Polarization of a conductor

Claimed by Margaret Tikhonovsky Fall 2016

The Main Idea

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

Ionic Solutions

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

Figure taken from Matter and Interactions 4th Edition

A Mathematical Model: Drift Speed

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

[math]\displaystyle{ \bar{v}={u}{E_{net}} }[/math]

where [math]\displaystyle{ \bar{v} }[/math] is the drift speed, [math]\displaystyle{ u }[/math] is the mobility of the charge [math]\displaystyle{ {\frac{(m/s)}{(N/C)}} }[/math], and [math]\displaystyle{ E_{net} }[/math] is the net electric field applied to the ionic solution. The proportionality constant, [math]\displaystyle{ u }[/math], is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

Polarization Process in Ionic Solution

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

Metals

Mobile Electron Sea

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by [math]\displaystyle{ \bar{v} }[/math], the drift speed, as mentioned earlier. For metals this equation is:

[math]\displaystyle{ \bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} }[/math]

where [math]\displaystyle{ \bar{v} }[/math] is the drift speed, [math]\displaystyle{ e }[/math] is the charge of the electron, [math]\displaystyle{ {E_{net}} }[/math] is the net applied electric field, [math]\displaystyle{ {Δt} }[/math] is the average time between collisions, and [math]\displaystyle{ {m_{e}} }[/math] is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where [math]\displaystyle{ {Δp} = {F_{net}}{Δt} }[/math] and [math]\displaystyle{ {F} = {e}{E_{net}} }[/math].

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

Examples

1. Simple

The mobility of electrons in copper is [math]\displaystyle{ {0.0045}{\frac{(m/s)}{(N/C)}} }[/math]. How much net electric field would be needed in order to give the electrons in copper a drift speed of [math]\displaystyle{ {0.0015}{\frac{m}{s}} }[/math]?

Solution

This is a simple problem using the formula given for drift speed

[math]\displaystyle{ \bar{v}={u}{E_{net}} }[/math]

Thus, [math]\displaystyle{ {E_{net}} = {\frac{\bar{v}}{u}} }[/math] = [math]\displaystyle{ \frac{0.0015}{0.0045} }[/math] = [math]\displaystyle{ {0.333}{\frac{N}{C}} }[/math].

This also makes sense from a units perspective, as our solution is in [math]\displaystyle{ \frac{N}{C} }[/math], the speed is [math]\displaystyle{ \frac{m}{s} }[/math], and the mobility is given in [math]\displaystyle{ \frac{(m/s)}{(N/C)} }[/math].

2. Middling

In a simple metal lattice structure, an electric field of [math]\displaystyle{ {10}{\frac{N}{C}} }[/math] is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

Solution

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

[math]\displaystyle{ \bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} }[/math]

All of the quantities are known. [math]\displaystyle{ {e} = {1.6}{*}{10^{-19}}{C} }[/math], [math]\displaystyle{ {E_{net}} = {10}{\frac{N}{C}} }[/math], [math]\displaystyle{ {m_{e}} = {9.1}{*}{10^{-31}} }[/math].

However, [math]\displaystyle{ {Δt} }[/math] is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the [math]\displaystyle{ {Δt} }[/math] in the equation. [math]\displaystyle{ {Δt} }[/math] refers to the AVERAGE time between collisions. In this case, the [math]\displaystyle{ {Δt} }[/math] would therefore be 7.5 seconds.

Plugging these numbers in, you get: [math]\displaystyle{ {1.3}{*}{10^{13}}{\frac{m}{s}} }[/math]. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

3. Difficult

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges Polarization Equilibrium Excess Charge

Solution

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

Real World Application

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

See also

External links

YouTube Video: [1]

IEEE Report: [2]

References

Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2015. Print.