# Magnification of an image

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?

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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.

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.

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

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

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
Staff Emeritus
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.
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.

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

Drakkith
Staff Emeritus
What about multiple reflections in two plane mirrors? There the images are the same size but their visual angle is less.
The path the light takes is the same distance as the image appears to be at, so its the same as before. No magnification.

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
Staff Emeritus
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.
How does it appear larger?

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

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