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

Options

A. `b((u-f)/(f))^(1/2)`
B. `b((f)/(u-f))^(1/2)`
C. `b((u-f)/(f))`
D. `b((f)/(u-f))^(2)`

Explanation

To find the size of the image of a short linear object of length \( b \) that lies along the axis of a concave mirror with focal length \( f \) at a distance \( u \) from the pole of the mirror, we can follow these steps:

### Step 1: Understand the Geometry
Consider the object as a short linear rod of length \( b \) positioned along the axis of the concave mirror. Let point A and point B be the two ends of the rod. The distance from the mirror's pole to the object is \( u \).

### Step 2: Use the Mirror Formula
The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \) of the mirror:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
From this, 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 3: Calculate the Magnification
The magnification \( m \) of the mirror is given by:
\[ m = -\frac{v}{u} \]
Substituting the expression for \( v \):
\[ m = -\frac{fu}{u - f} \cdot \frac{1}{u} = -\frac{f}{u - f} \]

### Step 4: Find the Size of the Image
The size of the image \( h' \) can be calculated using the magnification:
\[ h' = m \cdot h \]
Where \( h \) is the size of the object, which is \( b \):
\[ h' = -\frac{f}{u - f} \cdot b \]
Since we are interested in the approximate size of the image, we can ignore the negative sign (which indicates the image is inverted):
\[ h' \approx \frac{f}{u - f} \cdot b \]

### Final Answer
Thus, the size of the image is approximately:
\[ h' \approx \frac{f}{u - f} \cdot b \]

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