The Free Electron Model

As we reviewed in the previous section, the current, $I$, through a material is proportional to the voltage, $V$, across it. That is: $I \propto V$. We can use either the resistance $R$ or conductance $G$ as the proportionality constant.

$$I=\frac{V}{R}$$

or

$$I=GV$$

In materials science and engineering, we are interested in why materials are resistive/conductive. On this page you will use a simple version of the free electron model to start exploring some of the factors that affect resistance and conductance that are not related to the material itself and one factor that is. In the free electron model, we assume there are electrons in the material that are free to move around, which is true of metals (and some other materials). On a later page, we will introduce qualitative models to explain why some materials have free electrons and others don't.

Exercise 13.4.1: Geometry-related Contributions to Resistance
Due: Fri, Oct 18, 12:30 PM

Consider NetLogo model 13.4.1 and spend about 5-10 minutes on this exercise.

  1. The electron-density slider models how many free electrons there are per unit volume of a material. How is electron density related to current at a given voltage? Why is that?

    Remember that current is defined as the amount of charge moving through some area per time. In the model, it is the number of electrons moving through the anode (positive battery terminal) per time step.

    Note: you need to click setup each time you change the electron-density for it to take effect.

  2. The resistance of a piece of material is affected by the geometry of the material. Use NetLogo model 13.4.1 above to:

    1. Determine how current changes with wire cross sectional area.
    2. Explain the relationship.
    3. Conceptualizing this as part of the resistance, how is the resistance, $R$ related to cross sectional area $A$?

  3. Now use the model to:

    1. Determine how current changes with wire length.
    2. Explain the relationship (recall the analogy between voltage and tilting a pipe with water from Figure 13.3.2
    3. Conceptualizing this as part of the resistance, how is resistance $R$, related to the wire length, $l$?