A point object is placed at a distance 20 cm from the focus of a concave mirror of radius of curvature 20 cm. Find the distance (in cm) of the image from the focus.

Options

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Explanation

To solve the problem step by step, we will use the mirror formula and the properties of concave mirrors.

### Step 1: Identify the given data
- Radius of curvature (R) = 20 cm
- Distance of the object from the focus (d) = 20 cm

### Step 2: Calculate the focal length (f) of the concave mirror
The focal length (f) of a concave mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Step 3: Determine the object distance (U)
The object is placed at a distance of 20 cm from the focus. Since the focus is at 10 cm from the mirror, the object distance (U) from the mirror is:
\[ U = \text{distance from the focus} + f = 20 \, \text{cm} + 10 \, \text{cm} = 30 \, \text{cm} \]
(Note: In mirror convention, U is taken as negative for real objects, so U = -30 cm.)

### Step 4: Apply the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{V} + \frac{1}{U} \]
Substituting the values of f and U:
\[ \frac{1}{-10} = \frac{1}{V} + \frac{1}{-30} \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ \frac{1}{V} = \frac{1}{-10} + \frac{1}{30} \]

### Step 6: Finding a common denominator and solving for V
The common denominator for -10 and 30 is 30:
\[ \frac{1}{V} = \frac{-3}{30} + \frac{1}{30} \]
\[ \frac{1}{V} = \frac{-3 + 1}{30} = \frac{-2}{30} \]
Thus,
\[ V = \frac{-30}{2} = -15 \, \text{cm} \]

### Step 7: Calculate the distance of the image from the focus
The image distance (V) is -15 cm, which means the image is located 15 cm in front of the mirror. To find the distance of the image from the focus:
\[ \text{Distance from focus} = |V| - f = 15 \, \text{cm} - 10 \, \text{cm} = 5 \, \text{cm} \]

### Final Answer
The distance of the image from the focus is **5 cm**.

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