Question

If the errors in the measurements of diameter, length and electrical resistance of a wire are 1%, 0.5% and 2% respectively, then percentage error in the determination of the resistivity of material of the wire is:

• A. 3%
• B. 4%
• C. 4.5%
• D. 2.5%

Hint:

Whenever diameter is involved, its error is multiplied by 2 in area calculations.

Answer 1

The Correct Option is C

Step 1: Resistivity of a wire is given by: \[ \rho = \frac{RA}{L} \] where $R$ is resistance, $A$ is area, and $L$ is length.

Step 2: For a wire of circular cross-section: \[ A = \frac{\pi d^2}{4} \Rightarrow \frac{\Delta A}{A} = 2\frac{\Delta d}{d} \]
Step 3: Given percentage errors: \[ \frac{\Delta d}{d} = 1%, \quad \frac{\Delta R}{R} = 2%, \quad \frac{\Delta L}{L} = 0.5% \]
Step 4: Error in area: \[ \frac{\Delta A}{A} = 2 \times 1% = 2% \]
Step 5: Total percentage error in resistivity: \[ \frac{\Delta \rho}{\rho} = \frac{\Delta R}{R} + \frac{\Delta A}{A} + \frac{\Delta L}{L} \] \[ = 2% + 2% + 0.5% = 4.5% \] Hence, percentage error in resistivity is $4.5%$.