To find index error `e` distance between object needle and poler of the concave mirror is `20 cm`. The separation between the indices of object needle and mirror was observed to be `20.2 cm` in some observation, the observed image distance is `20.2 cm` and the object distance is `30.2 cm` find
(a) the index error e.
(b) focal length of the mirror `f`.

Explanation

To solve the problem step by step, we will find the index error \( e \) and the focal length \( f \) of the concave mirror.

### Step 1: Calculate the Index Error \( e \)

The index error is defined as the difference between the observed value and the actual value.

Given:
- Observed distance between the object needle and the pole of the mirror = 20.2 cm
- Actual distance between the object needle and the pole of the mirror = 20 cm

Using the formula:
\[ e = \text{Observed value} - \text{Actual value} \]
Substituting the values:
\[ e = 20.2 \, \text{cm} - 20 \, \text{cm} = 0.2 \, \text{cm} \]

### Step 2: Calculate the Actual Image Distance \( v \)

The observed image distance is given as 20.2 cm. To find the actual image distance, we subtract the index error from the observed image distance.

Given:
- Observed image distance = 20.2 cm

Using the formula:
\[ \text{Actual image distance} = \text{Observed image distance} - e \]
Substituting the values:
\[ v = 20.2 \, \text{cm} - 0.2 \, \text{cm} = 20 \, \text{cm} \]

### Step 3: Calculate the Actual Object Distance \( u \)

The observed object distance is given as 30.2 cm. To find the actual object distance, we subtract the index error from the observed object distance.

Given:
- Observed object distance = 30.2 cm

Using the formula:
\[ \text{Actual object distance} = \text{Observed object distance} - e \]
Substituting the values:
\[ u = 30.2 \, \text{cm} - 0.2 \, \text{cm} = 30 \, \text{cm} \]

### Step 4: Apply the Mirror Formula to Find the Focal Length \( f \)

The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Since we are dealing with a concave mirror, both \( u \) and \( v \) will be negative:
- \( v = -20 \, \text{cm} \)
- \( u = -30 \, \text{cm} \)

Substituting the values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-20} + \frac{1}{-30} \]

Finding a common denominator (which is 60):
\[ \frac{1}{f} = \frac{-3}{60} + \frac{-2}{60} = \frac{-5}{60} \]

Taking the reciprocal to find \( f \):
\[ f = \frac{-60}{5} = -12 \, \text{cm} \]

### Final Answers
(a) The index error \( e = 0.2 \, \text{cm} \)

(b) The focal length of the mirror \( f = -12 \, \text{cm} \)

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