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According to the thin lens equation $$\frac{1}{f} = \frac{1}{a} + \frac{1}{b}$$ and magnification is given as $$M = \frac{b}{a}$$ where b is image and a is object distance. So in order to get $M > 1$ we need $b>a$ and $a>f$ (for convex lens) to get a real image. My question is how do modern objectives achieve high magnifications of 20x and more on such small distance? Is it just lining up enough lenses to get a small enough effective $f$ or is there another principle behind it that makes it possible?

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we use multiple lenses to magnify the image and can use a liquid (special oil) with a very low refraction index such that the light coming from our specimen diverges less, which enables us to collect more light.

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" Is it just lining up enough lenses to get a small enough effective f or is there another principle behind it that makes it possible?"

For a 40x or 100x objective, the working distance (=a) is less than a mm . The image distance (=b) is typically 160mm (finite conjugate) to 200 mm (infinite conjugate + tube lens). (pls note the length of the objective , 20 to 40mm, should be accounted for somewhere)

So yes the magnification is achieved by shrinking the focal distance so that the working distance gets small enough compared to the image distance (limited by the dimentions of the microscope) in order to get the desired magnification.

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