A converging lens of focal length 6.25 cm is used as a magnifying glass. If the near point of the observer is 25 cm from the eye and the lens is held close to the eye, calculate (i) the distance of the object from the lens and (ii) the angular magnification. Find the angular magnification when the final image is formed at infinity.

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript and apply the relevant formulas.

### Step 1: Calculate the distance of the object from the lens

Given:
- Focal length of the lens, \( f = +6.25 \, \text{cm} \)
- Distance of the near point from the eye, \( v = -25 \, \text{cm} \) (negative because the image is formed on the same side as the object for a virtual image)

Using the lens formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( u \) is the object distance (which we need to find).

Rearranging the lens formula gives:
\[ \frac{1}{u} = \frac{1}{v} - \frac{1}{f} \]

Substituting the known values:
\[ \frac{1}{u} = \frac{1}{-25} - \frac{1}{6.25} \]

Calculating the right-hand side:
\[ \frac{1}{u} = -\frac{1}{25} - \frac{1}{6.25} \]

Finding a common denominator (which is 25):
\[ \frac{1}{u} = -\frac{1}{25} - \frac{4}{25} = -\frac{5}{25} = -\frac{1}{5} \]

Thus:
\[ u = -5 \, \text{cm} \]

### Step 2: Calculate the angular magnification

The formula for angular magnification \( M \) when the lens is used as a magnifying glass is given by:
\[ M = \frac{D}{|u|} \]
Where:
- \( D \) is the distance of the near point from the eye, which is \( 25 \, \text{cm} \).
- \( |u| \) is the absolute value of the object distance, which is \( 5 \, \text{cm} \).

Substituting the values:
\[ M = \frac{25}{5} = 5 \]

### Step 3: Find the angular magnification when the final image is formed at infinity

When the final image is formed at infinity, the object is placed at the focal length of the lens. Thus:
- \( u = -f = -6.25 \, \text{cm} \)

Using the same formula for angular magnification:
\[ M = \frac{D}{|u|} = \frac{25}{6.25} \]

Calculating this gives:
\[ M = 4 \]

### Final Answers:
1. The distance of the object from the lens is \( -5 \, \text{cm} \).
2. The angular magnification when the image is formed at the near point is \( 5 \).
3. The angular magnification when the final image is formed at infinity is \( 4 \).