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

[Rainbow]

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.

 

[Hootenanny]

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.

 

[Rainbow]

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.

 

[mgb_phys]

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.

 

[Doc Al]

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.

 

[Rainbow]

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.

 

[Doc Al]

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.

 

[Rainbow]

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.

 

[Doc Al]

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.

 

[jtbell]

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 f = - R / 2. Then they move on to thin lenses, and introduce or derive the lensmaker's formula:

\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)

(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.

 

[Rainbow]

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.

 

[Doc Al]

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.