Question

Given two resistance are $R_{1}=5.0\pm0.2\Omega\text{and} \, R_{2}=10.0\pm0.1\Omega. \, $ What is total resistance in parallel with possible percentage error?

• A. $15\Omega\pm2\%$
• B. $3.3\Omega\pm3\%$
• C. $15\Omega\pm3\%$
• D. $3.3\Omega\pm7\%$

Answer 1

Answer: (B)

In parallel,
$R_{p}=\frac{R_{1} R_{2}}{R_{1} + R_{2}}$ $=\frac{5.0 \times 10.0}{5.0 + 10.0}$ $=\frac{50}{15}=3.3\Omega$
Also,
$R_{p}=3.3\Omega$
$\frac{1}{R_{p}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$
Differentiating both sides we get,
$\Longrightarrow $ $\frac{\Delta R_{p}}{R_{p}^{2}}=\frac{\Delta R_{1}}{R_{1}^{2}}+\frac{\Delta R_{2}}{R_{2}^{2}}$
$\frac{\Delta R_{p}}{R_{p}}=\frac{50}{15}\left(\frac{0.2}{25} + \frac{0.1}{100}\right)$
$\%$ error = $\frac{50}{15}\left(\frac{0.8 + 0.1}{100}\right)\times 100=3\%$