A capacitor discharges through a resistor

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory

Computational Physics, Ch Seeing the answer: plots that can be trusted

Problem

A capacitor discharges through a resistor. A lab fits the voltage record

V(t)

with a straight line

V(t) = 12.0 - 0.85t

(V, with

t

in s) and plots the residuals

V_{\rm obs}-V_{\rm fit}

underneath:

t=0\,{\rm s}:\ +0.30\,{\rm V}

;

t=2\,{\rm s}:\ -0.05\,{\rm V}

;

t=4\,{\rm s}:\ -0.28\,{\rm V}

;

t=6\,{\rm s}:\ -0.27\,{\rm V}

;

t=8\,{\rm s}:\ -0.03\,{\rm V}

;

t=10\,{\rm s}:\ +0.29\,{\rm V}

. Repeated readings of the instrument at fixed

t

scatter by only

0.03\,{\rm V}

. Someone argues: "none of the residuals exceeds

0.30\,{\rm V}

, which is under

3\%

of the

12\,{\rm V}

full scale, so the straight-line model is an excellent fit." ((a)) Explain what is wrong with judging the fit by the size of the residuals relative to the full scale. ((b)) State what the shape of the residuals (not just their size) implies about the true discharge law, and give the factor by which the residuals exceed the instrument's random scatter.

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