The focal lengths of the objective and eyelens of a microscope are 1.6 cm and 2.5 cm respectively. The distance between the two lenses is 21.7 cm. If the final image is formed at infinity, the distance between the object and the objective lens is :

Options

A. 1.8 cm
B. 1.70 cm
C. 1.65 cm
D. 1.75 cm

Explanation

To solve the problem step by step, we will use the lens formula and the given information about the focal lengths and distances.

### Step 1: Identify the given values
- Focal length of the objective lens, \( F_0 = 1.6 \, \text{cm} \)
- Focal length of the eyepiece lens, \( F_e = 2.5 \, \text{cm} \)
- Distance between the two lenses, \( L = 21.7 \, \text{cm} \)

### Step 2: Understand the image formation
Since the final image is formed at infinity, the image formed by the objective lens (first image) must be at the focal point of the eyepiece lens. Therefore, the distance from the eyepiece lens to the first image is equal to the focal length of the eyepiece.

### Step 3: Calculate the distance of the first image from the objective lens
Let \( V_0 \) be the distance of the first image from the objective lens. The relationship can be expressed as:
\[ V_0 = L - F_e \]
Substituting the values:
\[ V_0 = 21.7 \, \text{cm} - 2.5 \, \text{cm} = 19.2 \, \text{cm} \]

### Step 4: Use the lens formula for the objective lens
The lens formula is given by:
\[ \frac{1}{V} - \frac{1}{U} = \frac{1}{F_0} \]
Where:
- \( V = V_0 = 19.2 \, \text{cm} \) (the image distance for the objective lens)
- \( U \) is the object distance we need to find.

Rearranging the lens formula to find \( U \):
\[ \frac{1}{U} = \frac{1}{V_0} - \frac{1}{F_0} \]
Substituting the values:
\[ \frac{1}{U} = \frac{1}{19.2} - \frac{1}{1.6} \]

### Step 5: Calculate \( \frac{1}{U} \)
Calculating the individual fractions:
\[ \frac{1}{19.2} \approx 0.05208 \]
\[ \frac{1}{1.6} = 0.625 \]
Now substituting these values:
\[ \frac{1}{U} = 0.05208 - 0.625 = -0.57292 \]

### Step 6: Calculate \( U \)
Taking the reciprocal to find \( U \):
\[ U = \frac{1}{-0.57292} \approx -1.75 \, \text{cm} \]

### Conclusion
The distance between the object and the objective lens is approximately \( 1.75 \, \text{cm} \) (the negative sign indicates that the object is placed in the opposite direction of the light).

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