magnification of an image

rishch

We're learning about magnification and they say how magnification is the ratio of the visual angle while looking through the instrument to that with the naked eye and then they say,

For small angles, magnification is defined as,

m= height of the image/height of the object

Why only for small angles, I thought this works for any angle? Is it an approximation.
I thought that height of image/height of object was the definition of magnification in the first place, how can it not apply for large angle or is this only when we're looking for magnification as it appears to our eye?

rishch

And also, in a simple microscope where they use a single convex mirror and an object within the focus of the convex lens so that it produces a virtual image, there they use the formula v/u to get the magnification, but I feel that even though the image is bigger, it's also behind which reduces the visual angle, so magnification (in terms of how much bigger the object APPEARS to us) won't just be v/u.

rishch

http://www.citycollegiate.com/magnifying_glass.htm This site contains more detailed info on what they were talking about. But I don't get the math. Just because alpha and beta are small angles why should alpha be equal to tan alpha?

xAxis

Hmm.. It's true. You can check it with your calculator. Sine of a small angle is equal to that angle (in radians). The best way to see that aproximation is to look at trigonometric circle. You reed the value of sine of an arc on Y axis. You can see how as angle aproches 0, the length of an arc (the angle) aproches the length of it projection on Y axis, which is sine of that angle.
For tan, because tanx = sinx/cosx, you know that for small x cosx aproaches 1, so tanx aproaches sinx/1, so tanx aproximately = sinx.

rishch

Okay but how did they get that angular magnification is height of image / height of object?

sophiecentaur

Because that is how the object and image seem to compare. If you focus your telescope to place the image at a great distance then the image will actually appear to be, say, ten times the height of the object. You can actually see this, with some skill, by looking at a brick wall with a telescope (binocular) on one eye and the other eye looking at the unmagnified image. Once you get used to it, you can actually count ten* unmagnified bricks superimposed on just one brick seen through the scope. (* or whatever the magnification happens to be)

It's harder to do the same thing with a microscope but the same principle applies.

rishch

But when looking at angular magnification other factors like distance from the eye come into play as well, and that formula does not take them into account. For example, if the object and image are of same size, then according to that formula m=1 but if the image is farther away it is actually diminished as it will subtend a smaller angle at our eyes.

Drakkith

The only way for the object and image to be the same size would be to have a flat piece of glass that doesn't magnify. And then you could say that the image is placed just as far away as the object is. Hence no magnification.

rishch

What about multiple reflections in two plane mirrors? There the images are the same size but their visual angle is less.

Drakkith

The path the light takes is the same distance as the image appears to be at, so its the same as before. No magnification.

rishch

But it appears to be bigger and that's what angular magnification is. Just like trees farther away are smaller, or railway tracks converge at the horizon.

Drakkith

How does it appear larger?

rishch

Because it's closer to you. Just like the trees right? The size of the image on the retina is bigger.

Drakkith

I don't think so. I just tested it in my bathroom mirror using my phone as a mirror. The size of my head in my bathroom mirror was larger than it was when I looked at the reflection off of my cell phone in the mirror. (AKA the light went from me to the mirror, then to my phone, then back to the mirror, then to my eye) I'd guess that it was about the same size as it would be if I were twice as far from the bathroom mirror. I think it's about how far the light has traveled, not about how far away the object you are looking at is.

sophiecentaur

When you try to assess 'size' in an objective way, your brain has a real difficulty. We still see Jumbo Jets lumbering at a leisurely speed across the skies of London because our brain tells us that something 'that big' couldn't be up there. So we 'see' a smaller aircraft going much slower than the real speed and assume it's only just above the roof tops.
There are many similar illusions. The only way to describe 'magnification' reliably has to be to describe the 'angle subtended' by object and image - as if the two were side by side at the same distance from us. The image we see in a telescope of microscope is only virtual, in any case and we can place it where we like by adjusting the eyepiece. This, we have to talk of angular size and not real size - the image of a flea in a microscope, adjusted to be 1km away, would be a hundred metres long - (logically) but it is a meaningless to talk in terms of length.

@risch: Talking about the size of the image on your actual retina is a good approach because that implies angular size - the most objective description. Your thumb, at arms length really is the same size as the Sun (in those terms) - fair enough.