Optics Homework: Converging Lens and Varying Refractive Index

[AdityaDev]

Homework Statement

A rectangular slab of length l=20cm and thickness d=4cm is placed in left of a converging lens of focal length f=20cm. A screen is placed in the focal plane plane of lens (right side of lens).Refractive index of the material of slab increases linearly from u₀ at the bottom by an amount ##du=2*10^-4)## between bottom and top faces.A parallel beam is made to incident on the left side of slab. Prove that only a bright spot forms on the screen? Find its position

Homework Equations

$$\Delta x=n \lambda$$
$$y=n \lambda /d$$

The Attempt at a Solution

Here as the refractive index varies, speed of light varies. It will be faster at bottom and slower at top.
$$v=c/ \mu$$
$$ \mu = \mu_o + y \delta \mu $$
For any two rays at y₁ and y₂ once the slowest ray comes out of slab,
$$t=l/v => t = l/ ( \mu_0 + d \delta \mu)$$
Hence $$ \Delta x = l(\mu_0 + d \delta \mu) \frac {(y_2-y_1) \delta \mu}{(\mu_0 + y_1 \delta \mu)(\mu_0 c + y_2 \delta \mu)} $$

 

[haruspex]

I'm not sure what you have calculated. What is Δx? The difference in the two path lengths?
I feel it is more useful to describe how the two light beams will be changed by the slab. Will they be bent through the same angle?
(It might help to recognise that a rectangular slab with uniformly changing index is equivalent to a more familiar piece of optics equipment.)

 

[AdityaDev]

Close the thread. I have solved the problem.

 

[AdityaDev]

I'm not sure what you have calculated. What is Δx? The difference in the two path lengths?
I feel it is more useful to describe how the two light beams will be changed by the slab. Will they be bent through the same angle?
(It might help to recognise that a rectangular slab with uniformly changing index is equivalent to a more familiar piece of optics equipment.)

I have solved the problem. The question is of high level. Thank you for the response.
Actually $$\mu_1 = \mu_0 + \delta \mu$$
Find the time taken by slowest ray to come out of slab. Let it be t.
$$t=l \mu_1/c$$
During that time find the distance traveled by fastest ray(at bottom.
Now the situation is like fraunhoffer single slit diffraction.

 

[HalfLostAndSearching]

Thank you for your response. I would like to suggest a few things to improve your solution.

Firstly, it would be helpful to define the variables used in the solution, such as l, d, f, u0, du, etc. This will make it easier for others to follow your solution.

Secondly, it would be beneficial to provide a brief explanation of the physical principles involved in this problem, such as Snell's law and the concept of refractive index. This will provide context for your solution and help others understand the problem better.

Additionally, it is important to use proper units for all the variables and equations. In this case, the units for l, d, and f should be in centimeters, and the units for refractive index should be dimensionless.

Finally, it would be helpful to provide a diagram or figure to illustrate the setup of the problem. This will make it easier for others to visualize the problem and understand your solution.

Overall, your solution is well thought out and demonstrates a good understanding of the problem. By incorporating these suggestions, you can make your solution even clearer and more comprehensive.