# Dispersion/total internal reflection

• cerberus9
In summary, the conversation discusses a common mirage formed by superheated air above a roadway where a truck driver experiences the illusion of seeing water ahead. The task is to find the index of refraction of the air just above the road surface, treating it as a problem involving total internal reflection. The solution involves solving for the critical angle using the equation n1*sin(theta 1) = n2*sin(theta 2) and then using this value to calculate the index of refraction.
cerberus9

## Homework Statement

Consider a common mirage formed by superheated air just above a roadway. A truck driver whose eyes are 2.00 m above the road, where n=1.0003 looks forward. She has the illusion of seeing a patch of water ahead on the road, where her line of sight makes an angle of 1.20 degrees below the horizontal. Find the index of refraction of the air just above the road surface. (Hint: Treat this as a problem one involving total internal reflection)

## Homework Equations

$$\Theta$$c=sin-1(n1/n2)

n1sin$$\Theta$$1=n2sin$$\Theta$$2

## The Attempt at a Solution

So since the textbook said to treat it as a problem with total internal reflection, i figured that I would solve for $$\Theta$$c and so I did

$$\Theta$$c=sin-1(n1/n2)
$$\Theta$$c=sin-1(1/1.0003)
$$\Theta$$c= 88.6

and I just have absolutely no idea what to do from here.

n1*sin(theta 1) = n2*sin(theta 2)
In the problem n1, theta 1 and theta 2 is known. Find n2.

I would first clarify the problem by defining the variables and providing context. In this scenario, the index of refraction of the air just above the road surface is unknown, and the angle of incidence and angle of refraction are given. The problem also mentions a mirage caused by superheated air, which is important to consider in understanding the phenomenon.

Next, I would approach the problem by using the Snell's law equation, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction of the two media. In this case, the two media are air and the superheated air above the roadway. The angle of incidence is the angle at which the truck driver's line of sight meets the road surface, and the angle of refraction is the angle at which the light is bent as it passes through the boundary between the two media.

Using this equation, we can rewrite it as n1sin\Theta1=n2sin\Theta2, where n1 is the index of refraction of air and n2 is the index of refraction of the superheated air. We know that the angle of incidence is 1.20 degrees and the angle of refraction is 90 degrees (since the light is refracted along the surface of the road). Substituting these values into the equation, we get:

n1sin(1.20)=n2sin(90)

Solving for n2, we get:

n2=n1sin(1.20)/sin(90)

Now, we need to find the value of n1. We can use the given information that the truck driver's eyes are 2.00 m above the road, where n=1.0003. This value of n is the index of refraction of air at standard temperature and pressure. We can use this to find the index of refraction of air at the height of the truck driver's eyes, which is:

n1=n(2.00/2.00)=1.0003

Substituting this value into the equation for n2, we get:

n2=1.0003sin(1.20)/sin(90)

Solving for n2, we get:

n2=1.0003(0.0209)/1=0.0209

Therefore, the index of

## What is dispersion?

Dispersion is the phenomenon where different wavelengths of light travel at different speeds through a medium, causing the separation of white light into its component colors.

## What causes dispersion?

Dispersion is caused by the different refractive indices of a medium for different wavelengths of light. This can be due to the varying speeds at which different wavelengths of light travel through the medium.

## How does total internal reflection work?

Total internal reflection occurs when a ray of light travels from a medium with a higher refractive index to a medium with a lower refractive index, and the angle of incidence is greater than the critical angle. This causes the light to reflect back into the higher refractive index medium, rather than refracting into the lower refractive index medium.

## What are the applications of total internal reflection?

Total internal reflection has many practical applications, including fiber optics, where light is transmitted through thin fibers using total internal reflection. It is also used in prisms and mirrors to redirect light, and in diamond cutting to create a brilliant sparkle.

## How does dispersion affect optical instruments?

Dispersion can cause distortion in optical instruments, as different wavelengths of light will bend at different angles. This can cause chromatic aberration, where colors appear fringed or distorted in telescopes and microscopes. To minimize this effect, lenses with different refractive indices can be used to correct for dispersion.

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