The resistance of a metal is given by `R = V//I` , where `V` is potential difference and `I` is current . In a circuit , the potential difference across resistance is `V = ( 8 +- 0.5) V` and current in resistance , `I = ( 4 +- 0.2)A`. What is the value of resistance with its percentage error?

Options

A. `( 2 +- 5.6%) Omega`
B. `( 2 +- 0.7%) Omega`
C. `( 2 +- 35%) Omega`
D. `( 2 +- 11.25%) Omega`

Explanation

To solve the problem, we need to calculate the resistance \( R \) using the formula \( R = \frac{V}{I} \), where \( V \) is the potential difference and \( I \) is the current. We also need to determine the percentage error in the resistance based on the errors in the measurements of \( V \) and \( I \).

### Step 1: Calculate the resistance \( R \)

Given:
- \( V = 8 \, \text{V} \)
- \( I = 4 \, \text{A} \)

Using the formula:
\[ R = \frac{V}{I} = \frac{8 \, \text{V}}{4 \, \text{A}} = 2 \, \Omega \]

### Step 2: Determine the errors in \( V \) and \( I \)

Given:
- Error in \( V \) (denoted as \( \Delta V \)) = \( 0.5 \, \text{V} \)
- Error in \( I \) (denoted as \( \Delta I \)) = \( 0.2 \, \text{A} \)

### Step 3: Calculate the relative error in resistance \( R \)

The formula for the relative error in \( R \) is given by:
\[ \frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I} \]

Substituting the values:
\[ \frac{\Delta R}{R} = \frac{0.5}{8} + \frac{0.2}{4} \]

Calculating each term:
\[ \frac{0.5}{8} = 0.0625 \]
\[ \frac{0.2}{4} = 0.05 \]

Adding these values together:
\[ \frac{\Delta R}{R} = 0.0625 + 0.05 = 0.1125 \]

### Step 4: Calculate the absolute error in \( R \)

Now, we can find the absolute error \( \Delta R \):
\[ \Delta R = R \times \frac{\Delta R}{R} = 2 \times 0.1125 = 0.225 \, \Omega \]

### Step 5: Calculate the percentage error in \( R \)

The percentage error is given by:
\[ \text{Percentage error} = \left( \frac{\Delta R}{R} \right) \times 100 = 0.1125 \times 100 = 11.25\% \]

### Final Result

Thus, the resistance with its percentage error is:
\[ R = 2 \, \Omega \pm 11.25\% \]