MAGNETIC AND ELECTRIC MIXING WITH STRESS

John W. Dooley, Physics Department, Millersville University

Some Other Properties:

OHM'S LAW

The perturbation-response picture can also be applied to the flow of electricity in a conductor. In this case we call the electric field the perturbation and the current the response. In one dimension, for a conductor of length, , we can integrate the electric field to find the electrical potential difference (the "Voltage", ) from one end to the other. We put a collector at the "down-stream" end and measure how much charge flows out in one second. This amount is the Current, . Ohm's law relates these two measurements:

where is the resistance of the sample, or

,

where is the conductance of the sample; .

The second form bears a striking resemblance to Poiseuille's law for flow of a viscous fluid in a pipe:

where is analogous to the electrical current, , and is analogous to the potential difference, .

Electrical flow in three dimensions is similar to fluid flow, and is often thought of in fluid terms. Mathematically, the electrical flow also looks like heat flow in the presence of temperature gradients. Electrical current flows in the direction opposite to the gradient of the electrical potential field. (That is, it flows perpendicular to surfaces of constant electrical potential.) The electrical current flow equation in an isotropic medium is

where is the electrical conductivity of the medium and is charge per second through a 1 meter square window.

is called the current density. The units of are Amp per square meter. The units of the electrical conductivity are

.

The minus sign indicates that the current flows towards lower electrical potential.

Compare this to the heat flow equation:

For copper,

,

and the thermal conductivity is

The ratio of thermal to electrical conductivity for copper is

At room temperatures, this ratio is approximately the same for many different metals, suggesting that the thermal and electrical conduction mechanisms in metals have the same origin. This fact is called the Wiedemann-Franz law in honor of its discoverers.

An interesting effect, which we will not study, is the thermoelectric effect: In the proper setting, a temperature gradient along a wire can cause electrical currents. One practical application of this effect is a temperature measuring device called a thermocouple. Another is a thermoelectric refrigerator.

CROSS COUPLING

We have studied three static perturbation-response systems: Stress/Strain, Electric Field/Electric Polarization, and Magnetic Field/Magnetic Polarization.

We studied them one at a time, as though (for example) a magnetic material could not experience stress. There are many important practical cases in which this one-at-a-time approach fails. For example, the large magnets used to accelerate particles in "atom smashers" suffer mechanical stress due to the interaction of their magnetic fields with the surroundings.

In a simpler case, an iron rod placed in a magnetic field changes length. The magnetic field induces a strain. This effect is called "magnetostriction." If the (unmagnetized) iron is stressed, no observable magnetization is produced. If the iron is stressed in the presence of an external magnetic field, then the polarization of the iron is changed.

The news is that the stress strain relationship is more complicated than we had hoped.

Magnetostrictive effects link magnetic and strain behavior:

The stress in a magnetized material is related to both the strain and the magnetic polarization:

The third order tensor, , is characteristic of the material. The phenomenon of elastic distortion in response to an applied magnetic field is called magnetostriction.

Electrostrictive effects link electrical and strain behavior:

In samples of some materials, an applied electric field produces a change in length of the sample.

The stress in an electrically polarized material is related to both the strain and the electric field:

The tensor, is characteristic of the material. The phenomenon of elastic distortion in responses to an applied electric field is called electrostriction.

Conversely, an applied stress produces a polarization. The electrical polarization of a sample which is subject to both an applied electric field and an applied stress is:

The tensor, is characteristic of the material. The phenomenon of polarization in response to an applied stress is called piezoelectricity.

Piezoelectric materials also exhibit electrostrictive effects. For piezoelectric materials, the constants d and d' are called piezoelectric constants. Piezoelectrics are widely used for sound generation and detection, for precise positioning of laser mirrors, and for changing the shape of telescope mirrors (to improve the focus).