Define the magnifying power of a compound microscope when the final image is formed at infinity. Why must both the objective and the eyepiece of a compound microscope have short focal lengths? Explain.

Magnifying Power of a Compound Microscope
The magnifying power (M) of a compound microscope when the final image is formed at infinity is the ratio of the angular size of the image to the angular size of the object when viewed with the naked eye. Mathematically, it is given by the product of the magnifying powers of the objective lens and the eyepiece.
The formula for magnifying power is:
M=Dfo×LfeM = \dfrac{D}{f_o} \times \dfrac{L}{f_e}
Where:
• DD is the least distance of distinct vision (typically 25 cm).
• fof_o is the focal length of the objective lens.
• LL is the length of the microscope tube (the distance between the objective and the eyepiece).
• fef_e is the focal length of the eyepiece.
Explanation of Terms:
1. Objective Lens:
o The objective lens is responsible for creating a magnified real image of the object placed at a short distance from it. The closer the object is to the focal point of the objective, the larger and clearer the image.
o The objective lens has a short focal length to create a large, real, and magnified image at a relatively small distance from the lens, which is then further magnified by the eyepiece.
2. Eyepiece:
o The eyepiece functions as a magnifier for the image formed by the objective. It is used to view the real image formed by the objective as a virtual image at a comfortable viewing distance.
o A short focal length for the eyepiece ensures that the final image is formed at a comfortable distance (usually at infinity) so that the eye does not strain.
Why Both the Objective and the Eyepiece Have Short Focal Lengths:
1. Objective Lens:
o The objective lens needs a short focal length to form a large and real image at a distance close to the objective. This allows the eyepiece to further magnify this image. The larger the image produced by the objective, the better the overall magnification of the microscope.
o A short focal length for the objective allows it to capture light from a small area (the object) and focus it into a large image, maximizing the initial magnification.
2. Eyepiece:
o The eyepiece needs a short focal length so that the image formed by the objective is viewed as a virtual image at a comfortable distance, typically at infinity. This ensures that the observer's eye does not experience fatigue or strain while focusing on the image.
o A short focal length of the eyepiece enables the microscope to produce a larger angular size of the final image when viewed by the eye. A longer focal length of the eyepiece would result in lower magnification.
Conclusion:
• The magnifying power of a compound microscope when the final image is formed at infinity is given by the product of the magnifying powers of the objective and the eyepiece.
• Both the objective lens and the eyepiece need to have short focal lengths:
o The objective lens has a short focal length to create a large, real image close to the objective, which can be magnified by the eyepiece.
o The eyepiece has a short focal length to magnify the real image created by the objective and to form a virtual image at a comfortable viewing distance, usually at infinity, for the observer's eye.