Section 43–4 Ionic conductivity

Ionized gas / Electric current / Ohm’s law

 

In this section, Feynman begins with an ionized gas and uses the motion of its charged particles to derive an expression for electric current and show that, under simplifying assumptions, it obeys Ohm’s law. The logical progression is thus from ionized gas → drift velocity and mobility → electric current → electrical conductivity → Ohm’s law. Although the section is titled “ionic conductivity,” it is somewhat narrower than its actual scope. “Electrical Conductivity in an Ionized Gas” is an appropriate alternative title because it is broad enough to encompass the physical system, drift velocity, electrical mobility, electric current, and Ohm’s law.

 

1. Ionized gas

“Suppose we have a gas in a vessel in which there are also some ions—atoms or molecules with a net electric charge. We show the situation schematically in Fig. 43–2. If two opposite walls of the container are metallic plates, we can connect them to the terminals of a battery and thereby produce an electric field in the gas. The electric field will result in a force on the ions, so they will begin to drift toward one or the other of the plates. An electric current will be induced, and the gas with its ions will behave like a resistor. By computing the ion flow from the drift velocity we can compute the resistance (Feynman et al., 1963).”

 

A dilute ionized gas can be approximated as an ohmic resistor under three idealized conditions that suppress capacitive, inductive, and nonlinear plasma effects. First, the applied electric field must be strictly constant, so that all time-dependent effects vanish—displacement (capacitive) currents and induced electromotive forces become negligible, leaving conduction current as the sole transport mechanism. Second, the electric field must be sufficiently weak that the drift velocity of the ions remains linearly proportional to the field, avoiding nonlinear phenomena like avalanche ionization and electrical breakdown. Third, the current density must remain low enough to prevent space-charge accumulation, which would distort the electric field and alter the ions’ drift velocity. In most practical high-voltage discharges and plasma applications, however, these idealizations no longer hold, and the gas exhibits both resistive and reactive (capacitive and inductive) behavior.

 

Feynman’s characterization of an ionized gas as a resistor is therefore a valid first-order approximation for ion drift under steady DC conditions. From the broader perspective of classical electromagnetism and circuit theory, however, a dilute ionized gas confined between two electrodes is more accurately modeled as an equivalent RLC circuit. The gas possesses a finite resistance (or conductance) arising from collisional momentum transfer between drifting ions and neutral molecules. At the same time, the electrode-gas system has a finite capacitance originating from the electrode geometry and the formation of Debye sheaths near the electrode surfaces (Nation & Simpson, 1965). In addition, the current flowing through the gas generates a magnetic field, giving rise to a typically small self-inductance. Under time-varying conditions, both capacitive and inductive contributions become significant and must be included to accurately describe the system’s electrical response.

 

A more accurate introduction to the section might read: We now consider a dilute, weakly ionized gas confined between two metallic electrodes. The gas consists of neutral molecules together with a small concentration of ions. When a weak, constant electric field is applied, the ions drift through the neutral background molecules while undergoing frequent collisions, giving rise to an electric current. Under these conditions—constant ion density, constant mobility, and negligible space-charge effects—the gas behaves approximately as an ohmic conductor, allowing the resistance to be derived from the microscopic properties of the ions and their collisions. This wording makes explicit the assumptions that are only implicit in Feynman's presentation and clarifies that his derivation is intended for a weakly ionized dilute gas, not for a general plasma or strongly ionized gas.

 

2. Electric current

“The electric current to one of the plates is given by the total charge of the ions which arrive at the plate in a unit of time… If there are n_i ions per unit volume, the number which reach the plate in the time T is n_i⋅A⋅v_drift⋅T. The current I is the charge collected in T divided by T, so I=qn_iAv_drift (Feynman et al., 1963).”

