# Optics Problem: Find Smallest Refractive Index of Slab

• arpon
In summary, the question asks for the minimum refractive index of a rectangular slab that will allow all incident light to emerge from the opposite face. It is found that the maximum value for the refractive index is infinite, but this is not the answer. By using trigonometric functions and knowing that air has a refractive index close to 1, it is determined that the maximum value for the angle of incidence is equal to the arc-sine of the refractive index, and the critical angle can be expressed as the arc-sine of the inverse of the refractive index. From this, it is found that the minimum refractive index is approximately 1.41.
arpon

## Homework Statement

Light falls on the surface AB of a rectangular slab from air. Determine the smallest refractive index n that the material of the slab can have so that all incident light emerges from the opposite face CD.

## The Attempt at a Solution

There must be total internal reflection at Q and S.
That means, ##90-r>\theta _c## [##\theta _c## is the critical angle]
## sin(90-r)>sin \theta _c##
##cos r > \frac{1}{n}## [## sin \theta _c = \frac {1}{n}##]
##n>sec(r)##
But, the maximum value of ##sec(r)## is infinite.
That means n should be infinite.
But that is not the answer.

Try it again. Write down the maximum ##r## in terms of arc-sin of something. Write down ##\theta_C## in terms of arc-sin of something. Then write down ##90-r > \theta_C## in terms of these arc-sin formulas. Don't forget that air has an index of refraction very close to 1. See what you get.

I have got the answer. :)

..

Last edited:

## 1. What is the purpose of finding the smallest refractive index of a slab in optics?

The purpose of finding the smallest refractive index of a slab is to determine the maximum angle at which light can enter the slab and still be refracted, also known as the critical angle. This information is important in understanding the behavior of light when it passes through different materials and can be used in various applications such as designing optical devices.

## 2. How is the smallest refractive index of a slab calculated?

The smallest refractive index of a slab can be calculated by using Snell's law, 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 refractive indices of the two materials. By rearranging this equation, the smallest refractive index can be solved for.

## 3. What factors can affect the smallest refractive index of a slab?

The smallest refractive index of a slab can be affected by the material properties of the slab, such as its density and composition. The angle of incidence and the wavelength of light can also affect the smallest refractive index. Additionally, the surrounding medium and any external forces acting on the slab can also impact the value of the smallest refractive index.

## 4. Can the smallest refractive index of a slab be negative?

No, the smallest refractive index of a slab cannot be negative. Refractive index is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium, and since the speed of light cannot be negative, the refractive index cannot be negative either.

## 5. What are some real-life applications of knowing the smallest refractive index of a slab?

Knowing the smallest refractive index of a slab can be useful in designing and optimizing optical devices such as lenses, prisms, and fiber optics. It is also important in understanding the behavior of light in different materials, which can have practical applications in fields such as medicine, communications, and electronics.

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