Write an expression for the Optical path length

In summary, the conversation is about a person seeking help with a problem involving a converging lens. They provide equations for calculating the Optical path length (OPL) of two different rays passing through the lens and try to solve for the distance x. However, they encounter a discrepancy in their calculations and ask for clarification. Another person responds with a correction and suggests using the equation (a + b)^2 = a^2 + 2ab + b^2 for solving the problem.
  • #1
goodo
Hi,

I'm hoping I can get some help. (I'm a first time poster, so if there's something that isn't quite clear, please let me know).

A converging lens (n=1.355) in air has diameter 4.0cm, thickness 0.50cm and zero thickness at its edges.

A point object at 'So' = 18cm has an image at 'Si'= x cm. (where 'So' and 'Si' are distances from the center of the lens)

(a) Write an expression for the Optical path length (OPL) of the ray passing through the edge of the lens and (b) an expression for the OPL of the ray passing through the centre of the lens, and (c) determine the distance x.

(sqrt= square root, sq = squared) For (a) I used pythagorus. OPL = sqrt(18 sq + 2 sq) + sqrt(x sq + 2 sq)

(b) OPL = 17.75 +(x-0.25) + 0.5n
=17.5+x+.6775
=18.1775+x
=18.18+x

(c) Now I have both in term of x. Since OPL1=OPL2,
sqrt(18 sq + 2 sq) + sqrt(x sq + 2 sq)=18.18+x
(square both sides) 18 sq + 4 + x sq + 4 = 18.18 sq + x sq
332 + x sq = 330.5 + x sq

I can't think of any other way to do it, yet something must be wrong as I have to calculate x. Is there a reason why I'm 1.5 off other than rounding? Wouldn't the x sq both cancel out leaving me with nothing?

Thanks
goodo
 
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  • #2
Welcome goodo,
I think everything's OK up to here:
sqrt(18 sq + 2 sq) + sqrt(x sq + 2 sq)=18.18+x
(square both sides) 18 sq + 4 + x sq + 4 = 18.18 sq + x sq
The square of a sum is not the sum of squares.
Remember: (a + b)2 = a2 + 2ab + b2.

BTW, the 1st root has no x, so you can calculate its value. Makes things easier...
 
Last edited:
  • #3
Thansk for your response arcnets.
 

1. What is the definition of optical path length?

The optical path length is the distance that light travels through a medium, taking into account any changes in the speed or direction of the light.

2. How is the optical path length calculated?

The optical path length is calculated by multiplying the physical distance traveled by the refractive index of the medium.

3. What is the difference between optical path length and physical distance?

Optical path length takes into account any changes in the speed or direction of light, while physical distance only considers the actual distance traveled.

4. How is the optical path length related to the refractive index?

The optical path length is directly proportional to the refractive index. As the refractive index increases, the optical path length also increases.

5. Can the optical path length be negative?

No, the optical path length cannot be negative as it represents the distance that light travels and distance cannot be negative.

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