Problem 97 - Arihant AIEEE Physics

Question

A compound microscope has an eye piece of focal length \(10 \mathrm{~cm}\) and an objective of focal length \(4 \mathrm{~cm}\). The magnification, if an object is kept at a distance of \(5 \mathrm{~cm}\) from the objective and final image is formed at the least distance of distinct vision \((20 \mathrm{~cm})\), is : (a) 10 (b) 11 (c) 12 (d) 13

Step-by-Step Solution

Answer

The magnification of the microscope is 12.

Step 1: Understand the Setup

We are dealing with a compound microscope, which has two lenses: an objective lens with a focal length of \( f_o = 4 \, \text{cm} \) and an eyepiece lens with a focal length of \( f_e = 10 \, \text{cm} \). The object is placed \( u_o = 5 \, \text{cm} \) from the objective lens, and the final image is at the least distance of distinct vision \( D = 20 \, \text{cm} \).

Step 2: Calculate Objective Lens Magnification

First, calculate the magnification produced by the objective lens. The formula for linear magnification by the objective is \( m_o = \frac{v_o}{u_o} \), where \( v_o \) is the image distance from the objective lens.We use the lens formula: \( \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \), which leads to:\[ \frac{1}{4} = \frac{1}{v_o} - \frac{1}{5} \]Solving this, we find \( v_o = 20 \, \text{cm} \). Therefore, \( m_o = \frac{20}{5} = 4 \).

Step 3: Calculate Eyepiece Lens Magnification

The magnification produced by the eyepiece lens is given by \( m_e = 1 + \frac{D}{f_e} \). Substituting the given values, we find:\[ m_e = 1 + \frac{20}{10} = 3 \].

Step 4: Calculate Total Magnification of the Microscope

The total magnification \( M \) of the microscope is the product of the magnifications by the objective and eyepiece lenses:\[ M = m_o \times m_e = 4 \times 3 = 12 \].

Key Concepts

Objective Lens Magnification

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 \( m_o = \frac{20}{5} = 4 \).

Eyepiece Lens Magnification

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.

Least Distance of Distinct Vision

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.