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Wize University Physics Textbook (Master) > Geometric Optics

Microscopes and Telescopes

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Angular Size



The human eye sees a maximum angular size at the near point (2525 cm):



The maximum angular size the eye can focus is:

 φmax=tan⁡−1(ho25 cm)≈(ho25 cm) \boxed{ \ \varphi_{max}=\tan^{-1}\left(\dfrac{h_o}{25 \ \text{cm}}\right)\approx \bigg(\dfrac{h_o}{25 \ \text{cm}}\bigg) \ }


where we have used d0=25 cmd_0=25 \ \text{cm}, and also the small angle approximation tan⁡θ≈θ\tan\theta\approx\theta.



The angular magnification is defined as:

 m=φlensφ \boxed { \ m=\frac{\varphi_{lens}}{\varphi} \ }


Combining this with the thin-lens equation yields:

 mmax=25 cmf+1 \boxed{ \ m_{max}=\frac{25~\rm cm}{f}+1 \ }



Microscopes


A compound light microscope uses two converging lenses to amplify magnification.

  • The first lens (objective) has a short focal length, so that objects placed outside this focal length form a real, inverted, magnified image.
  • The second lens (ocular) has a longer focal length.
  • The image formed by first lens is located inside the focal length of the second, so the second lens produces a further magnified, virtual image that is then seen by the eye.





The total magnification mm is the product of the magnifications mobjectivem_{objective} & meyepiecem_{eyepiece}produced by the objective & eyepiece lenses:

 mtotal=mobjective×meyepiece \boxed{ \ m_{total}=m_{objective}\times m_{eyepiece} \ }


This can also be written as:

 mtotal=(−Lfo)(25 cmfe) \boxed{ \ m_{total}=\bigg(-\dfrac{L}{f_{o}}\bigg)\bigg(\dfrac{25 \ \text{cm}}{f_{e}}\bigg) \ }

  • LL is the length of the microscope's tube (the distance between the lenses)
  • fof_{o} is the focal length of the objective lens
  • fef_{e} is the focal length of the eyepiece lens

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Telescopes




Telescopes are optical instruments used to see objects at far distances. Their construction can be of two types: reflecting or refracting.

Refracting telescopes

Refracting telescopes work with the same mechanism as the compound microscope. In this case, the distance between the objective and ocular lens is the length of the telescope tube.

Reflecting telescopes

Reflecting telescopes work by gathering as much light as possible to provide a detailed image. They consist of a large, concave, parabolic mirror. Light from very distant objects is focused in the mirror, producing a magnified, inverted virtual image.

Reflecting telescopes are useful for looking at stars and planets in space, where enough detail cannot be seen with a refracting telescope.

Example: Angular Magnification


Which of the following option improves the largest angular magnification for the compound microscope?

A) Increase the focal length for both the objective and the eyepiece
B) Increase the focal length of the objective, but decrease that of the eyepiece
C) Decrease the focal length of the objective, but increase that of the eyepiece
D) Decrease the focal length of both the objective and the eyepiece


M=(−Lfo)(1+25 cmfe)M=\left(-\frac{L}{f_o}\right)\left(1+\frac{25~\rm cm}{f_e}\right)
Decreasing either the focal length of the objective or that of the eyepiece increases the magnification. Therefore, decreasing both will improve the magnification is the most.

The answer is D).




Solvia wants her new compound microscope to be able to have a magnification of 100. She already bought an eyepiece with a focal length of 1 cm and an objective of focal length 5 cm. What must be the distance between the two lenses for the maximum magnification to be as she desires?