On the axis of a spherical mirror of focal length `f` a short linear object of length `L` lies on the axis at a distance `mu` from the mirror. Its image has an axial length `L'` equal to

Options

A. `L[(f)/(u-f)]^(1//2)`
B. `L[(u+f)/(f)]^(1//2)`
C. `L[(u-f)/(f)]^(2)`
D. `L[(f)/(u-f)]^(2)`

Explanation

To find the axial length \( L' \) of the image formed by a spherical mirror when a short linear object of length \( L \) is placed at a distance \( \mu \) from the mirror, we can follow these steps:

### Step 1: Understand the setup
We have a spherical mirror with focal length \( f \) and a linear object of length \( L \) placed on the axis at a distance \( \mu \) from the mirror. The object can be represented as having two ends, which we will denote as point 1 and point 2.

### Step 2: Determine the object distances
For the two ends of the object:
- The distance of point 1 from the mirror is \( u_1 = \mu - \frac{L}{2} \)
- The distance of point 2 from the mirror is \( u_2 = \mu + \frac{L}{2} \)

### Step 3: Use the mirror formula to find image distances
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where \( v \) is the image distance and \( u \) is the object distance.

For point 1:
\[ \frac{1}{f} = \frac{1}{v_1} + \frac{1}{u_1} \]
Rearranging gives:
\[ v_1 = \frac{fu_1}{u_1 - f} \]

For point 2:
\[ \frac{1}{f} = \frac{1}{v_2} + \frac{1}{u_2} \]
Rearranging gives:
\[ v_2 = \frac{fu_2}{u_2 - f} \]

### Step 4: Calculate the image distances
Substituting \( u_1 \) and \( u_2 \):
1. For point 1:
\[ v_1 = \frac{f(\mu - \frac{L}{2})}{\mu - \frac{L}{2} - f} \]

2. For point 2:
\[ v_2 = \frac{f(\mu + \frac{L}{2})}{\mu + \frac{L}{2} - f} \]

### Step 5: Find the axial length of the image
The axial length \( L' \) of the image is given by the difference in the image distances:
\[ L' = v_2 - v_1 \]

### Step 6: Substitute the values of \( v_1 \) and \( v_2 \)
Substituting the expressions for \( v_1 \) and \( v_2 \):
\[ L' = \frac{f(\mu + \frac{L}{2})}{\mu + \frac{L}{2} - f} - \frac{f(\mu - \frac{L}{2})}{\mu - \frac{L}{2} - f} \]

### Step 7: Simplify the expression
To simplify \( L' \):
1. Find a common denominator.
2. Combine the fractions.
3. Factor out common terms.

After simplification, we find:
\[ L' = \frac{f^2 L}{(\mu - f)^2} \]

### Final Result
The axial length of the image \( L' \) is given by:
\[ L' = \frac{L f^2}{(\mu - f)^2} \]