Rajendra conducted an experiment using a concave mirror with a focal length of 30 cm. He places an object at a distance of 20 cm in front of the mirror. The image is formed

Options

A. 20 cm in front of the mirror
B. 10 cm in front of the mirror
C. 30 cm benind the mirror
D. 60 cm behind the mirror

Explanation

To solve the problem step by step, we will use the mirror formula and the sign conventions for concave mirrors.

### Step 1: Identify the given values
- Focal length (f) of the concave mirror = -30 cm (negative because it is a concave mirror)
- Object distance (u) = -20 cm (negative as per sign convention since the object is in front of the mirror)

### Step 2: Write down the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) = focal length
- \( v \) = image distance
- \( u \) = object distance

### Step 3: Substitute the known values into the mirror formula
Substituting the values of \( f \) and \( u \):
\[ \frac{1}{-30} = \frac{1}{v} + \frac{1}{-20} \]

### Step 4: Rearrange the equation to find \( \frac{1}{v} \)
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{-30} + \frac{1}{20} \]

### Step 5: Find a common denominator and solve for \( v \)
The common denominator for -30 and 20 is 60. Thus, we can rewrite the fractions:
\[ \frac{1}{-30} = \frac{-2}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Now substituting these values:
\[ \frac{1}{v} = \frac{-2}{60} + \frac{3}{60} = \frac{1}{60} \]

### Step 6: Solve for \( v \)
Taking the reciprocal gives:
\[ v = 60 \text{ cm} \]

### Step 7: Determine the position of the image
Since \( v \) is positive, this indicates that the image is formed on the same side as the object, which means it is formed behind the mirror.

### Conclusion
The image is formed 60 cm behind the mirror.

### Final Answer
The correct option is: **60 cm behind the mirror.**

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