A small lamp of power radiates light uniformly in all directions

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

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

Problem

A small lamp of power

P = 150\,\mathrm{W}

radiates light uniformly in all directions, giving an intensity

I(d) = P/(4\pi d^2)

at distance

d

from the lamp. The distance is measured as

d = 3.20\,\mathrm{m}

with a relative uncertainty of

0.80\%

. (a) Compute the condition number

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

I

as a function of

d

. (b) Use

\kappa

to bound the relative uncertainty in the computed intensity

I(d)

, and quote

I(d)

with its absolute uncertainty. (c) Suppose

N = 1.00\times10^4

independent readings, each equal to

I(d)

to within rounding, are summed in a counted loop as part of a longer calculation. Using the worst-case rounding bound

\tfrac12 N\epsilon_{\rm mach}

, give the resulting relative error from summation alone, and say whether it can compete with the uncertainty found in part (ii).

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