When a current of `(2.5 +- 0.5)` ampere flows through a wire, it develops a potential difference of `(20 +- 1)` volt, the resistance of the wire is

Options

A. `(8 +- 2) Omega`
B. `(10 +- 3) Omega`
C. `(18 +- 4) Omega`
D. `(20 +- 6) Omega`

Explanation

To solve the problem of finding the resistance of a wire when given a current and potential difference with their respective uncertainties, we will follow these steps:

### Step 1: Use Ohm's Law
Ohm's Law states that the resistance \( R \) can be calculated using the formula:
\[ R = \frac{V}{I} \]
where \( V \) is the potential difference and \( I \) is the current.

### Step 2: Substitute the Values
Given:
- Potential difference \( V = 20 \, \text{V} \)
- Current \( I = 2.5 \, \text{A} \)

Substituting these values into the formula:
\[ R = \frac{20 \, \text{V}}{2.5 \, \text{A}} = 8 \, \Omega \]

### Step 3: Calculate the Uncertainty in Resistance
To find the uncertainty in resistance \( \Delta R \), we will use the formula derived from the differentiation of the resistance equation:
\[ \frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I} \]
Rearranging gives:
\[ \Delta R = R \left( \frac{\Delta V}{V} + \frac{\Delta I}{I} \right) \]

### Step 4: Substitute the Uncertainties
Given:
- Uncertainty in potential difference \( \Delta V = 1 \, \text{V} \)
- Uncertainty in current \( \Delta I = 0.5 \, \text{A} \)

Now substituting the values:
- \( R = 8 \, \Omega \)
- \( \Delta V = 1 \, \text{V} \)
- \( V = 20 \, \text{V} \)
- \( \Delta I = 0.5 \, \text{A} \)
- \( I = 2.5 \, \text{A} \)

Calculating the relative uncertainties:
\[ \frac{\Delta V}{V} = \frac{1}{20} = 0.05 \]
\[ \frac{\Delta I}{I} = \frac{0.5}{2.5} = 0.2 \]

### Step 5: Calculate \( \Delta R \)
Now substituting these into the uncertainty formula:
\[ \Delta R = 8 \left( 0.05 + 0.2 \right) = 8 \times 0.25 = 2 \, \Omega \]

### Step 6: Final Result
The resistance with its uncertainty is:
\[ R \pm \Delta R = 8 \pm 2 \, \Omega \]

### Conclusion
Thus, the resistance of the wire is \( 8 \pm 2 \, \Omega \).

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