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I think compound microscope will also work if focal length of eyepiece is kept smaller than that of objective lens. I am just a beginner in physical optics. I understand ray diagram of compound microscope. Kindly explain in an easy way.

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\Diagram for a Compund Microscope

[ Image from http://www.schoolphysics.co.uk/age16-19/Optics/Optical%20instruments/text/Microscope_/index.html]

Theoretically speaking, there is no problem at all which of the focal lengths is greater as the formula for magnification is given by:- $$M = \frac{D}{f_e-1}\left[\frac{v}{f_O}-1\right]$$ Therefore whether $f_O > f_e$ or $f_O < f_e$, you can still acheive magnification.

However, usually when you are usually observing with a compound microscope, conventionally your sample is extremely close to the objective and hence we use a small focal length for the objective and in most cases, the eye-piece ends up having a greater focal length. The reverse case will also work theoretically, but as I pointed out there are practical limitations.

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The objective lens is placing an image in the right place for the eyepiece to magnify. This can be much closer to the eyepiece than would be easy to achieve physically, and placed much more precise. So it can give higher magnification overall. The objective lens can be large and collect a large amount of light, creating a bright highly magnified image. The eyepiece lens is designed to work on the scale of a human pupil and deliver the image to the retina.

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Eye piece have moderate focal length. If it is decreased after a certain limit, the focus of eye piece would be very close to the optical centre of the eye piece and therefore the eye piece would suffer spherical aberration. The image formed by the eye piece would no longer be clear.

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