Obtain the equation of magnification for compound microscope.

Explanation

Linear magnification due to the objective,
`m_0=(h.)/(h)`
where h. is dimension of first image.
`h_0` is dimension of object.
Here,tan`beta=(h)/(f_0) implies h= f_0tan beta ` … (1) and
`tan beta=(h.)/(B_1B.)`
`implies h.=L tan beta` ...(2)
[`because B_1B.=L]`
`therefore m_0=(h.)/(h)=(Ltan beta)/(f_0tan beta)=(L)/(f_0)`
`therefore m_0 =(L)/(f_0)` ...(3)
When the final image is formed at the near point is,
`m_e=1+(D)/(f_e)`
when the final image is formed at infinity, the angular magnification due to the eyepiece,
`m_e=(D)/(f_e)` ...(4)
Thus, the total magnification,
`m=m_0m_e=((L)/(f_0))((D)/(f_e))`
To achieve a large magnification of a small object (hence the name microscope), the objective and eyepiece should have small focal lengths.
Various other factors such as illumination of the object contribute to the quality and visibility of the image.
In modern microscopes, multicomponent lenses are used for both the objective and the eyepiece to improve image quality by minimising various optical aberrations (defects) in lenses.