A needle 10 cm long is placed along the axis of a convex lens of focal length 10 cm such that the middle point of the neddle is at a distance of 10 cm from the lens. Find the length of the image of the neddle.

Answer

To find the length of the image of a needle placed along the axis of a convex lens, we will follow these steps:

### Step 1: Understand the setup
- We have a needle that is 10 cm long.
- The middle point of the needle is placed 10 cm away from the lens.
- The focal length of the convex lens is also 10 cm.

### Step 2: Determine the positions of the ends of the needle
- Since the needle is 10 cm long, if the middle point is at 10 cm from the lens, the ends of the needle will be:
- Point A (one end) is at \(10 \, \text{cm} - 5 \, \text{cm} = 5 \, \text{cm}\) from the lens.
- Point B (the other end) is at \(10 \, \text{cm} + 5 \, \text{cm} = 15 \, \text{cm}\) from the lens.

### Step 3: Use the lens formula to find the image positions
The lens formula is given by:
\[ \frac{1}{V} - \frac{1}{U} = \frac{1}{F} \]
Where:
- \(V\) is the image distance,
- \(U\) is the object distance (negative for real objects),
- \(F\) is the focal length.

#### For Point A:
- The object distance \(U_A = -5 \, \text{cm}\) (since it's on the same side as the object).
- The focal length \(F = 10 \, \text{cm}\).

Using the lens formula:
\[ \frac{1}{V_A} = \frac{1}{U_A} + \frac{1}{F} \]
\[ \frac{1}{V_A} = \frac{1}{-5} + \frac{1}{10} \]
\[ \frac{1}{V_A} = -\frac{2}{10} + \frac{1}{10} = -\frac{1}{10} \]
\[ V_A = -10 \, \text{cm} \]

#### For Point B:
- The object distance \(U_B = -15 \, \text{cm}\).

Using the lens formula:
\[ \frac{1}{V_B} = \frac{1}{U_B} + \frac{1}{F} \]
\[ \frac{1}{V_B} = \frac{1}{-15} + \frac{1}{10} \]
\[ \frac{1}{V_B} = -\frac{2}{30} + \frac{3}{30} = \frac{1}{30} \]
\[ V_B = 30 \, \text{cm} \]

### Step 4: Calculate the length of the image
The length of the image of the needle is given by the distance between the images of points A and B:
\[ \text{Length of image} = |V_B - V_A| = |30 - (-10)| = |30 + 10| = 40 \, \text{cm} \]

### Conclusion
The length of the image of the needle is \(40 \, \text{cm}\).

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