In `u-v` method to find focal length of a concave mirror, if object distance is found to be `10.0 cm` and image distance was also found to be `10.0 cm`, then find maximum permissible error in (f).

Explanation

To find the maximum permissible error in the focal length (f) of a concave mirror using the `u-v` method, we can follow these steps:

### Step 1: Identify the given values
- Object distance (u) = -10.0 cm (negative for concave mirror)
- Image distance (v) = -10.0 cm (negative for concave mirror)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of u and v:
\[ \frac{1}{f} = \frac{1}{-10} + \frac{1}{-10} \]
\[ \frac{1}{f} = -\frac{1}{10} - \frac{1}{10} = -\frac{2}{10} \]
Thus,
\[ f = -5 \text{ cm} \]

### Step 3: Differentiate the mirror formula
To find the maximum permissible error in f, we differentiate the mirror formula:
\[ -\frac{dF}{F^2} = -\frac{dV}{V^2} - \frac{dU}{U^2} \]
Taking the absolute values, we have:
\[ \frac{dF}{F^2} = \frac{dV}{V^2} + \frac{dU}{U^2} \]

### Step 4: Determine the least count (error) in measurements
Assuming the least count of the measuring instrument is 0.1 cm:
- \(dV = 0.1 \text{ cm}\)
- \(dU = 0.1 \text{ cm}\)

### Step 5: Substitute the values into the error formula
Substituting the values into the differentiated equation:
\[ dF = F^2 \left( \frac{dV}{V^2} + \frac{dU}{U^2} \right) \]
Where:
- \(F = -5 \text{ cm}\)
- \(V = -10 \text{ cm}\)
- \(U = -10 \text{ cm}\)

Calculating \(dF\):
\[ dF = (-5)^2 \left( \frac{0.1}{(-10)^2} + \frac{0.1}{(-10)^2} \right) \]
\[ dF = 25 \left( \frac{0.1}{100} + \frac{0.1}{100} \right) \]
\[ dF = 25 \left( 0.001 + 0.001 \right) = 25 \times 0.002 = 0.05 \text{ cm} \]

### Step 6: Conclusion
The maximum permissible error in the focal length (f) is:
\[ \Delta f = 0.05 \text{ cm} \]
Thus, the reading of the focal length can be expressed as:
\[ f = -5 \pm 0.05 \text{ cm} \]