The intensity of light a fixed distance from a point source follows the…

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

The intensity of light a fixed distance

r

from a point source follows the inverse-square law,

I(r) \propto r^{-2}

. A distance is measured as

r = 250.0\,\mathrm{m}

by an instrument whose relative uncertainty is

0.20\%

, i.e.\

|\Delta r/r| = 2.0\times10^{-3}

. (a) Write

f(r) = r^{-2}

and compute the condition number

\kappa = |r f'(r)/f(r)|

. (b) State the bound on the relative error introduced purely by storing

r=250.0\,\mathrm{m}

as a binary64 number. (c) Use

\kappa

to bound the relative error of the computed intensity

I

arising from the measurement uncertainty in

r

, and state which of the two error sources, storage or measurement, dominates and by roughly what factor.

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