Optics: 5) Magnifier and microscope

today I will finally continue to write about optics. Last time I was talking about dioptre and today I will explain how magnifier and microscope works.

Angle of view

Angle of view plays important part in magnifier, microscope and so on. The problem why we can not see individual cells in leaf with naked eye is that we are not able to distinquish things that are too close to each other. Human eye is able to distinquish things that are about 1′ (arc minute=1/60° degrees) apart.

You can easily try it when you draw something on paper and then walk away from it. Or you are not able to tell trees apart when you are hundreds meters away from them.

(On the picture you can see the angle of view for camera, it can be measure horizontaly, verticaly or diagonally).

To make this angle bigger so we can distinquish everything we can walk towards stuff. But when we have a leaf we can get only limitely close to its surface and our eye can not adjust to something so close. Look at your thumb when you put it three centimeters from your nose. It will be blurry even if you try your best, this is because your eye does not have enough dioptre to make the image clear, plus your eye is going to hurt because of the muscles in eye stretching to make the optical power of your eye bigger. Conventional visual distance is distance for which human eye has to release least effort, this is about 25 centimeters.

To know the distance two objects must be apart to distinquish them we can use tangens:

tg τ=y/d

τ is the angle of view which is for human 1′.
y is the distance of two objects which you are trying to distinquish.
d is the distance from you to the objects.

y=d * tg τ

So the limit of our eye is that it is not able to be powerful enough so we need something which will help us and it has to work the same way as our eye, magnifier!

The light rays are going too much away from each other and your eye is not able to change their direction to create picture.


There is thing called angular magnification.

γ=τ’/τ = tg τ’/tg τ =  y/f/y/d = d/f

γ is the angular magnification and is the distance to focal point.

Angular magnification says to us how much our magnifier is strong. The formula above works for objects that are right in focal point, otherwise there would be “a” which is the distance to the object. If the object is right in focal point our eye does not need anything to do and as it gets closer the light rays are more and more going apart so that at one point you will need better magnifier and then it just wont be enough so you will have to use microscope.


There are two lenses in microscope. The first one is close to the object and it has the largest dioptre possible, making its focal point small as possible. It is called objective lens.

The second one is not so strong and its role is to make finally adjustment of light rays so they create image in your eye.

Optics 5, Pic 1

The picture above which I drew is horrible wrong but I can describe what is going on there. On the right you have the small object (brown). There is light coming from it in all various angle but important is that the lens has enough dioptres to use them all. F’ is the focal point of second lens, this point should be at the distance where all the rays from first lens converge into one point but I was not able to draw it properly. This is the way microscope is designed. Those rays start to go apart again but soon they hit the second lens, converging again and entering the eye in proper angles so that they hit all the spots on red line creating much bigger image.

Microscope is not unlimited source of magnification since when you will try to make bigger something too small you will get into problem with the wavelength of light.


Optics: 4) Measuring dioptre

today I was doing the best thing in optics to date. I was measuring the dioptre of my glasses (yes I wear glasses) and also I measured the dioptre of my magnifier (yes I measured it but then I figured out that I did it wrong so I will skip it).

Ok, before I get to the measuring and how I did it I will explain how lenses work because in last episodes what I did were only mirrors.

The difference between mirrors and lenses is that mirror reflect light while lenses let it through while changing its direction of travel.

There are several types of lenses which can be sorted to two main groups Pic1, Optics 4of convex lens and concave lens.

On the huge picture you can see the six types. The first row are convex lenses. First one is called biconvex lens then planoconvex lens and the third is concave-convex lens. You can see than there is always convex which hints for the first row, for convex type.

It is similar with the second type, those are biconcave lens, planoconcave lens and convex-concave lens.

I know this is cool, what can we do with this?
This equation which you can see on the left is the equation for lens which is thin. This means that there is no space between the arcs of the lens by this I mean that the arcs touch . Those arcs you can see on the right of the first picture. r1 is radius of the first arc and r2 of the second. f is the focal distance, the distance from focal point to the middle of the lens. The thing here is that lens has two focal distances, that is because it is made of two parts separeted by the vertical axis as you can see on the next picture. Also this whole equation not only equals to 1/f but also to φ(phi). The unit of φ is dioptre so φ=1/f. If f increases dioptre decreases logicly. So if someone has glasses with 4 dioptre his focal distance is 25 centimeters because dioptre is measured in meters!
This equation can be used both for concave and convex lenses of course (but concave lens will have r negative).

n1 and n2 are the refractive index of the glass which is around 1.6 and of the stuff where the lens is in, air, water or something else (n2 is the higher one).

You can find lot of problems on this equation and I did some from one book. It is good to exercise some of them because then you will feel much better on the stuff you are actually doing.

Now last thing before I get to the glasses, lets see how convex lens react to the three main rays which I mentioned in earlier postsPic2, Optics 4 (I will do the concave lens next time because I did not get to it yet).
When the candle is in about twice the distance of the focal point you can see that the size is fairly similar and what concave lens does, is that those light rays which are going from each other will be headed back towards the same point where the image will be formed. Of course the problem is that you wont see the picture of something when you put your lens from your glasses on the paper. It is because there is whole other bunch of rays from all different sides that will disturb any image that could be made.

Pic3, Optics 4When you look on the picture above, you can see that blue and green line were not able to touch anywhere which is the same thing that happened with the mirror when you put something between the mirror and focal point.

This image is enlarged and not true image since the rays are not actually going that way but our eye thinks so.

I was measuring the dioptres of my glasses. For the right eye I have -2 dioptres. You see it is very important that it is minus because that is what is saying that it is concave lens.
I took the glasses and drew line on the paper of their bottom side, which I then expanded and tried as accurately as possible to find out the radius of this circle 9.2 for the inside of concave lens and 12.8 for the outer part.

Do not forget that those glasses are concave convex lens which also means that the inside is -9.2 because it is “negative” of the glass.

When I gave it to the equation I found out that focal distance was 54.52 centimeters and dioptres -1.8342 which is not very close but since the way I was doing this was not meant to be very accurate I could not get anything better. (I took the refractive index of glass to be 1.6).


PS. this was my 100th post!
PPS. I will update about those glasses because I am not totally sure yet how they work so stay tuned.
Picture of equation
Picture of magnifier