A resistor with

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Mathematical Methods for Physics, Ch Probability, random variables and distributions

Problem

A resistor with R=1000 ΩR = 1000\ \Omega, measured to δR=20 Ω\delta R = 20\ \Omega, is placed in series with a capacitor with C=4.00 μFC = 4.00\ \mu\mathrm{F}, measured to δC=0.10 μF\delta C = 0.10\ \mu\mathrm{F}. The two measurements are independent, and the time constant of the circuit is τ=RC\tau = RC. ((a)) Using the total differential of τ(R,C)\tau(R,C) and the rule for combining independent uncertainties, show that (δττ)2=(δRR)2+(δCC)2,\left(\frac{\delta\tau}{\tau}\right)^2 = \left(\frac{\delta R}{R}\right)^2 + \left(\frac{\delta C}{C}\right)^2, then compute the numerical values of τ\tau, written τ0\tau_0 below, and δτ\delta\tau. ((b)) A single repeat measurement tt of the time constant is modelled by the Gaussian density f(t)=Aexp⁡ ⁣[−(t−τ0)22σ2],f(t) = A\exp\!\left[-\frac{(t - \tau_0)^2}{2\sigma^2}\right], with τ0\tau_0 and σ=δτ\sigma=\delta\tau found in part (a). Using the Gaussian integral ∫−∞∞e−ax2 dx=π/a\int_{-\infty}^{\infty}e^{-ax^2}\,\mathrm{d}x = \sqrt{\pi/a}, find the normalisation constant AA and state its units. ((c)) Using the one-sigma fraction for a Gaussian, state the probability that a single measurement tt falls within δτ\delta\tau of τ0\tau_0.

Browse the problem library