Here's a question on light ray optics and focal length.

In summary: So, overall, I think it was a valuable exercise.In summary, In physics class we were taught that the focal length of a lens is half the radius of curvature, but this isn't always true. The focal length of a lens of non-negligible thickness is indeed dependent on the refractive index of the material among other factors.
  • #1
Rainbow
62
0
Here's a question on ray optics.

We say that the focal length of a lens is half times the radius of curvature, right. Now, consider any lens(say convex), made of any material(say glass). Now a beam of light parallel to the principle axis, passes through the lens. All the rays in the beam will converge at a point, which we call the focus. For now, let us assume that the focal length of this particular lens is half time the radius of curvature of the lens. Now, take another lens whose structure is identical to that of the previous lens, i.e., it has the same radius of curvature and everything, the only difference being that this time the lens is made of a material that is denser than that of the previous lens. This time, if we pass a beam of light parallel to the principle axis through the lens, all the rays in the beam will converge at a different focus, with focal length shorter than that in case of the previous lens. So, we can see that, this time the focal length is not equal to half times the radius of curvature. This shows that saying that the focal length of a lens is half times the radius of curvature is not always true.

Please comment.
 
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  • #2
Rainbow said:
We say that the focal length of a lens is half times the radius of curvature, right.

Actually, the focal length of a spherical mirror is half the radius of curvature of the mirror. The focal length of a lens of non-negligible thickness is indeed dependent on the refractive index of the material among other factors.
 
  • #3
Hootenanny said:
Actually, the focal length of a spherical mirror is half the radius of curvature of the mirror. The focal length of a lens of non-negligible thickness is indeed dependent on the refractive index of the material among other factors.

It doesn't matter. The situation that I have mentioned, is equally applicable to spherical lens.
 
  • #4
Hootenanny said the equation you quoted is only for MIRRORS - you are correct that the focal length of any lens must depend on it's refractive index, but the focal length of a mirror doesn't.
 
  • #5
The focal length of a thin spherical lens is given by the Lensmaker Formula; it's not simply R/2 as for a spherical mirror--it depends on both radii and the indices of refraction.
 
  • #6
But then why in most places it is written that, f=R/2? As, you already mentioned that the f is given by the lens-maker's formula, and that it depends on both radii and the indices of refraction.
 
  • #7
Rainbow said:
But then why in most places it is written that, f=R/2?
Where have you seen this?

As, you already mentioned that the f is given by the lens-maker's formula, and that it depends on both radii and the indices of refraction.
Exactly. Which is certainly not f = R/2.
 
  • #8
I have been taught this in my physics class. And I also found it many books like 'Fundamentals of Physics- by Halliday, Resnick and Walker', and my school textbook, and many other references.
 
  • #9
I don't think so. Why don't you scan in the page in Halliday and Resnick where it says that, so we know what it is you're reading.
 
  • #10
Or give us a short quotation from a specific book (authors and edition), along with the chapter and page number. If we're lucky, one of us will have the book and can look it up.

The introductory textbooks that I've seen all discuss spherical mirrors first, and give their focal length as [itex]f = - R / 2[/itex]. Then they move on to thin lenses, and introduce or derive the lensmaker's formula:

[tex]\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)[/tex]

(for a lens in vacuum, or to a good approximation, in air)

In my experience, it's common for students to glom onto the first formula they see for a certain quantity, and try to use it everwhere thereafter, even in situations where it doesn't apply.
 
  • #11
jtbell said:
In my experience, it's common for students to glom onto the first formula they see for a certain quantity, and try to use it everwhere thereafter, even in situations where it doesn't apply.

You're correct. And I just realized that I am one of those students. The formula f=R/2 is for a spherical mirror and not a lens.

I'm sorry to waste your time for this stupid question. But thank you for having enough patience and reminding me of the fact.
 
  • #12
I don't think it was a waste of time. True, as we suspected, you mixed up mirrors with lenses. But you showed excellent reasoning in your first post to prove that that formula didn't make sense for lenses. And now you've gotten it straight.
 

1. What is focal length?

Focal length is the distance between the center of a lens or curved mirror and its focal point. It is a measure of how strongly the lens or mirror bends light rays.

2. How is focal length related to light ray optics?

Focal length is a key concept in light ray optics because it determines the size and location of an image formed by a lens or mirror. Light rays passing through a lens or reflecting off a mirror are bent according to the focal length, resulting in either a magnified or reduced image.

3. What factors affect the focal length of a lens?

The curvature of the lens or mirror and the refractive index of the material it is made of are the main factors that affect focal length. The shape and placement of the lens or mirror in an optical system can also impact the focal length.

4. How is focal length measured?

Focal length is typically measured in millimeters (mm) and can be determined by placing an object at a known distance from the lens or mirror and measuring the distance between the object and the resulting image. It can also be calculated using the lens equation: 1/f = 1/u + 1/v, where f is the focal length, u is the object distance, and v is the image distance.

5. Can focal length be changed?

Yes, focal length can be changed by altering the shape or placement of the lens or mirror in an optical system. This can be achieved by using different lenses or mirrors with varying degrees of curvature, or by adjusting the distance between the lens or mirror and the object or image.

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