Question

Two resistors with nominal resistance values $R_1$ $R_2$ have additive uncertainties $\Delta R_1$ and $\Delta R_2$ respectively. When these resistances are connected in parallel, the standard deviatioin of the error in the equivalent resistance $R$ is

• A. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_1}\Delta R_1\right\}}^2+{\left\{\frac{\partial R}{\partial R_2}\Delta R_2\right\}}^2}$
• B. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_2}\Delta R_1\right\}}^2+{\left\{\frac{\partial R}{\partial R_1}\Delta R_2\right\}}^2}$
• C. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_2+{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_1}$
• D. $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_1+{\left\{\frac{\partial R}{\partial R_1}\right\}}^2\Delta R_2}$

Answer 1

The correct option is A $\pm \sqrt{{\left\{\frac{\partial R}{\partial R_1}\Delta R_1\right\}}^2+{\left\{\frac{\partial R}{\partial R_2}\Delta R_2\right\}}^2}$

${\sigma }_{res}=\sqrt{{\left(\frac{\partial R}{\partial R_1}\right)}^2\sigma {{}_1}^2+{\left(\frac{\partial R}{\partial R_2}\right)}^2{\sigma }_2^2}$
${\sigma }_{res}=\sqrt{{\left(\frac{\partial R}{\partial R_1}\right)}^2\Delta R{{}_1}^2+{\left(\frac{\partial R}{\partial R_2}\right)}^2\Delta R_2^2}$