In U - V method, the object distance is 30.0cm and image distance is 60.0cm from pole of concave mirror. The percentage error in measurement of focal length of mirror is `(n)/(18)%`. Find the value of n

Options

A. `5`
B. `4`
C. `3`
D. `2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Object distance (u) = -30.0 cm (negative for concave mirror)
- Image distance (v) = -60.0 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 \(v\):
\[ \frac{1}{f} = \frac{1}{-60} + \frac{1}{-30} \]

### Step 3: Calculate the focal length (f)
Calculating the right-hand side:
\[ \frac{1}{f} = -\frac{1}{60} - \frac{1}{30} \]
Finding a common denominator (which is 60):
\[ \frac{1}{f} = -\frac{1}{60} - \frac{2}{60} = -\frac{3}{60} = -\frac{1}{20} \]
Thus, the focal length \(f\) is:
\[ f = -20 \text{ cm} \]

### Step 4: Differentiate the mirror formula
To find the percentage error in the focal length, we differentiate the mirror formula:
\[ \frac{d}{df} \left(\frac{1}{f}\right) = \frac{d}{dv} + \frac{d}{du} \]
This gives:
\[ -\frac{df}{f^2} = \frac{dv}{v^2} + \frac{du}{u^2} \]

### Step 5: Substitute the errors
Let \( \Delta u \) and \( \Delta v \) be the errors in the measurements of \( u \) and \( v \). Assuming the least count of the measuring instrument is 0.1 cm:
\[ \Delta u = 0.1 \text{ cm}, \quad \Delta v = 0.1 \text{ cm} \]
Substituting these values into the differentiated equation:
\[ -\frac{\Delta f}{f^2} = \frac{0.1}{(-60)^2} + \frac{0.1}{(-30)^2} \]
Calculating the right-hand side:
\[ -\frac{\Delta f}{f^2} = \frac{0.1}{3600} + \frac{0.1}{900} \]
Finding a common denominator (which is 3600):
\[ -\frac{\Delta f}{f^2} = \frac{0.1}{3600} + \frac{0.4}{3600} = \frac{0.5}{3600} \]

### Step 6: Solve for \(\Delta f\)
Now, substituting \(f = -20\):
\[ -\frac{\Delta f}{(-20)^2} = \frac{0.5}{3600} \]
Calculating:
\[ -\frac{\Delta f}{400} = \frac{0.5}{3600} \]
Cross-multiplying gives:
\[ \Delta f = -\frac{0.5 \times 400}{3600} \]
Thus:
\[ \Delta f = -\frac{200}{3600} = -\frac{1}{18} \text{ cm} \]

### Step 7: Calculate the percentage error
The percentage error in the focal length is given by:
\[ \text{Percentage error} = \left(\frac{\Delta f}{f}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage error} = \left(\frac{\frac{1}{18}}{20}\right) \times 100 = \frac{100}{360} = \frac{5}{18} \% \]

### Step 8: Identify \(n\)
From the problem, the percentage error is given as \(\frac{n}{18}\%\). Therefore, we have:
\[ \frac{n}{18} = \frac{5}{18} \]
Thus, \(n = 5\).

### Final Answer
The value of \(n\) is \(5\).

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