Problem Sheet 7

The numbers in square brackets refer to the number of the problem in your textbook.

[1] [E14.3] It is much easier to measure the electrical conductivity of a material than to measure its thermal conductivity. Use the Wiedemann-Franz law to find the thermal conductivities of: (a) An alloy that has an electrical resistivity $\rho_e$ of 28$\mu \Omega . cm$. (b) Tungsten of the kind used for lamp filaments that has an electrical conductivity of $\kappa_e$ of 9.9 x 10^66^^ S/m.
[2] [E14.4] The metal zinc has free electron concentration per unit volume, $n_v$, of 1.3 x 1029/m3 and an electron mobility $\mu_e$ of 8 x 10-4 m2/V.s. The charge carried by an electron, e, is 1.6 x 10-19 coulomb. Based on this information, what is the electrical conductivity of zinc? Handbooks list the measured resistivity of zinc as 5.9 $\mu \Omega . cm$. Is this consistent with your calculation?
[3] [E14.5]A power line is to carry 5 kA at 11kV using pylons 460 m apart. The dip d in a wire of weight m1 per unit length strung between pylons L apart at a tension T is given by $d = L^{2} m_1/8T$. The maximum tension allowed is 0.8 of the yield stress, $\sigma_y$. if the maximum allowable dip is 6m, which of the materials below could be used?

Material Electrical resistivity $\rho_e (\Omega m)$ Yield stress $\sigma_y (M Pa)$ Density $\rho (kg/m^3)$
Aluminium 1.7 x 10-8 102 2700
Copper 1.5 x 10-8 300 8900
Carbon steel 55 x 10-8 510 7800

[4] [E14.6] A material is required for a transmission line that gives the lowest full-life cost over a 20-year period. The total cost is the sum of the material cost and the cost of the energy dissipated in joules. The cost of electricity $C_E$ is 6x10-3 $/MJ. Material prices are listed in the table. Derive an expression for the total cost per metre of the cable in terms of the cross-sectional area A (which is a free parameter), the material and electrical costs and the material parameters. Show that the minimum cost occues when the two contributions to the cost are equal. Hence derive a performance index for the material and decide on the best of the materials in the table. Material Electrical resistivity$\rho_e (\Omega m)$Price$C_m$($/kg) Density $\rho (kg/m^3)$
Aluminium 1.7 x 10-8 1.6 2700
Copper 1.5 x 10-8 5.2 8900
Carbon steel 55 x 10-8 0.5 7800

[5] [E14.8] Roughly 50% of all cork that is harvested in Portugal ends up as cork dust, a worthless by-product. As a materials expert, you are approached by an entrepreneur who has the idea of making useful products out of cork dust by compacting it and heating it, using microwave heating. The loss factor L of cork is 0.21. The entrepreneur reckons he needs a power density P of at least 2 kW per m3 for the process to be economic. If the maximum field E is limited to 102V/m, what frequency f of microwaves will be needed?

page revision: 9, last edited: 03 Mar 2009 06:33