the results.

Contextual Notes

Participants highlight the potential confusion arising from the definitions of speed and the conditions under which the derivative is analyzed. The discussion includes unresolved questions about the implications of setting dT/dx to zero and the nature of phase velocity.

Prannoy Mehta
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(8:53, to be precise)

They have taken a derivative to analyze the minima of the total time taken. dT/dx = 0. But in reality should it not be 1/v which is a non zero value. What am I missing ?
 
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Prannoy Mehta said:
(8:53, to be precise)

They have taken a derivative to analyze the minima of the total time taken. dT/dx = 0. But in reality should it not be 1/v which is a non zero value. What am I missing ?


Kinda confused by what you mean, they set the derivative to zero to analyze the graph. The derivative itself is not zero. 1/v would also not be the correct derivative.

Can you elaborate on what you mean?
 
I think he means that the dimensions of dT/dv are the same as 1/speed. But which speed would it be?
 
Prannoy Mehta said:
They have taken a derivative to analyze the minima of the total time taken. dT/dx = 0. But in reality should it not be 1/v which is a non zero value. What am I missing ?
Yes, it is the speed (or better, rate) of change of the total time when x changes. At the minimum that rate is zero.
Note that not everything with dimensions distance/time is the speed of some object (or wave). Here you have such an example.
 
Sorry, for the confusion. Over there they have dT/dx as 0, that is right, if you follow the graph and want to find out the minima. But then dx/dt = v. Which is less then or equal to the speed of light, clearly not infinity. So how can there be any point in the graph where speed is infinity, that is when dT/dx=0 (dT/dx = 1/v).
 
There is no dx/dt in that derivation. There is a dT/dx.
The minimum of that expression, T (x), is not related to any speed. Or inverse speed.

Even if you take the inverse, dx/dT, this is not a speed of anything so is not limited by speed of light.
 
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In addition to what Nasu has said,

If you look at Snells law completely you can clearly see that the speed of light is either c, or 0 at the bounds when we set n between 1 and infinity.

Your v is the phase velocity, and it can be infinite, however this doesn't break known physics as the light still can only transmit information at c. This superlum v begins to happen with n's lower than 1.
 
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