# Snell's law and inverse function

• Karol
But the text in the box clearly states: "The second condition is that the sine must be smaller than or equal to 1. The sine is at most 1, when the angle is 90 degrees." So, i supposed that my answer is correct.
Karol

## Homework Statement

Snell's law is:
$$\frac{\sin\theta_1}{c_1}=\frac{\sin\theta_2}{c_2}$$
$$\frac{c_1}{c_2}=n_{12}$$
Express ##\theta_2## as a function of ##\theta_1##
Find the largest value of ##\theta_1## for which the expression for ##\theta_2## that you just found is defined (for larger values of ##\theta_1## than this the incoming light will be reflected).

## Homework Equations

Inverse sine: ##y=\sin^{-1}(x)~\rightarrow~\sin(y)=x##

## The Attempt at a Solution

$$\sin ( \theta_2 )=\frac{\sin( \theta_1 ) }{n_{12}}~\rightarrow~\sin^{-1}\left( \frac{\sin( \theta_1 )}{n_{12}} \right)=\theta_2$$
##\theta_2## can be ##\frac{\pi}{2}## at the max:
$$\sin^{-1}\left( \frac{\sin( \theta_1 )}{n_{12}} \right)=\frac{\pi}{2}~~\rightarrow~~\sin\left( \frac{\pi}{2} \right)=\frac{\sin(\theta_1)}{n_{12}}$$
$$\Rightarrow~\sin(\theta_1)=n_{12}\sin\left( \frac{\pi}{2} \right)=n_{12}$$
$$\theta_1<\arcsin(n_{12})$$
I didn't use at all the definition of inverse function in the second question, i feel what i have done isn't what it's meant from the chapter of inverse functions

Karol said:
I didn't use at all the definition of inverse function in the second question
Not sure what you mean by that. The answer you gave involved an inverse function, and your final step involved inverting the sine function. Can you be more specific about what it is that you feel you have not used?
By the way, the diagram doesn't match the text. It clearly shows a case with c2>c1, whereas the text assumes the reverse.

haruspex said:
the diagram doesn't match the text. It clearly shows a case with c2>c1, whereas the text assumes the reverse.
How does the text show anything about the diagram? only the last formula: ##\theta_1<\arcsin(n_{12})## to my opinion, may show something in that direction.
if ##c_2>c_1~~\rightarrow~\sin(\theta_2)>\sin(\theta_2)~\rightarrow~\theta_2>\theta_1## and the diagram shows the inverse.
The diagram for ##c_2>c_1## must be:

for ##c_2>c_1~\rightarrow~n_{12}<1## and ##\theta_1## has a definite value and that's correct.

Karol said:
How does the text show anything about the diagram?
I confused you by incorrectly describing the mismatch.
The diagram shows an example of c1>c2, as you say. But in that case there is no limit (before π/2) on θ1. n12>0, so the arcsin of it is not defined.
Your second diagram, with c2>c1, makes more sense. E.g. total internal reflection can occur from water (low c) to air (high c), but not the other way around.

haruspex said:
The diagram shows an example of c1>c2, as you say. But in that case there is no limit (before π/2) on θ1. n12>0, so the arcsin of it is not defined.
Your second diagram, with c2>c1, makes more sense. E.g. total internal reflection can occur from water (low c) to air (high c), but not the other way around.
True, that is also why i thought my answer wasn't right, since there is no limit for ##\theta_1## if ##n_{12}>1##

## 1. What is Snell's law?

Snell's law, also known as the law of refraction, describes the relationship between the angle of incidence and angle of refraction when a light ray passes through a boundary between two different materials.

## 2. How is Snell's law expressed mathematically?

Snell's law can be expressed as n1sinθ1 = n2sinθ2, where n1 and n2 are the refractive indices of the two materials and θ1 and θ2 are the angles of incidence and refraction, respectively.

## 3. What is the inverse function of Snell's law?

The inverse function of Snell's law is known as the inverse Snell's law. It describes the relationship between the angle of incidence and the refractive index of a material when light passes through the boundary between two materials.

## 4. How is the inverse Snell's law expressed mathematically?

The inverse Snell's law can be expressed as sinθ1/sinθ2 = n2/n1, where n1 and n2 are the refractive indices of the two materials and θ1 and θ2 are the angles of incidence and refraction, respectively.

## 5. What is the significance of Snell's law and its inverse function in science?

Snell's law and its inverse function are fundamental principles in the study of optics and are commonly used to predict the behavior of light as it passes through different materials. They are also used in various fields such as engineering, medicine, and astronomy to design and analyze optical systems and devices.

Replies
6
Views
2K
Replies
4
Views
1K
Replies
2
Views
1K
Replies
7
Views
2K
Replies
15
Views
1K
Replies
7
Views
3K
Replies
20
Views
676
Replies
3
Views
1K
Replies
10
Views
5K
Replies
15
Views
5K