Question

A large solid region in three dimensions represents a big rock. Part of the surface of the rock is kept at a prescribed temperature - maybe hotter at one point than at another. The remainder of the surface is perfectly insulated. You wait until the temperature inside the rock settles into its steady state condition. Say that temp[x, y, z] represents the steady state temperature at a position {x, y, z} inside the rock. In the steady state, no point inside the rock and not on the surface can be a source of new heat flow or a sink for old heat. Why does this tells you that

ΔTemp[x, y, z]= γ^2 temp/ γx^2 + γ^2 temp/γy^2 + γ^2 temp/γz^2 =0

at each point {x, y, z} inside but not on the surface of the rock? Explain why the hottest and the coldest locations of the rock must be on the outside skin of the rock, and not inside the rock.

Answer 1

From the information given, in the steady state, no point inside the rock and on the surface can be a source of new heat flow on sink fan heat flow, which implies that there is no flow of heat with changing position

i.e Δ temp(x,y,z) = 0.

Further part of the surface of the rock is kept at a prescribed temperature and the remainder is insulated, hence these part. This implies, temperature at the surface will be hotter than any other part.

Step-by-step explanation: