Figure 5.1: Current is electrical charges passing through a conductor.
Much of our determination of the properties of systems involving charges has been with the implicit assumption that those charges were static. Just as in mechanics, we can learn a lot by understanding the static conditions of objects. Note that static here did not imply that objects didn't move. The equation for motion apply to any object that has constant acceleration, i.e. the only thing that had to be static about the system was its acceleration. We could determine a number of things about the world just from this. To truly understand motion, however, we needed Newton's Laws as a way of describing dynamics. This opened up a whole new universe of understanding which went well beyond Galileo's equations for motion. Phenomena that were complete mysteries before, e.g. the orbits of planets, suddenly had precise descriptions in terms of Newton's Equation of Motion.
We have the same situation with electricity. We will see later in the course that electrodynamics offers explanations of phenomena that could not even be guessed at with just the applications of Gauss' and Coulomb's Laws. To get to the point of understanding this though, we have to start from the beginning.
We begin by formally defining current as the time rate of change of charge through a conductor. Thus, if we have a conductor material of any cross-section, we can define the current in that material as the amount of charge that passes through a hypothetical plane that cuts all the way through the material.
In the figure above, each of the hypothetical planes above sees the same current, i = dq/dt, go by it. We define the unit of current as the Ampere and abbreviate it as A. 1 A = 1 coulomb/sec. Since charge is conserved, note that current, the time rate of change of charge, must also be a conserved quantity. If were not, then charges could accumulate or disappear at some point in the material. This is assumed not to happen for conducting material nor for electrical devices. Currents flow as the result of potential differences between positions.
Currents are not dependent solely on potential differences. An object capable of carrying current can be characterized by its resistance where resistance is defined as
| (5.1.2.1) |
The resistance of an object depends on its geometry and its resistivity. Resistivity is a property of the material itself and is the same no matter what the shape or dimension of the material. The symbol for resistivity is the Greek letter rho. For an object made from material with resistivity rho,
| (5.1.2.2) |
where L is the length of the object and A is its cross-sectional area which is assumed to be constant throughout the object. If A it is not constant, then we need to do an integration over infinitesimally small distances of the material where we can assume that the cross-sectional area is a constant.
For this course, unless explicitly stated otherwise, you can always assume that there is a linear connection between potential difference, current, and resistance in which
| (5.1.2.3) |
Although this appears to be identical to the definition of R, keep in mind that this relationship assumes that potential difference and current are linearly related over some reasonably large set of possible V's and i'. Semiconductor materials, as an example, display remarkably (and most usefully) non-linear relationships between potential difference and current. The above equation of V = i*R is called Ohm's Law because it holds reasonably well for a large number of objects used as electrical devices.
One final definition is needed. When we talk about rates of energy change in an electrical circuit, we deal with two notions of the term, power. The power provided by sources of electromotive force, emf's, is given by the formula derived as follows. EMF's provide potential differences that move charges. The energy provided by the emf can be calculated by looking at infinitesimal charges moved through the potential difference
| (5.1.3.4) |
Thus the power provided by an emf is the current times the potential difference it maintains between its positive and negative poles.
In any material or device that has electrical resistance, the rate of change of energy for current flowing through it is lost or rather transformed into thermal energy. To determine the relationship for this, note that Ohm's Law tells us that the potential difference across a material or device with resistance R is VR = iR. Therefore, the power necessary to maintain current i going through the material or device is
| (5.1.3.5) |
As stated before, this power is the rate at which electrical energy is turned into thermal energy in the resistance.
Send comments to larryg@upenn5.hep.upenn.edu.
This page was last modified on 02/19/2003 at 13:53:57 (EST).
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