with the help of a neat labelled ray diagram, show the image formation by a compound microscope when the final image is formed at the least distance of distinct vision. write the expression for its magnifying power in terms of parameters related to the microscope.
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When ever high magnification is desired, a compound microscope is used. It consists of two convex lenses, one placed before the object calledObjective and other placed before the eye called Eye-piece.
The objective has very small focal length  and small aperture, whereas eye-piece has relatively large focal length  and large aperture.
Working
When a small object OP is placed at distance pjust before the focus of objective then a real, inverted and magnified image IQ is formed at distance qwhich falls within the focus of eye-piece. The eye-piece is used as magnifying glass and it takes up the image IQ as an object and since it lies within the focal length of eye-piece then it forms a virtual and much magnified final image I'Q' as shown below.
 Fig Ray diagram of Compound Microscope
Magnification
If  and  are the visual angles subtended by the object and image respectively when formed at the least distance of distinct vision. Then magnifying power of microscope will be
 
where , when object OP is placed at least distance of distinct vision in front of eye.  for small angle.
And in right angle triangle ∆Q'XI'


Now putting the values of  and  in equation (1)we get,


It can be written as,

 
where
M1 is the magnification of objective
M2 is the magnification of eye-piece
As,

And since eye-piece is used as magnifying glass then,

Therefore equation(2) can be written as,
 
This expression gives the magnification of eye-piece.
As the object is placed very close to the focus of objective then,

And since IQ is formed very close to the eye-piece then,

Where L is the distance between objective and eye-piece and it is also known as length of the microscope. Now equation(3) may be written as,

The limit to which a microscope can be used to resolve details, depends on the width of the objective. A wider objective and use of blue light of short wavelength produces less diffraction and allows more details to be seen.
The objective has very small focal length  and small aperture, whereas eye-piece has relatively large focal length  and large aperture.
Working
When a small object OP is placed at distance pjust before the focus of objective then a real, inverted and magnified image IQ is formed at distance qwhich falls within the focus of eye-piece. The eye-piece is used as magnifying glass and it takes up the image IQ as an object and since it lies within the focal length of eye-piece then it forms a virtual and much magnified final image I'Q' as shown below.
 Fig Ray diagram of Compound Microscope
Magnification
If  and  are the visual angles subtended by the object and image respectively when formed at the least distance of distinct vision. Then magnifying power of microscope will be
 
where , when object OP is placed at least distance of distinct vision in front of eye.  for small angle.
And in right angle triangle ∆Q'XI'


Now putting the values of  and  in equation (1)we get,


It can be written as,

 
where
M1 is the magnification of objective
M2 is the magnification of eye-piece
As,

And since eye-piece is used as magnifying glass then,

Therefore equation(2) can be written as,
 
This expression gives the magnification of eye-piece.
As the object is placed very close to the focus of objective then,

And since IQ is formed very close to the eye-piece then,

Where L is the distance between objective and eye-piece and it is also known as length of the microscope. Now equation(3) may be written as,

The limit to which a microscope can be used to resolve details, depends on the width of the objective. A wider objective and use of blue light of short wavelength produces less diffraction and allows more details to be seen.
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