An infinitely long rod lies along the axis of a concave mirror of focal length f. The near end of the rod is distance `u gt f` from the mirror. Its image will have length

Options

A. `(uf)/(u-f)`
B. `(uf)/(u+f)`
C. `(f^(2))/(u+f)`
D. `(f^(2))/(u-f)`

Answer

To solve the problem of finding the image length of an infinitely long rod placed along the axis of a concave mirror, we can follow these steps:

### Step 1: Understand the Setup
We have a concave mirror with a focal length \( f \) and an infinitely long rod positioned such that the near end of the rod is at a distance \( u \) from the mirror, where \( u > f \).

### Step 2: Use the Mirror Formula
The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \):
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
Since we are dealing with a concave mirror, the object distance \( u \) will be taken as negative:
\[ \frac{1}{v} - \frac{1}{|u|} = -\frac{1}{f} \]
Thus, we can rearrange this to find \( v \):
\[ \frac{1}{v} = \frac{1}{|u|} - \frac{1}{f} \]

### Step 3: Solve for Image Distance \( v \)
Rearranging gives:
\[ v = \frac{1}{\frac{1}{|u|} - \frac{1}{f}} = \frac{uf}{f - u} \]
Here, we have used the fact that \( |u| = u \) since \( u \) is given as positive and greater than \( f \).

### Step 4: Determine the Image Length
The image of the infinitely long rod will also be infinitely long, but we need to find the distance from the mirror to the image of the near end of the rod. The image length \( L' \) can be calculated as:
\[ L' = v - f \]
Substituting the expression for \( v \):
\[ L' = \frac{uf}{f - u} - f \]
To simplify this:
\[ L' = \frac{uf - f(f - u)}{f - u} = \frac{uf - f^2 + uf}{f - u} = \frac{2uf - f^2}{f - u} \]

### Step 5: Final Image Length
Thus, the length of the image of the rod is given by:
\[ L' = \frac{f^2}{u - f} \]

### Conclusion
The image length of the infinitely long rod is:
\[ \frac{f^2}{u - f} \]