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In a compound microscope, the objective lens plays a crucial role in magnifying an object close to it. When you place an object near this lens, it creates an enlarged image. The extent of this enlargement is the "magnification" provided by the objective lens.
To calculate this magnification, we use the formula for linear magnification:

• \( m_o = \frac{v_o}{u_o} \)

Here, \( m_o \) is the magnification by the objective lens, \( v_o \) is the image distance from the objective lens, and \( u_o \) is the object distance from the objective lens.
Using the lens formula, we can find \( v_o \):

• \( \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \)

where \( f_o \) is the focal length of the objective lens. For example, if \( u_o = 5 \text{ cm} \) and \( f_o = 4 \text{ cm} \), solving gives us \( v_o = 20 \text{ cm} \). This gives an objective lens magnification of \( m_o = \frac{20}{5} = 4 \).

The eyepiece lens in a compound microscope is responsible for magnifying the image formed by the objective lens. Think of it as a simple magnifying glass, further enlarging the already magnified image.
The magnification of the eyepiece lens is determined using:

• \( m_e = 1 + \frac{D}{f_e} \)

In this equation, \( m_e \) is the magnification by the eyepiece lens, \( D \) is the least distance of distinct vision (often taken as 25 cm for normal human vision), and \( f_e \) is the focal length of the eyepiece lens.
In practice, if \( D = 20 \text{ cm} \) and \( f_e = 10 \text{ cm} \), you can solve for \( m_e \) as follows:
\[ m_e = 1 + \frac{20}{10} = 3 \] This effectively implies that the eyepiece lens magnifies the image formed by the objective lens three times.

The least distance of distinct vision is an important factor in mastering how microscopes work. It is the minimum distance at which the human eye can see an object clearly, without any strain, typically around 25 cm. However, in certain problems and environments, this value might differ slightly, like in our example where it is 20 cm.
This distance is important for calculating eyepiece magnification because it serves as a reference point for forming the final image by the eyepiece. It affects how comfortably you can focus on an image with the most clarity.
When visualizing through a microscope:

• The image must appear at or around this least distance to be clearly distinct.
• This is why \( D \) is used in the eyepiece magnification formula \( m_e = 1 + \frac{D}{f_e} \).

Essentially, it helps ensure the final image is at a distance where your vision is optimized for perception, leading to precise and effortless viewing through the microscope.