MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2000

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 3: Due Fri. Sept. 29, Before 5PM in 13-5026

Exercise 3.1

The bulk modulus, $K$, of an isotropic linear elastic solid is defined by the dialation, $\Delta V/\ensuremath{{V}_\circ}$, response to hydrostatic pressure $P$:

$$\frac{\Delta V}{\ensuremath{{V}_\circ}} = \frac{V - \ensuremath{{V}_\circ}}{\ensuremath{{V}_\circ}} = -\frac{P}{K}$$ (1)

Typical values of $K$ for an ionic crystal are about $100$ GPa. (GPa = 1 Gigapascal, 1 atm $\approx 10^{-1}$ MPa)

The permittivity of vacuum, $\ensuremath{{\kappa}_\circ}$ is $8.85 \times 10^{-12} \mbox{C}^2/\mbox{J m}$. Typical values of the dielectric susceptibility, $\chi$ ( $\vec{P} = \ensuremath{{\kappa}_\circ}\chi \vec{E}$), of an ionic crystal are about 50 (unitless).

The permittivity of vacuum, $\ensuremath{{\mu}_\circ}$, is $4 \pi 10^{-7} \mbox{T}^2 \mbox{m}^3/J$ (T is a tesla). The magnetic susceptibility, $\psi$ ( $\vec{I} = \ensuremath{{\mu}_\circ}\psi \vec{H}$), of a typical paramagnetic ionic crystal is about 10 (unitless).

Calculate all the ratios of: stored elastic energy, stored polarization energy, and stored magnetic energy in a typical ionic crystal at 1 atm, 220 volts/m, and in the earth's magnetic field.