Chapter 3 Electrical Potential Energy

Connected to two different metals, a dead frog’s legs twitch. Did energy flow to or from the frog? What is unique about recently dead tissue that make it appear to come back to life?

This experiment was first conducted by the Italian scientist Luigi Galvani who in 1791 took this discovery as evidence that the frog’s leg muscle, like a Leyden jar, retained some electric charge even after death.[1] By touching the frog, he asserted, this animal electric fluid becomes discharged and the leg twitches.

Mary Wollstonecraft Shelley spent the summer of 1816 vacationing in Switzerland, where she read Galvani’s report of latent animal electric fluid making the frog’s legs twitch. At the same time, she composed an idea for a novel about a scientist bringing dead tissue back to life. As Shelley’s focus was on her protagonist, Victor Frankenstein, she did not specifically mention electricity as the means by which the dead corpse is revived.[2] In our modern interpretations of both Galvani’s experiment and Shelley’s story, electricity is the cause of both the dead frog’s twitching leg and the creature’s resuscitation.

Alessandro Volta coined the term galvanism in honor of his colleague Galvani, but had a different opinion as to the source of electricity. He maintained that the source of the electricity was due to the contact of the dissimilar metals. To prove his point, Volta built a device to store electrical energy, with no animal tissue.

Volta stacked bimetallic disks of copper and zinc plates, layered with salt-water soaked paper. This voltaic pile became the first practical source of steady electrical current.[3] Like the frog’s leg in Galvani’s experiment, the salt-water provides an electrolytic solution for ions to travel from one plate to the other. Once the voltaic pile was invented, it became relatively straightforward to stack them to arbitrary height, which made it possible to supply fairly steady electrical current to wires forming an electric circuit.

Volta’s achievement spurred remarkable progress in the subjects of electricity and magnetism. In 1820, the Danish physicist Hans Christian Ørsted used a voltaic pile to pass a steady current near a magnetized compass needle causing the needle to deflect, and therefore demonstrated that electricity and magnetism are related. Using Ørsted’s result, French mathematician and physicist Andre Marie Ampere and German physicist and chemist Johann Schweigger independently developed the earliest galvanometer to measure current in terms of the torque exerted on a compass needle suspended near the current. In the same year that Ørsted announced his discovery of electromagnetism, for example, Michael Faraday employed the steady currents produced by Volta’s cells, or batteries, to construct the first electric motor.

Beginning in 1825, Georg Ohm used voltaic cells and a galvanometer to begin his investigation of how different metals conduct electric current. In his first two sets of experiments, he inserted metal wires of different lengths and materials between the terminals of a voltaic cell made of copper and zinc, and varied the number of plates to control the strength of the current. Among other things, he determined that the current flowing through a wire is related to the length of the wire and the type of metal from which it is made. But the voltaic cells of the early 1820’s were far from perfect, and Ohm found that he could not obtain a sufficiently steady current to make his measurements as precise as he felt they needed to be. For the third experiment, Ohm replaced the voltaic cell with a thermocouple, for which a temperature difference between two dissimilar metals causes current to flow, so that he could produce the steadier and more reliably measurable currents required to establish a relationship between current flowing in a conductor and the length of the conductor. He found that the current was inversely proportional to the length of the wire plus a constant and directly proportional to the temperature difference in the thermocouple source.[4]

Ohm leaned heavily on the work of Joseph Fourier, whose The Analytical Theory of Heat proved so influential for nineteenth century physics, and proposed that the flow of electrical charge in a conductor was analogous to the flow of heat. He also suggested that the cause of current flow was analogous to the temperature difference or gradient that in Fourier’s theory is responsible for heat flow. Ohm asserted that the length of the wire, plus the constant term that he conceived of as the “effective length” of the voltaic cell or the thermocouple source, is proportional to the electrical resistance of the wire plus the “internal resistance” of the current source, much like the thermal resistance to heat flow in a conductor.[5]

Although Ohm was guided by an analogy with Fourier’s theory of heat flow, the connection between his discovery and other electrical quantities only become clear as the concept of energy became more developed. For example, Gustav Kirchhoff pointed out in 1850 that it is the difference in voltage or electrostatic potential, which is defined as the work per unit charge to transport a unit charge from one location to another in an electric field, between the terminals of the wire that is ultimately responsible for the flow of current. This is true whether the current source is a voltaic cell or a thermocouple. Therefore, Kirchhoff asserted, the electric current density is proportional to a vector derivative, the gradient in this case, of the electrostatic potential, which in turn is directly proportional to the electric field. With this, Kirchhoff helped link electrostatics with electrodynamics.

Of course, Kirchhoff also used conservation of charge and conservation of energy to develop his famous rules for currents flowing in a circuit. His “junction rule” states that the sum of the currents entering the point where two or more wires meet must equal the sum of the currents leaving that point or junction. His second rule, often referred to as the “loop rule,” requires that the total energy around a closed loop of current carrying wire must be conserved. It is impressive that he formally derived these two rules, which are now a standard part of undergraduate physics and electrical engineering courses, while still an undergraduate student![6]

In this chapter we make a distinction between conservative and nonconservative forces. If you do work against a conservative force, the force will be able to the same work on something else in the future. However, the work done against a nonconservative force goes into the random kinetic energy we call heat. In that case, we say the energy is dissipated.

We explore energy concepts by first reviewing work and energy in mechanical systems. Next we define voltage, and then we cover circuits with batteries and resistors. This, in turn, leads to Kirchhoff’s loop and junction rules for an electric circuit, which are a consequence of energy and charge conservation respectively.

[1] See, for example, J.L. Heilbron, Electricity in the17th and 18th centuries: a study of early Modern physics. (University of California Press, 1979), 491.

[2] Mary Shelley, Frankenstein (1818 text), ed. Marilyn Butler (Oxford: Oxford University Press, 1993), p. 195. See also the following article that situates Shelley’s novel within the debate between Volta and Galvani over the nature of so-called animal electricity: R. C. Sha, “Volta’s Battery, Animal Electricity, and Frankenstein,” European Romantic Review, 23(1) (2012), 21-41.

[3] A. Volta, “On the Electricity Excited by the Mere Contact of Conducting Substances of Different Kinds,” Phil. Trans. R. Soc. Lond. 90 (1800), 403-431. This article is in French, but an English translation of it may be found under the same title in Philosophical Magazine, 7:2 (1800), 289-311. The figure on this page is a plate from near the end of Volta’s original paper.

[4] Details of Ohm’s experimental set up and analysis can be found in Joseph F. Keithley, The Story of Electrical Measurements from 500 BC to the 1940s, (New York: IEEE Press, 1999), 92-103.

[5] While the analogy between current and heat flow was useful to Ohm in formulating the law that now bears his name, it also misled him into making erroneous predictions. For example, based on the formal analogy with Fourier’s theory of heat, Ohm predicted that a conductor with charge would be in equilibrium if the charge were uniformly distributed throughout the volume of the conductor. It is easy to show (using Gauss’s law, for example, see Chapter 4) that this is not the case. In static equilibrium, the excess or net charge resides on the surface of the conductor.

[6] Kalil T. Swain Odham, The doctrine of description: Gustav Kirchhoff, classical physics, and “the purpose of all science” in 19th-century Germany, (PhD., University of California, Berkeley), 137-38.