An object is kept at a distance of 60 cm from a concave mirror. For getting a magnification of `1/2`, focal length of the concave mirror required is

Options

A. 20 cm
B. 40 cm
C. - 20 cm
D. 30 cm

Answer

To solve the problem step by step, we will use the mirror formula and the magnification formula.

### Step 1: Understand the given information
- The object distance (U) is given as 60 cm. Since it is a concave mirror and the object is real, we take U as -60 cm.
- The magnification (m) is given as \( \frac{1}{2} \).

### Step 2: Use the magnification formula
The magnification (m) for mirrors is given by the formula:
\[ m = -\frac{V}{U} \]
Where:
- \( V \) is the image distance
- \( U \) is the object distance

Substituting the known values:
\[ \frac{1}{2} = -\frac{V}{-60} \]

### Step 3: Solve for V
Rearranging the equation to find \( V \):
\[ \frac{1}{2} = \frac{V}{60} \]
Cross-multiplying gives:
\[ V = 60 \times \frac{1}{2} = 30 \text{ cm} \]
Since the image is formed on the same side as the object in a concave mirror, we take \( V \) as -30 cm.

### Step 4: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Substituting the values of \( V \) and \( U \):
\[ \frac{1}{F} = \frac{1}{-30} + \frac{1}{-60} \]

### Step 5: Calculate \( \frac{1}{F} \)
Finding a common denominator (which is 60):
\[ \frac{1}{F} = \frac{-2}{60} + \frac{-1}{60} = \frac{-3}{60} \]
Thus:
\[ \frac{1}{F} = -\frac{1}{20} \]

### Step 6: Solve for F
Taking the reciprocal gives:
\[ F = -20 \text{ cm} \]

### Final Answer
The focal length of the concave mirror required is \( -20 \text{ cm} \).

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