Potential, Current, and Their Sources

Potential

The concept of potential arises in association with conservative force fields, such as gravitational or electrical. As the name suggests the potential is associated with a "potential" ability to perform work. For gravitational force field on the earth surface, where the vertical acceleration is g , the work performed in lifting a mass m from ground to a height h is equal to gmh . The potential of the level h above ground is equal to that work divided by the mass, i.e., gh. This potential is obviously defined with respect to the earth surface and should be properly labeled as the potential difference, rather than just potential. The zero potential level is actually quite arbitrary and therefore only the potential difference possesses a good physical meaning. The electrical potential difference is defined in a similar way as the gravitational, i.e., as the work performed when moving an electric charge q between two potential levels, divided by that charge. A fresh flashlight battery of type D, for example, has a potential difference of about 1.5 Volts between its terminals regardless of what their absolute potentials really are. If an electron were allowed to move from the negative terminal of such a battery to the positive one it would gain a kinetic energy of 1.5 electron Volts (no more than a quarter of pico microWatt-seconds) but its potential would drop by 1.5 Volts. The potential energy has been converted to kinetic. If we wanted to move the electron back to the negative electrode we would have to perform work of 1.5 electron Volts because we would be moving the electron against the repelling force of excess electrons (excess over neutralizing positive charges) residing in the negative terminal. But the electron would gain 1.5 Volts in potential, i.e., its ability to perform work would be restored.

If we move an electron against the repelling force of excess electrons the electron gains in potential. Therefore we associate a higher potential with the electrode that has excess electrons. Conversely, the electrode that has lower density of electrons than that of positive charges attracts electrons and reduces their potential by inviting them to expend it on energy of motion towards it. Therefore the electrode with lower density of electrons is defined to have lesser potential. These conclusions are important for understanding of the distinction between voltage and current sources and we will return to it again.

Current

The term current may be associated with the flow of material particles or with the motion of electrically charged particles. Both share a common definition of flux which is the flow divided by the mass m or the charge q of the flowing particle. The flux is directly proportional to the velocity of the particles and to their number, as one may expect. If N particles move at velocity v the corresponding flux is vN and the current is mvN for material particles of individual mass m and qvN for particles carrying individually a quantity of charge q. Because it is the product of velocity and density that determines the current we can have the same electric current with fast and less dense electrons as with slow and closer spaced electrons. In a single electron formation the flux is simply equal to the velocity of electrons divided by their spacing.

Ideal Current Source

An ideal current source is defined as having the ability to force its nominal current into any load. Its internal resistance is defined to be infinite which enables it to support any kind of externally imposed potential difference across its terminals. An approximation to an ideal current source is a battery of very high voltage V in series with a very large resistance R. Such approximation would supply a current V/R into any load that has a resistance much smaller than R. In view of what we said about the potential and the current, a current source with a potential imposed on it would have in its positive electrode rarefied electrons moving fast and on the negative electrode denser electrons moving slower. Only the product of velocity and density must be identical. This also tells us that in positive wires electrons move faster than in negative ones if they belong to the same circuit. Only the ratios of velocities and respective spacings must be identical on both wires which assures uniform flow around the loop. If this seems disturbing let it be said that the difference in velocities is very small (in the 13-th decimal place for copper wire of 1mm diameter at a current of 1 Ampere) but it is there and none of the observed behaviour in conductors can be explained without it. Of course, the current source may have also zero potential across its terminals which means that the spacings of electrons on both terminals are equal to the spacings of the neutralizing atoms. Such would be the case if the current source were to drive a shorted wire but that state could not persist for long as we will see later. We intend to justify the statements and claims made here.

Ideal Voltage Source

A flashlight battery may be considered a voltage source but not an ideal one. The ideal voltage source is defined to have the nominal potential difference between its terminals and a zero internal resistance. As such it should be able to supply any amount of current desired.

In terms of the microscopic events we can define the voltage source as a device which can withdraw electrons from one of its terminals and deposit them onto the other. This can be done by mechanical means as in a generator or by chemical means as in a battery. A voltage source must be able to maintain an excess electron density on one electrode and an equal but opposite rarefaction on the other, regardless of how many electrons may be leaving any one terminal. Because the excess density is inversely proportional to the electron spacing one could also define the voltage source as a device that enforces a difference of electron spacings between its terminals. On one electrode the spacing of electrons is larger than that of atoms and on the other it is smaller. The actual current flow from such a source is solely determined by the external conditions.

Two voltage sources of the same nominal potential difference do not necessarily maintain the same spacing of electrons on their respective electrodes. There is a direct relationship between the electric excess charge on the source terminals and the potential difference between them. The proportionality constant C between them is a function of the terminal geometry and of the nature of the surrounding matter. For example a fresh D cell and a fresh AA cell have identical potential difference across their electrodes but the excess charges on their electrodes may be quite different. Their geometry determines the proportionality constant C which, in turn, determines the amount of excess charge. We can therefore also state that the spacing of electrons for a given potential difference depends on the proportionality constant C and we will exploit this fact when studying the reflections at discontinuities in wires.