 

A key assumption underlying Feynman’s expression for electric current, I = n_i⋅A⋅v_d⋅q is that the ion density (n_i) is a fixed property of the gas—a prescribed constant independent of the applied voltage and resulting current. In a real ionized gas, however, ion density is a dynamic quantity governed by competing processes of ionization, recombination, and collection at the electrodes (Raizer & Allen, 1997). As ions drift toward the electrodes, they are neutralized upon contact. Unless the ions are continually replenished by ionization mechanisms—such as ultraviolet radiation, cosmic rays, or thermal ionization—the overall ion density will gradually decrease. Furthermore, increasing the applied voltage modifies not only the drift velocity—it accelerates ion collection at the electrodes, depleting the ion density at a faster rate. This depletion breaks the linear relationship between the electric current and applied voltage, giving rise to nonlinear conduction.

“An electric current I is the flow of charge in a unit time. The electric current to one of the plates is given by the total charge of the ions which arrive at the plate in a unit of time. If the ions drift toward the plate with the velocity v_drift, then those which are within a distance (v_drift⋅T) will arrive at the plate in the time T. If there are n_i ions per unit volume, the number which reach the plate in the time T is n_i⋅A⋅v_drift⋅T (Feynman et al., 1963).”

Feynman’s definition of electric current as “the flow of charge in a unit time” is correct at an introductory level, but it could be more precise. Formally, electric current is the time rate at which net electric charge passes through a specified surface: I = dQ/dt. However, this definition does not state the transport mechanism. In Maxwell’s electrodynamics, the total current may include both conduction current (arising from the motion of charged particles) and displacement current (associated with time-varying electric fields). Furthermore, in media containing multiple charge carriers—such as electrons, positive ions, and negative ions—the net current density is the vector sum of contributions from all charge carriers. Experimentally, the electric current can be measured by various instruments including ammeters based on the magnetic, resistive, Hall effect, or thermal principles, depending on the magnitude and nature of the current.

3. Ohm’s law

“We find that the current is proportional to the voltage, which is just the form of Ohm’s law, and the resistance R is the inverse of the proportionality constant: 1/R=μq²n_i(A/b). (43.20) (Feynman et al., 1963).”

 

Ohm’s law states that the current through a conductor is directly proportional to the voltage across it, provided the resistance remains constant. This proportionality is not a fundamental law of physics but an empirical relation that holds only under specific physical conditions. First, the conductor must be strictly ohmic, meaning its charge carrier density and mobility are independent of the applied electric field, so that the drift velocity scales linearly with the field strength. Second, the temperature must remain constant, because heating increases the lattice vibrations and collision rates, thereby modifying the resistance and current-voltage relationship. Third, the applied voltage must remain weak enough to avoid high field phenomena such as impact ionization, or space-charge injection, which introduce nonlinearities (Raizer & Allen, 1997). While Ohm’s law provides an accurate description under these idealized conditions, any non-ohmic devices—such as semiconductors, diodes, or electrolytes—requires a non-linear or complex impedance model.

 

Note: Ohm’s law is not strictly valid on very short timescales because it assumes that electrons reach a steady drift velocity instantaneously. In Griffith’s (2013) words, “N. Ashby, Am. J. Phys. 43, 553 (1975), points out that for good conductors t is absurdly short (10^-19 s, for copper, whereas the time between collisions is t_c = 10^-14 s). The problem is that Ohm's law itself breaks down on timescales shorter than t_c; actually, the time it takes free charge to dissipate in a good conductor is of order t_c, not t (p. 412).”

 

An ionized gas generally does not obey Ohm’s law for three fundamental reasons. First, the number of charge carriers is not fixed: increasing the applied voltage can produce additional electrons and ions through ionization, so the ion density becomes a function of the electric field. Second, the mobility of the ions is field-dependent: higher voltage (electric field) increases the ions’ kinetic energies and collision frequencies, making the drift velocity nonlinearly dependent on the electric field. Third, space-charge effects cause electrons and ions drift at different speeds, distorting the local electric field and producing non-uniform voltage drops across the gas. Since the ion density, mobility, and local field distribution all depend on the applied voltage, the current-voltage relationship is fundamentally nonlinear, and the ionized gas is, in general, a non-ohmic medium.

 

Key Takeaways:

1. An ionized gas conducts electricity because charged particles—electrons and ions—drift through the gas under an applied electric field while collisions with other particles impede their motion.

2. Electric current is more precisely defined as the time rate of net electric charge flow across a specified area (I = dq/dt). In general, electric current encompasses both the physical drift of charge carriers (conduction current) and time-varying electric fields (displacement current).

