A metal plate is held with its temperature at T(x,y) = 22 + 0.15x^2 + 0.40y^2 ^ C

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Gradient, divergence and curl

Problem

A metal plate is held with its temperature at

T(x,y) = 22 + 0.15x^2 + 0.40y^2\ {}^{\circ}\mathrm{C},

with

x

and

y

in centimetres measured from one corner. ((a)) Find

\nabla^2T

. ((b)) Is the temperature at

(4.0\ \mathrm{cm},\ 3.0\ \mathrm{cm})

above or below the average temperature round a small circle on the plate centred there? Adapt the chapter's sphere-averaging argument to a circle, and find the difference for a circle of radius

0.50\ \mathrm{cm}

. ((c)) Could this be the steady temperature of a plate heated only at its edges? Give the reason.

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