A point source of light of power produces an intensity at distance from it

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Partial differentiation, chain rules and Jacobians

Problem

A point source of light of power

P

produces an intensity

I(P,r) = P/(4\pi r^2)

at distance

r

from it. At some instant

P = 100\,\mathrm{W}

and

r = 2.0\,\mathrm{m}

. ((a)) Find

(\partial I/\partial P)_r

and

(\partial I/\partial r)_P

at this instant, naming the held variable in each. ((b)) The source is turned up by

\Delta P = +5\,\mathrm{W}

and the detector is moved out by

\Delta r = +0.02\,\mathrm{m}

. Use the total differential to estimate the resulting change in intensity. ((c)) Suppose instead

P(t) = 100 + 2t\,\mathrm{W}

and

r(t) = 2.0 + 0.1t\,\mathrm{m}

(with

t

in seconds). Find

\mathrm{d}I/\mathrm{d}t

t=0\,\mathrm{s}

using the chain rule along this path. ((d)) New labels

s = P

and

w = r^2

are introduced, so that

P = s

and

r = \sqrt{w}

. Find the Jacobian

\partial(P,r)/\partial(s,w)

at this instant, and say what it tells you about the area of a small rectangle in the

sw

plane compared with the area of its image in the

Pr

plane.

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