Lens Calculations using Refractive Index

In summary, the author discusses how to find Δ using h, r, and d. However, they make an error in their calculations and suggest using the sagitta theorem, which the author learned in Jr High School.
  • #1
nao113
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Homework Statement
Consider a lens made of glass with refractive index 𝑛𝑛=1.5 is placed in the air (refractive index =1.0), as shown in the figure. The first surface is plane and the second surface is spherical convex shape whose radius of curvature is 𝑟𝑟=100 [mm]. z-axis is called optical axis. The first surface perpendicularly intersects the optical axis at O, and second surface intersects the optical axis at Q. The thickness of the lens OQ = 𝑑𝑑 = 20 [mm]. A light beam parallel to the optical axis is incident to the lens, where the distance (height) of the beam from the optical axis is h. The refracted beam intersects the optical axis at X. Derive the distance between the lens and the intersection X, Q
Relevant Equations
n1theta1 = n2 theta2
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Screen Shot 2022-06-20 at 14.48.32.png


Answer:
I already found the answer for Sin theta 2 like the pic below
20220620_155107.jpg
But, I am still not sure about how to derive delta here. Can anyone show me they way? thank you
 
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  • #2
Can you find ##\theta_3## in terms of ##\theta_1## and ##\theta_2##? Then use Snell's law to eliminate ##\theta_2##.
 
  • #3
kuruman said:
Can you find ##\theta_3## in terms of ##\theta_1## and ##\theta_2##? Then use Snell's law to eliminate ##\theta_2##.
WhatsApp Image 2022-06-20 at 11.21.05 PM.jpeg
WhatsApp Image 2022-06-20 at 11.21.06 PM.jpeg
WhatsApp Image 2022-06-20 at 11.21.06 PM-2.jpeg


I put it like this, how is it?
 
  • #4
nao113 said:
It doesn't look good. How about finding a formula for Δ in terms of ##h##, ##r## and ##d##? Then instead of doing three separate calculations, you can substitute three separate values in the one formula. Even better, load the formula on a spreadsheet. That way it is less likely that you will get an inconsistent answer, you and we can troubleshoot your work more easily.

I see one glaring error that makes your numerical answers incorrect. That is your assertion that $$\sin\theta_3=\sin\theta_2-\sin\theta_1.~~\leftarrow~~\text{This is not correct}$$Why do you think that the sines are related this way? The way to do this problem is to answer the four questions in the order (2), (3), (4) and (1). So let's start with (2), "What is the relation between ##\theta_1##, ##\theta_2## and ##\theta_3##?"
 
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  • #5
kuruman said:
How about finding a formula for Δ in terms of h, r and d?
Hint: This relationship is a standard piece of lens/mirror plane geometry known as the sagitta theorem. Learn it.
 
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  • #6
hutchphd said:
Hint: This relationship is a standard piece of lens/mirror plane geometry known as the sagitta theorem. Learn it.
I didn't know it had a theorem name. I always (re)derived it as needed using the Pythagorean theorem.
 
  • #7
I remember it because I ground a telescope mirror in Jr High school.
Now I am lucky to remember to put on pants...
 
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FAQ: Lens Calculations using Refractive Index

What is the refractive index and how does it relate to lens calculations?

The refractive index is a measure of how much a material can bend light. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material. In lens calculations, the refractive index is used to determine the amount of bending that occurs when light passes through a lens, which is essential for determining the focal length and power of the lens.

How do you calculate the refractive index of a material?

The refractive index of a material can be calculated by dividing the speed of light in a vacuum by the speed of light in the material. This can be done using the formula n = c/v, where n is the refractive index, c is the speed of light in a vacuum (3 x 10^8 m/s), and v is the speed of light in the material.

What is Snell's law and how is it used in lens calculations?

Snell's law describes the relationship between the angles of incidence and refraction when light passes through a boundary between two materials with different refractive indices. In lens calculations, Snell's law is used to determine the angle of refraction and the amount of bending that occurs when light passes through a lens.

How does the refractive index affect the power of a lens?

The refractive index of a material directly affects the power of a lens. A material with a higher refractive index will bend light more, resulting in a lens with a higher power. This is why lenses made of materials with higher refractive indices, such as glass, are often used for stronger prescriptions.

Can the refractive index of a material change?

Yes, the refractive index of a material can change depending on factors such as temperature, pressure, and the wavelength of light passing through it. This is why it is important for lens calculations to use the correct refractive index for the specific material and conditions.

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