A person with a normal near point (25cm) using a compound microscope with objective of focal length 8.0 mm and an eyepiece of focal length 2.5cm can bring an object placed at 9.0mm from the objective in sharp focus. What is the separation between the two lenses? Calculate the magnifying power of the microscope,

To solve this problem, we need to determine two key aspects: the separation between the two lenses of the compound microscope and its magnifying power. Let's break this down step by step.

Understanding the Components

A compound microscope consists of two main lenses: the objective lens and the eyepiece. The objective lens is responsible for forming a real image of the object, while the eyepiece magnifies this image for the viewer. The focal lengths of these lenses are given as follows:

• Focal length of the objective lens (f_o): 8.0 mm
• Focal length of the eyepiece (f_e): 2.5 cm (which is 25 mm)

Finding the Separation Between the Lenses

The object is placed at a distance of 9.0 mm from the objective lens. To find the image distance (v_o) for the objective lens, we can use the lens formula:

1/f = 1/v + 1/u

Where:

• f = focal length of the lens
• v = image distance from the lens
• u = object distance from the lens (which is negative in lens convention)

Substituting the values for the objective lens:

1/8 = 1/v_o - 1/9

Rearranging gives:

1/v_o = 1/8 + 1/9

Finding a common denominator (72):

1/v_o = 9/72 + 8/72 = 17/72

Thus, v_o = 72/17 mm ≈ 4.24 mm.

The image formed by the objective lens acts as a virtual object for the eyepiece. The distance from the objective lens to the eyepiece (d) can be calculated as:

d = v_o + f_e

Substituting the values:

d = 4.24 mm + 25 mm = 29.24 mm.

Calculating the Magnifying Power

The total magnifying power (M) of a compound microscope can be calculated using the formula:

M = M_o × M_e

Where:

• M_o = magnification produced by the objective lens
• M_e = magnification produced by the eyepiece

First, we calculate M_o:

M_o = v_o / u = 4.24 / 9 = 0.471.

Next, we find M_e. The magnification of the eyepiece can be approximated as:

M_e = 1 + (D / f_e)

Where D is the near point distance (25 cm or 250 mm). Thus:

M_e = 1 + (250 / 25) = 1 + 10 = 11.

Now, substituting these values into the total magnification formula:

M = M_o × M_e = 0.471 × 11 ≈ 5.18.

Summary of Results

In conclusion, the separation between the two lenses of the microscope is approximately 29.24 mm, and the magnifying power of the microscope is around 5.18. This means that the microscope can make objects appear about 5.18 times larger than they actually are when viewed through the eyepiece.