An object is to be seen through a simple microscope of focal length 12 cm. Where should the object be placed so as to produce maximum angular magnification? The least distance for clear vision is 25 cm.

Explanation

To solve the problem of where the object should be placed to produce maximum angular magnification using a simple microscope with a focal length of 12 cm, we can follow these steps:

### Step 1: Understand the conditions for maximum angular magnification
For maximum angular magnification, the image should be formed at the least distance of distinct vision (D), which is given as 25 cm. Since the image is virtual, we take it as negative in the lens formula. Therefore, we have:
\[ v = -D = -25 \text{ cm} \]

### Step 2: Use the lens formula
The lens formula relates the object distance (u), image distance (v), and focal length (f) of the lens:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
We can rearrange this to find \( \frac{1}{u} \):
\[ \frac{1}{u} = \frac{1}{v} - \frac{1}{f} \]

### Step 3: Substitute the known values
Now, substitute \( v = -25 \) cm and \( f = 12 \) cm into the equation:
\[ \frac{1}{u} = \frac{1}{-25} - \frac{1}{12} \]

### Step 4: Find a common denominator and calculate
The common denominator for 25 and 12 is 300. Thus, we can rewrite the fractions:
\[ \frac{1}{u} = -\frac{12}{300} - \frac{25}{300} = -\frac{37}{300} \]

### Step 5: Solve for u
Now, take the reciprocal to find \( u \):
\[ u = -\frac{300}{37} \text{ cm} \]
Calculating this gives:
\[ u \approx -8.1 \text{ cm} \]

### Conclusion
The object should be placed approximately 8.1 cm away from the lens (on the same side as the object).

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