Resistance of a given wire is obtained by mea suring the current flowing in it and the voltage difference applied across it . If the percentage errors in the measurement of the current and the voltage difference are `3%` each , then errors in the value of resistance of the wire is

Options

A. `6%`
B. zero
C. `1%`
D. `3%`

Answer

To find the percentage error in the resistance of a wire when the percentage errors in the measurements of current and voltage are given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship**: The resistance \( R \) of a wire is given by Ohm's law, which states:
\[ R = \frac{V}{I} \]
where \( V \) is the voltage across the wire and \( I \) is the current flowing through it.

2. **Identify the errors**: We are given that the percentage error in the measurement of current \( I \) is \( 3\% \) and the percentage error in the measurement of voltage \( V \) is also \( 3\% \).

3. **Use the formula for error propagation**: The formula for the percentage error in resistance \( R \) when dealing with division is:
\[ \frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I} \]
where \( \Delta R \) is the absolute error in resistance, \( \Delta V \) is the absolute error in voltage, and \( \Delta I \) is the absolute error in current.

4. **Convert percentage errors to decimal form**: The percentage errors can be expressed as:
\[ \frac{\Delta V}{V} = 0.03 \quad \text{and} \quad \frac{\Delta I}{I} = 0.03 \]

5. **Substitute the errors into the formula**: Now, substitute these values into the error propagation formula:
\[ \frac{\Delta R}{R} = 0.03 + 0.03 = 0.06 \]

6. **Convert back to percentage**: To express this as a percentage, multiply by 100:
\[ \text{Percentage error in } R = 0.06 \times 100 = 6\% \]

### Final Answer:
The percentage error in the value of resistance of the wire is \( 6\% \).

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