A short linear object of length b lies along the axis of a concave mirror or focal length f at a distance u from the pole of the mirror. The size of the image is approximately equal to

Options

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

Explanation

To solve the problem of finding the size of the image formed by a concave mirror when a short linear object of length \( b \) is placed at a distance \( u \) from the pole of the mirror, we can follow these steps:

### Step 1: Understand the Mirror Formula
The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \) of the mirror. It is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

### Step 2: Differentiate the Mirror Formula
Since we are dealing with a small object and want to find the change in image size, we differentiate the mirror formula with respect to \( u \):
\[ 0 = -\frac{dv}{v^2} - \frac{du}{u^2} \]
This simplifies to:
\[ \frac{dv}{du} = -\frac{v^2}{u^2} \]

### Step 3: Relate Image Size to Object Size
The change in image size \( dv \) can be expressed in terms of the object size \( b \):
\[ dv = -\frac{v^2}{u^2} \cdot du \]
Substituting \( du = b \) (the length of the object), we get:
\[ dv = -\frac{v^2}{u^2} \cdot b \]

### Step 4: Express \( v \) in Terms of \( u \) and \( f \)
From the mirror formula, we can express \( v \) in terms of \( u \) and \( f \):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \implies v = \frac{fu}{u - f} \]

### Step 5: Substitute \( v \) into the Image Size Equation
Now, substitute \( v \) into the equation for \( dv \):
\[ dv = -\frac{\left(\frac{fu}{u - f}\right)^2}{u^2} \cdot b \]
This simplifies to:
\[ dv = -\frac{f^2 u^2}{(u - f)^2 u^2} \cdot b = -\frac{f^2}{(u - f)^2} \cdot b \]

### Step 6: Finalize the Size of the Image
The size of the image \( |dv| \) is given by:
\[ |dv| = \frac{f^2}{(u - f)^2} \cdot b \]

### Conclusion
Thus, the size of the image is approximately equal to:
\[ \text{Size of the image} = \frac{f^2}{(u - f)^2} \cdot b \]