A copper rod of length with insulated sides has temperature obeying

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter · Temperature, heat and the macroscopic behaviour of matter

Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed

Problem

A copper rod of length

0.50\,\mathrm{m}

with insulated sides has temperature

u(x,t)

obeying

u_t=Du_{xx}

. At

t=0

its temperature varies continuously along the rod and lies everywhere between

15\,{}^\circ\mathrm{C}

and

60\,{}^\circ\mathrm{C}

, with

20\,{}^\circ\mathrm{C}

x=0

and

40\,{}^\circ\mathrm{C}

x=0.50\,\mathrm{m}

. For

t>0

the two ends are held at those same values,

20

and

40\,{}^\circ\mathrm{C}

. Without solving the equation, find the lowest and the highest temperature that any interior point of the rod can have at any later time.

Browse the problem library