An object is placed at a distance of 20 cm from a concave mirror of focal lenght 10 cm. What is the image distance?

Options

A. `+20cm`
B. `+10 cm`
C. `-20 cm`
D. `-10 cm`

Answer

To find the image distance for an object placed in front of a concave mirror, we can use the mirror formula:

\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Where:
- \( f \) = focal length of the mirror
- \( v \) = image distance
- \( u \) = object distance

### Step 1: Identify the given values
- Focal length \( f = -10 \) cm (negative because it is a concave mirror)
- Object distance \( u = -20 \) cm (negative as per the sign convention for mirrors)

### Step 2: Substitute the values into the mirror formula
Using the mirror formula:

\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Substituting the known values:

\[ \frac{1}{-10} = \frac{1}{v} + \frac{1}{-20} \]

### Step 3: Rearranging the equation
Rearranging the equation to isolate \( \frac{1}{v} \):

\[ \frac{1}{v} = \frac{1}{-10} + \frac{1}{20} \]

### Step 4: Finding a common denominator
The common denominator for -10 and 20 is 20. Thus, we rewrite the fractions:

\[ \frac{1}{v} = \frac{-2}{20} + \frac{1}{20} \]

### Step 5: Simplifying the equation
Now, combine the fractions:

\[ \frac{1}{v} = \frac{-2 + 1}{20} = \frac{-1}{20} \]

### Step 6: Inverting to find \( v \)
Now, take the reciprocal to find \( v \):

\[ v = -20 \text{ cm} \]

### Conclusion
The image distance \( v \) is -20 cm, which indicates that the image is formed 20 cm in front of the mirror (on the same side as the object).