How does the sign convention work in ray optics?

[Mohammed Ayaz Quadri]

Homework Statement

I needed help to undestand some concepts in Ray Optics for my assignment. The topics which U am concerned are: Part 1: Mirror Equation, Refraction at singly curved surface, lens maker's equation, combination of thin lenses in contact.
Part 2: Magnification Power of simple Microscope, Compound Microscope and Astronomical Telescopes.
First of all please explain me the sign convention. I don't need what it is, but how it works. Yoy apply it to derive fornulae, then again apply it in numericals to solve problems. How does that work? But I have failed to understand sign convention in the Part 2 topics. I mean in the solved problems, sigb convention is used and not used at random. How is that to be done? I have uploaded an image of a formula, I know its correcr for lenses, but tell me if I should put a negative sign on left side or right side to make it correct for mirrors. Thank you.

Homework Equations

Attached.

The Attempt at a Solution

I tried solving questions and I guess that for MP sign convention should not be used for solving numericals. But why?[/B]

 

[Merlin3189]

There are different sign conventions in use. It does not matter which you choose, so long as you are consistent.
The one I use is 'Real is Positive' (It may have other names, but that's how I learned it.) In that, real objects and images have ositive distances from the lens, virtual objects and images have negative distances from the lens. Converging lenses form real images of objects at infinity, so have positive focal length. Diverging lenses form virtual images of an object at infinity, so have negative focal lengths.

For magnification ditances on one side of the optical axis are positive (usually up) and on the other side (down) are negative.
If the magnification is negative, that means the image is inverted.
But if you attach signs to magnification, the comment in your upload file should be talking about the magnitude of M (|M|) being greater or less than 1. If magnification turned out to be -2 say, the image would be enlarged even though M <1.

 

[Mohammed Ayaz Quadri]

The one I use is left side of mirror lens is negative right is positive. Above the principle axis positive below is negative. Acc to that I talked about negative sign in Linear Magnification formula, but for mirrors only. For lens the formula should be correct. And yes the comment is about magnitude of M. Thank you for your input. What about using sign convention in numerical for Mag Power of simple and compound microscope? It really confuses me. Thank you.

 

[Merlin3189]

... What about using sign convention in numerical for Mag Power of simple and compound microscope? ...

I don't think you would need nor want to worry about signs in a standard microscope. You just want to know the size of the magnification. It is negative because the image is inverted.

Microscopes are a bit peculiar, because a specialised field of microscopy has grown up over a long period. They have developed their own standards and conventions, such as that magnification of a compound microscope is equal to objective magnification x eyepiece magnification, and that the standard lenses do have a definite value of magnification (when used in a standard size microscope tube.) That is a convenient approximation.

It is a confusing topic, because after learning about magnification as the ratio of object size to image size, when you come to optical instruments like microscope and telescopes, you have to switch to angular magnification. That is because you have to deal with objects and images of indeterminately large distances and sizes (~infinity). You can read up about that yourself.

 

[Mohammed Ayaz Quadri]

Thank you. One last thing, we have studied ray optics with approximations. Like in mirror equation and Magnification power. For eg we have assumed tan of theta is appro theda for small values. Thus can have been able to derive formula. But is there any possible way to derive all those formulae with 100% accuracy irrespective of how complex they are?

 

[Merlin3189]

I'm not expert in this field, but I believe all curved mirror and lens formulae are approximate.

There is an element of mathematical convenience, but in fact you need to make some approximating assumptions to even have a formula!
The idea that for each object point there is a single image point where all the light rays will come to a focus, is not true.

For a spherical concave mirror there is no focus for a beam of parallel rays of light: it is only an approximation for rays close to the axis.
For a parabolic concave mirror there is a focus for all rays parallel to the axis, but not for parallel rays at an angle to the axis: so the focus is an approximation for rays nearly parallel to the axis.
For most optical systems we have to assume that rays are close to and nearly parallel to the axis.

There are several sorts of abberation in optical systems and these are the result of the fact that our ideal notion of focus is an approximation. There are many ingenious ways people have mitigated abberations by combining multiple lenses so that some of the approximations tend to cancel each other.

There are methods for calculating the path of any ray through an optical system (look for something like 'ray transfer matrix' ) and with computers to calculate paths for many rays, it is now possible to develop complex optical systems to optimise the image formation for a given situation, generally using aspherical lenses and mirrors. That is way beyond what you are studying now.

 

[Mohammed Ayaz Quadri]

Thank you everyone. I have mastered all the topics in my portion.