PHY103A: Analogy Between Electrostatics and Steady-State Diffusion - Physics Assignment Help

Assignment Task:

Task:

1. Minimum energy theorem: We have a collection of several conductors, each with some charge. We know that charges over each conductor are so distributed that their surfaces are equipotentials. Show that this is equivalent to the electric field produced being such that the electrostatic energy of the system is minimum.

2. Potential in some region of free space is expressed as V?x, y, z? ? 20⁄?xyz?. For this potential, find the energy stored in a cube 1 ? x, y, z ? 2.

3. Analogy between electrostatics and steady-state diffusion and steady-state heat conduction:

(a) Consider a spherical container of radius R filled with a substance that diffuses (diffusion coefficient D) into the medium around it. Concentration of the substance in the container is maintained at n?. When steady state is achieved:

(i) find the relationship between the diffusion current at the surface of the sphere and at a distance r from its centre. Does the relationship remind you of a law of electrostatics?

(ii) Using result of (i) and Fick’s equation, find the diffusion current at r.

(b) We now repeat problem in part (a) for temperature T?r? and heat current j??r?around a sphere maintained at temperature T?. Heat conductivity of the medium around the sphere is K.
4. Show that the solution V?x, y, z? of Laplace’s equation ∇?V?0 has no maximum or minimum and relate it to Earnshaw’s theorem.

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