Question

Question 3: A problem with analagous mathematical structure to electrostatics (12 marks) In the static (i.e. time-independent) case, the temperature throughout a material can be described by the scalar-valued function T(x, y, z). In addition, the flow of heat can be described by the vector-valued function h(x, y, z), the magnitude of which indicates the amount of heat energy that flows per unit area per unit time and the direction of which is that of the heat flow. (a) (2 marks) Is T a field? Explain. If so, what sort? If not, why not? (b) (2 marks) Is h a field? Explain. If so, what sort? If not, why not? The flow of heat satisfies the following two equations: v: h = S (3.1) h = -KVT (3.2) where S denotes the rate at which any sources present create or destroy heat energy and K denotes the thermal conductivity. In the most general case, S and K could both be scalar fields. (c) (3 marks) Under what conditions will the above equations guarantee that T satisfies Poisson's equa- tion? Under that condition, derive the form of Poisson's equation for T. (d) (5 marks) The surface of a good thermal conductor can be assumed to be an isotherm, i.e. it will have the same temperature at each point. Suppose a semi-infinite slab of good thermal conducting material occupies all space for 2 <0. A point source generating heat energy is positioned a distance d above the surface of this slab. Assuming that T does indeed satisfy Poisson's equation (as per part (c)), briefly discuss how you could solve for the temperature distribution in the equilibrium (i.e. time-independent) case. Note: You are not being asked to provide a detailed mathematical solution, just a discussion about the concepts and the process. Hint: To what extent is this problem equivalent to placing a point charge above a conducting surface (a configuration analysed in lectures)?

Answer 1

3) a> Field line is a locus that is defined by a vector. Here, T(nry, z) is a scalar-valued function that describes the temperature Shroughout a material. Therefore it is not a field. ** Analagous to electric potential V(nry, 2) in electrostatics. b> ħ is a vector - valued function that describes The flow of heat and its magnitude indicates The amount of heat energy. Due to its vector property, it is a field. ** Analogous to electric field in electrostatics whose magnitude indicates the amount of election -static energy.

7 The above equation garantees that I satisfies Poisson's equation only when there is some sources of heat present. If the sources is not present, then Poisson's equation becomes Laplace's equation. ces

Derivation - 3. ñ =s and r=-KDT Substituting ħ in .h as we get, 2. (207) = $ >> -* (0.01) -S - ²7 = -8/4 Poisson's equation DT. Note: If sao (0²1=0 Laplace's equation. ** Analagous to 2² ware/o in electrostatic.

22024 > Let aq be a to source generating i heat energy positioned distance d a from the semi-infinite za slab. To solve such problem we use solution analagous to image charge method in electrostatics. It is based on Uniqueness Sheorem according to which, if T satisfies the boundary condition then it is a unique solution. Boundary condition :- Tabove – Tbelow (in 20 general) & we simplify this problem by replacing the semi-infinite isothermal region with a heat source the slab One slab

lets say -q located at (0,0,-d). Such arran- gement will produce same heat at any point for zoo and will satisfy the boundary condition ame as that of slab would have produced. Now according to Uniqueness Cheorem, there is only one solution to Poisson's equation which satisfies a given boundary condition. Therefore by replacing the semi-infinite slab with image heat source (point source at distance d below the slab, such that One & scalared - valued function T of combination of original and image source would satisfy the boundary condition, we can solve for the temperature distribution in the equilibrium case. a condition, (to be satisfied here Boundary à at 2=0, T=0 by and at (u²+ y +2²)> d² To N.B. When we replace semi-infinite slab by image heat source, the problem reduces to two point source of equal magnitude but opposite nature placed at equal distance from the slab. z >O 2-0 niso ld 24