A child has near point at 10 cm. What is the maximum angular magnification the child can have with a convex lens of focal length 10 cm?

Explanation

To find the maximum angular magnification that a child can have with a convex lens of focal length 10 cm, given that the child's near point is at 10 cm, we can follow these steps:

### Step 1: Identify the given values
- Focal length of the convex lens (F) = +10 cm
- Near point of the child (D) = 10 cm (this is the distance at which the child can see clearly)

### Step 2: Use the lens formula
The lens formula is given by:
\[ \frac{1}{F} = \frac{1}{V} - \frac{1}{U} \]
Where:
- \( F \) = focal length of the lens
- \( V \) = image distance
- \( U \) = object distance

### Step 3: Substitute known values into the lens formula
Since the near point is at 10 cm, we can consider the image distance \( V \) to be -10 cm (the negative sign indicates that the image is formed on the same side as the object for a virtual image). Now substituting the values into the lens formula:
\[ \frac{1}{10} = \frac{1}{-10} - \frac{1}{U} \]

### Step 4: Solve for U
Rearranging the equation to solve for \( \frac{1}{U} \):
\[ \frac{1}{U} = \frac{1}{-10} - \frac{1}{10} \]
\[ \frac{1}{U} = -\frac{1}{10} - \frac{1}{10} = -\frac{2}{10} \]
\[ \frac{1}{U} = -\frac{1}{5} \]
Thus,
\[ U = -5 \text{ cm} \]

### Step 5: Calculate the magnification
The magnification \( m \) of a lens is given by the formula:
\[ m = \frac{V}{U} \]
Substituting the values we found:
\[ m = \frac{-10}{-5} = 2 \]

### Conclusion
The maximum angular magnification the child can have with the convex lens is **2**.

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