PHYS208 2/18 Class

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Resistivity and Ohm's Law

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What happens when the two ends of a conductor are held at different values of potential? That is, what happens when a potential difference is applied across a resistor? A battery is connected to a thin wire?

Charges experience an electric field and move accordingly. In a typical metal, each atom in the lattice will allow one of its electrons to move freely. (See Periodic Table.) So the metal can be thought of as a rigid lattice of positively-charged ion cores surrounded by a swarm of free electrons. Later in the semester we will see how the Hall effect demonstated that electrons are in fact the free charge carriers in simple metals.

Do the electrons accelerate indefinitely because of the applied potential? NO, on average they reach a terminal speed, much as a skydiver reaches a terminal velocity of about 120 mph falling through air, a viscous medium. The electrons tend to drift slowly through the conductor, on the order of 0.01 mm/s. The "viscosity" that the electrons experience is called resistance. Many conductors are referred to simply as resistors.

But what about the speed of propagation? Many of you know that electronic signals propagate near the speed of light. For example, the bits coming to this computer via the Ethernet travel at about 0.8 times the speed of light. We will examine this more fully very near the end of the semester.

For the time being you may think of the "water in hose" analogy. When the valve to a garden hose is opened, water "immediately" leaves the end of the hose. However the water actually passing through the valve takes much longer to reach the end of the hose. Here the pressure wave that propagates at the speed of sound in water would represent the electronic signal.

Also please keep in mind that although electrons are flowing through the conductor, there is no charge accumulation anywhere. The electrons flow in a continuous loop, known as a circuit. The electrons are continuously flowing past the positively-charged ion cores, with no change in the net charge anywhere. See animation below:

Charge Flow in a Conductor

So the result of a voltage difference across a resistor is that electric current flows. Precisely, current is the instantaneous rate at which charge passes through a cross-section of the conductor.

The current, represented by i, is measured in amperes (A or amp for short), which is a coulomb per second.

We will start with the study of steady currents, the so-called direct current, or dc for short. Later in the semester, we will study RC circuits and their decaying transient currents, followed by alternating (or ac) circuits.

Ohm's Law

Simple circuit experiment -- with better quality graph

Definition of Resistance
If a material obeys the relationship above, over a wide range of voltages, it is said to be ohmic. Not all materials are ohmic, and fortuately so, or we would not have the wonderful semiconductor-based devices to which we have become so accustomed. However over the next few classes we will be considering resistors to be made from ohmic materials.

The unit of resistance is volt/amp, known as an ohm, and represented by upper case omega.

Units of Resistance

Resistance controls the amount of current that results when voltage difference is applied to a resistor.

Ohm's Law

Resistivity

The resistance of a conductor depends on material and geometry. These effects may be separated by considering the resistivity of the material.

Refer to the Play-Doh Experiment

Resistance of a Cylindrical Conductor

The proportionality constant rho is the resistivity of the conductor, a material property, not a geometrical one. There is a wide range of resistivities available, from as low as 10-8 ohm-m for pure metals to as high as 1017 ohm-m for an excellent insulator.

Simplest Possible Circuit

 The circuit elements are specified: the power supply or battery provides a stated emf and is connected across a resistance R. The emf of the battery is applied directly across the resistor; i.e. the voltage provided by the battery is identical to the voltage difference between points a and b. Thus by applying Ohm's law to the single resistor, connected to a single emf:

 To the chalkboard for presentation of first simple circuit and derivation of Joule's law.

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