Deriving Snell's Law from an Optimization Problem Refraction is a physical phenomenon that happens when light, or more generally a wave, passes through two different mediums. In this problem, we consider two mediums separated by an horizontal line of length d (for example, this line could separate the water in a pool and the air above it). A photon is emitted from a source placed at a point A and detected at a point B. See Figure 1 for the placement of A and B. First Medium A h₁ d Second Medium B Figure 1: Photon emission and detection points. The speeds of the photon in the first and second medium are v₁ and #2, respectively. The refraction indexes are defined as n₁and n₂ = where c is the speed of light in vacuum. Assuming that the photon travels on the fastest path from A to B, prove Snell's law of refraction sin 01 ปา sin 02 v2 where 01 and 02 are the angles defined in Figure 2. First Medium A d Second Medium B Figure 2: Definition of 1 and 02. (1) Suggested steps: 1. Choose the variabler for the optimization as suggested in the Figure 3, and find the domain of r. 2. Compute the distances 1 and 2 traveled in the first medium and second medium as functions of r (and the parameters h₁, h₂, and d). 3. Compute the times t₁ and t₂ that the photons spends in the first and second medium. Remember that time is given by distance divided by speed. 4. Compute the total time. 5. Find the derivative of the total time. Argue that there are no singular points and set up the equation to find the critical points. 6. Observe that sin 01√+ and sin ₂ = d-z , and use it to show that the equation written at the step above is equivalent to sint 0 sin ₂ U2 7. Show that the single critical point obtained is the location of both a local and global (absolute) minimum for total time. 8. Conclude that the fastest time is the one identified by Snell's law. First Medium d-2 I d Second Medium B Figure 3: Definition of 2.
Deriving Snell's Law from an Optimization Problem Refraction is a physical phenomenon that happens when light, or more generally a wave, passes through two different mediums. In this problem, we consider two mediums separated by an horizontal line of length d (for example, this line could separate the water in a pool and the air above it). A photon is emitted from a source placed at a point A and detected at a point B. See Figure 1 for the placement of A and B. First Medium A h₁ d Second Medium B Figure 1: Photon emission and detection points. The speeds of the photon in the first and second medium are v₁ and #2, respectively. The refraction indexes are defined as n₁and n₂ = where c is the speed of light in vacuum. Assuming that the photon travels on the fastest path from A to B, prove Snell's law of refraction sin 01 ปา sin 02 v2 where 01 and 02 are the angles defined in Figure 2. First Medium A d Second Medium B Figure 2: Definition of 1 and 02. (1) Suggested steps: 1. Choose the variabler for the optimization as suggested in the Figure 3, and find the domain of r. 2. Compute the distances 1 and 2 traveled in the first medium and second medium as functions of r (and the parameters h₁, h₂, and d). 3. Compute the times t₁ and t₂ that the photons spends in the first and second medium. Remember that time is given by distance divided by speed. 4. Compute the total time. 5. Find the derivative of the total time. Argue that there are no singular points and set up the equation to find the critical points. 6. Observe that sin 01√+ and sin ₂ = d-z , and use it to show that the equation written at the step above is equivalent to sint 0 sin ₂ U2 7. Show that the single critical point obtained is the location of both a local and global (absolute) minimum for total time. 8. Conclude that the fastest time is the one identified by Snell's law. First Medium d-2 I d Second Medium B Figure 3: Definition of 2.
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Transcribed Image Text:Deriving Snell's Law from an Optimization Problem
Refraction is a physical phenomenon that happens when light, or more generally a wave, passes through
two different mediums. In this problem, we consider two mediums separated by an horizontal line of length d
(for example, this line could separate the water in a pool and the air above it). A photon is emitted from a
source placed at a point A and detected at a point B. See Figure 1 for the placement of A and B.
First Medium
A
h₁
d
Second Medium
B
Figure 1: Photon emission and detection points.
The speeds of the photon in the first and second medium are v₁ and #2, respectively. The refraction indexes
are defined as n₁and n₂ = where c is the speed of light in vacuum. Assuming that the photon travels
on the fastest path from A to B, prove Snell's law of refraction
sin 01
ปา
sin 02
v2
where 01 and 02 are the angles defined in Figure 2.
First Medium
A
d
Second Medium
B
Figure 2: Definition of 1 and 02.
(1)
Suggested steps:
1. Choose the variabler for the optimization as suggested in the Figure 3, and find the domain of r.
2. Compute the distances 1 and 2 traveled in the first medium and second medium as functions of r (and
the parameters h₁, h₂, and d).
3. Compute the times t₁ and t₂ that the photons spends in the first and second medium. Remember that
time is given by distance divided by speed.
4. Compute the total time.
5. Find the derivative of the total time. Argue that there are no singular points and set up the equation to
find the critical points.
6. Observe that sin 01√+ and sin ₂ =
d-z
, and use it to show that the equation written at
the step above is equivalent to sint
0
sin ₂
U2
7. Show that the single critical point obtained is the location of both a local and global (absolute) minimum
for total time.
8. Conclude that the fastest time is the one identified by Snell's law.
First Medium
d-2
I
d
Second Medium
B
Figure 3: Definition of 2.
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