Due 5pm on Friday 17 March.

Submit your problem set (typed, or hand written and photographed/scanned) via Blackboard.

Answers should include detailed comments and reasoning. A grade up to 5 can be obtained by answering only questions in Part A. Higher grades require all questions to be answered (with clear, well reasoned arguments showing insight and creative application of knowledge and skill).  Refer to the Grading Criteria document for details.

Part A

Problem 1 . 1

The height of an area of land relative to some reference point is given by the equation

h(x,y) = h0 + ax2 − by2  ,

where a and b are positive constants and h0  is the height at x = y = 0.

Choose your own values of constants a, b and h0  (but ensure that a  b).  For each of the following, make a skech of the field – this may be hand-drawn or drawn on a computer using your own script. Include a copy of your script(s), which can be based on the template files on 5-minute physics, if done on a computer.  Briefly describe the important aspects of the field in each case.

(a) The height h

(b) The gradient of the height.

(c) The divergence of the gradient.

Finally, what should the curl of the gradient be? Check this by performing the calculation. Problem 1 .2

A 120 mm long metallic rod with a diameter of 11 mm is insulated so that heat can only flow in and out of the rod at one end (defined as x = 0). This end of the rod is maintained at 18 ◦ C while the initial temperature in the rod takes the form of part of a sine wave with a maximum temperature of 65celsius.

(a) Show that a temperature profile of the form

T(x,t) = TA sin ( )e −t/τ + T0

is a solution of the temperature diffusion equation when the diffusion constant, D, is a function of one or more of the constants TA , T0 , λ and τ . Determine this relationship. Note that you are not asked to solve the diffusion equation - just to show that the form given satisfies the equation.

(b) The boundary condition at the ends of the rod must be

Constant temperature end:      = 0,          Insulated end:       '北=L = 0

where L is the length of the rod.  Give brief explanations of why these conditions apply for the current situation.

(c) Using the boundary and initial conditions provided, find numerical values for TA , T0  and λ .

(d) The rod cools over time as heat is lost through the non-insulated end.  Suppose it takes 25s for the peak temperature of the rod to drop to 32 ◦ C. Calculate a numerical value for τ and hence find D .