3. Ohm’s law is an empirical approximation valid only for ohmic conductors under strict conditions: constant resistance, constant temperature, and weak electric fields. While Feynman’s model holds for weak DC electric fields, real ionized gases are fundamentally non-ohmic—stronger fields can change the ion density and electrical mobility, whereas space-charge effects can distort the electric field.

 

The Moral of the Lesson: Physiological function as an electrical system

Ionic—or electrical—conductivity is far more than a laboratory curiosity: it keeps us alive. Every thought, heartbeat, and muscle movement depend on the controlled flow of ions across cell membranes. Sodium, potassium, chloride, and calcium ions within our cells and blood must be maintained in delicate balance. When an electrolyte imbalance occurs—whether from a poor diet, excessive sweating, or severe fluid loss—it affects nerve conduction, muscle contraction, and cardiac function.

 

Examples of Electrolyte Imbalances:

• Hyponatremia (Low sodium): Can cause headache, confusion, and seizures. Sodium is essential for generating action potentials across cell membranes.
• Hypokalemia (Low potassium): Can cause muscle weakness, cramps, and cardiac arrhythmias. Potassium plays a central role in establishing the resting membrane potential and repolarizing excitable cells.
• Hypocalcemia (Low calcium): Increases neuromuscular excitability, leading to muscle spasms, tingling, and tetany. Calcium is crucial for neurotransmitter release and muscle contraction.
• Hypochloremia (Low magnesium): May cause muscle weakness, tremors, and cramps. Magnesium is a cofactor for ATP-dependent enzymes and is essential for normal nerve and muscle function.

 

Dietary Sources and Maintenance

A healthy diet (e.g., Keto Diet as shown below) is your first line of defense against electrolyte imbalance. While severe imbalance cannot always be resolved by diet alone, daily food intake plays a vital role in maintaining normal electrolyte levels:

Source: Keto Diet Electrolytes: Why They Matter & How to Get Enough?

Eggs provide several minerals, including potassium, phosphorus, and calcium, along with protein and other nutrients—they may not be the top source of any single mineral or electrolyte, but they add variety and nutritional value to your diet. The secret to electrolyte balance isn't chasing one nutrient—it's eating a colorful, varied diet: fruits, vegetables, dairy or fortified alternatives, beans, whole grains, and protein-rich foods like eggs.

 

Conclusions:

Importantly, electrolytes are far more than dietary minerals—they are the body’s essential charge carriers whose concentrations and movement across cell membranes underlie much of the body’s physiological function. While a balanced diet and adequate hydration usually keep these ion concentrations within the acceptable range, severe imbalances can become life-threatening emergencies requiring medical intervention. In short, maintaining electrolyte balance is not merely an application of ionic conduction or healthy eating, but it is the foundation of life itself.

 

Review Questions:

1. Ionized Gas as a Resistor: Under what assumptions can a dilute ionized gas be treated as an ohmic resistor? Identify the conditions under which this approximation breaks down.

2. Definition of Electric Current: How would you define electric current, both conceptually and mathematically? Explain how the electric current I is related to the microscopic drift velocity of charge carriers.

3. Validity of Ohm's Law: Under what conditions does Ohm’s law hold, and what physical mechanisms cause deviations from linearity?

 

References

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Griffiths, D. J. (2013). Introduction to Electrodynamics (3^rd ed.). Prentice Hall.

Raizer, Y. P., & Allen, J. E. (1997). Gas discharge physics (Vol. 2). Berlin: Springer.

Kasper, D. L., Fauci, A. S., Hauser, S. L., Longo, D. L., Jameson, J. L., & Loscalzo, J. (2021). Harrison's Principles of Internal Medicine (21st ed.). McGraw-Hill.

Mahan, L. K., & Raymond, J. L. (2020). Krause and Mahan's Food and the Nutrition Care Process (15th ed.). Elsevier.

Nation, J. A., & Simpson, D. (1965). A measurement of the effective thickness of the plasma sheath at a cold electrode. British Journal of Applied Physics, 16(11), 1705-1709.