Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Section A / 2 / Solution

Write $X=1/(\omega C)=200\Omega$. The magnitude of the series RC impedance isFor independent small errors interpreted as standard uncertainties, propagation of uncertainty givesThe input uncertainties are $u_R=2\Omega$, $u_{\omega }/\omega =0.01$, and $u_C/C=0.01$. Thus $u(Z_m{)}^2=0.80+3.20+3.20=7.20{\Omega }^2$, andThe fractional uncertainty is $1.20\%$. If the stated errors are instead worst-case bounds, their first-order contributions add in magnitude, giving approximately $4.47\Omega$, or $2.0\%$, rather than the quadrature result. Both conventions require the error interpretation to be specified.