Question

Given $R_{1}=5.0\pm0.2\Omega , \, 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
$\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 = $\frac{50}{15}\left(\frac{0.8 + 0.1}{100}\right)\times 100=3\%$