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, follow these steps:

### Step 1: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
For a concave mirror, both object distance (u) and image distance (v) are negative. Given:
- Object distance, \( u = -10.0 \, \text{cm} \)
- Image distance, \( v = -10.0 \, \text{cm} \)

### Step 2: Substitute Values into the Mirror Formula
Substituting the values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-10} + \frac{1}{-10} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{1}{10} - \frac{1}{10} = -\frac{2}{10} = -\frac{1}{5} \]

### Step 3: Calculate the Focal Length
Taking the reciprocal to find \( f \):
\[ f = -5.0 \, \text{cm} \]

### Step 4: Determine the Errors in Measurements
The minimum measurable distance is given as \( \Delta u = 0.1 \, \text{cm} \) and \( \Delta v = 0.1 \, \text{cm} \).

### Step 5: Differentiate the Mirror Formula
Differentiating the mirror formula gives:
\[ \frac{\Delta f}{f^2} = \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \]

### Step 6: Rearrange for Maximum Error
Rearranging for \( \Delta f \):
\[ |\Delta f| = f^2 \left( \frac{|\Delta u|}{u^2} + \frac{|\Delta v|}{v^2} \right) \]

### Step 7: Substitute Known Values
Substituting the known values into the equation:
\[ |\Delta f| = (-5)^2 \left( \frac{0.1}{(-10)^2} + \frac{0.1}{(-10)^2} \right) \]
Calculating:
\[ |\Delta f| = 25 \left( \frac{0.1}{100} + \frac{0.1}{100} \right) \]
\[ |\Delta f| = 25 \left( 0.001 + 0.001 \right) = 25 \times 0.002 = 0.05 \, \text{cm} \]

### Step 8: Write the Final Result
The focal length with its maximum permissible error is:
\[ f = -5.0 \pm 0.05 \, \text{cm} \]