Question

The potential difference across the ends of a conductor is \( (30 \pm 0.3) V \) and the current through the conductor is \( (5 \pm 0.1) A \). The error in the determination of the resistance of the conductor is:

• A. \( 1% \)
• B. \( 2% \)
• C. \( 3% \)
• D. \( 4% \)

Hint:

When dividing two measured quantities, the percentage errors are added to obtain the overall percentage error.

Answer 1

The Correct Option is C

Step 1: Use the Formula for Resistance and Relative Error According to Ohm’s Law, resistance is given by: \[ R = \frac{V}{I}. \] The formula for the relative error in resistance is: \[ \frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I}. \]
Step 2: Substituting Given Values Given values: \[ V = 30V, \quad \Delta V = 0.3V, \quad I = 5A, \quad \Delta I = 0.1A. \] Calculating percentage errors: \[ \frac{\Delta V}{V} = \frac{0.3}{30} \times 100 = 1%. \] \[ \frac{\Delta I}{I} = \frac{0.1}{5} \times 100 = 2%. \] \[ \frac{\Delta R}{R} = 1% + 2% = 3%. \] % Final Answer Thus, the correct answer is option (3): \( 3% \).

Answer 2

To determine the percentage error in the calculation of resistance, we begin with Ohm's law, which states:

\( R = \frac{V}{I} \)

where:

• \( V = (30 \pm 0.3) \, \text{V} \) is the potential difference
• \( I = (5 \pm 0.1) \, \text{A} \) is the current

The absolute error in the resistance \( R \) can be determined using the formula for the propagation of relative errors in division:

\( \frac{\Delta R}{R} = \sqrt{\left(\frac{\Delta V}{V}\right)^2 + \left(\frac{\Delta I}{I}\right)^2} \)

Substituting the given values:

• \( \Delta V = 0.3 \, \text{V} \)
• \( V = 30 \, \text{V} \)
• \( \Delta I = 0.1 \, \text{A} \)
• \( I = 5 \, \text{A} \)

Calculate the relative error:

\( \frac{\Delta V}{V} = \frac{0.3}{30} = 0.01 \)

\( \frac{\Delta I}{I} = \frac{0.1}{5} = 0.02 \)

\( \frac{\Delta R}{R} = \sqrt{(0.01)^2 + (0.02)^2} = \sqrt{0.0001 + 0.0004} = \sqrt{0.0005} \)

Converting the absolute error into percentage error:

\( \frac{\Delta R}{R} \times 100 = \sqrt{0.0005} \times 100 \approx 0.02236 \times 100 \approx 2.236\% \)

Therefore, rounding to the appropriate significant figures, the error in the determination of the resistance of the conductor is approximately:

\( 3\% \)