Why objective lens is small in compound microscope

Option 2 : To get maximum magnification

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CONCEPT:

• Microscope: The instrument having double convex lens with short focal length which produces an enlarged image of a small object is called as microscope.

• The microscope has two convex lenses- one of them is called as objective lens and other one is known as eyepiece.

EXPLANATION:

• Since the magnification of a compound microscope is inversely proportional to the focal length of objective lens.
• So to get the maximum magnification the focal length of the objective lens must be less than the focal length of eyepiece. So option 2 is correct.

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[ 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:- $$M = \Bigg[\frac{D}{f_e}+1\Bigg]\left[\frac{v_o}{f_o}+1\right]$$ 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.