Electric field

Question: Define electric field and explain why, when defining the electric field, the test particle must be stationary.

The electric field can be defined as the electric force per unit charge acting on a test charge that is stationary. An alternative definition of the electric field is the electric force per unit charge experienced by an infinitely small test charge placed in a vacuum. Mathematically, it can be expressed by the equation E = F/q, where F is the electric force experienced and q is the infinitely small test charge. This is because a large test charge can influence the charge carriers in the surrounding and distort the electric fields.

It is surprising that a physics teacher provides the following answer for the need of a stationary test charge: “a moving charge will generate electromagnetic waves that can interfere with the original electric field.” Strictly speaking, it is the accelerating charge carrier that produces electromagnetic waves. However, it is possible to have gravitational fields, electric fields, and magnetic fields in a region in space. Thus, physicists may specify the condition that the test particle is stationary because the net force could include a magnetic force for the moving test particle in addition to the electric (or electrostatic) force.

How would Feynman answer?

Feynman may answer this question from the perspectives of definition of the electric field, the condition of stationary test charge, and problems of defining the electric field.

1. Definition of the electric field:

Feynman would first explain that the definition of the electric field is based on Coulomb’s law and we need to assume that all other charges remain in the same positions. In his own words, “[w]hen applying Coulomb’s law, it is convenient to introduce the idea of an electric field. We say that the field E(1) is the force per unit charge on q₁ (due to all other charges). Dividing Eq. (4.9) by q₁, we have, for one other charge besides q₁, E(1) = (1/4πϵ₀)q²/r². Also, we consider that E(1) describes something about the point (1) even if q₁ were not there — assuming that all other charges keep their same positions (Feynman et al., 1964, section 4–2 Coulomb’s law; superposition).” Importantly, Feynman considers the electric field to be due charges that come in packages like electrons and protons, and think of them as being spread out in a continuous distribution.

To explain the concept of electric field, Feynman says that “[t]his potentiality for producing a force is called an electric field. When we put an electron in an electric field, we say it is “pulled.” We then have two rules: (a) charges make a field, and (b) charges in fields have forces on them and move (Feynman et al., 1963, section 2.2 Physics before 1920).” In other words, Feynman visualizes a charged particle creates a “condition” in space such that when we put another charged particle there, it feels an electric force. Furthermore, the nature of electric field can be illustrated with an analogy as follows: when two corks (charge particles) are floating in a pool of water, we can exert a force on a floating cork by giving another cork a gentle push. We can explain that there is a disturbance in the water (electric field) caused by a cork, and the water then disturbs the other cork “directly.”

2. Stationary test charge:

According to Feynman, the electrical force does not precisely follow Coulomb’s law that decreases inversely as the square of the distance between charges. His notion of electrical force includes the concept of magnetic force which a moving charged particle experiences. Thus, Feynman explains that “the electrical forces depend also on the motions of the charges in a complicated way. One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect. That is why we call the subject ‘electromagnetism.’ It is true that when charges are standing still the Coulomb force law is simple, but when charges are moving about the relations are complicated by delays in time and by the effects of acceleration, among others (Feynman et al., 1964, section 1–1 Electrical forces). In a sense, the magnetic force is a relativistic effect that complicates the electrical force on the moving charge.

On the other hand, Feynman elaborates that “[w]e find, from experiment, that the force that acts on a particular charge—no matter how many other charges there are or how they are moving — depends only on the position of that particular charge, on the velocity of the charge, and on the amount of charge. We can write the force F on a charge q moving with a velocity v as F = q(E + v×B) (Feynman et al., 1964, section 1–1 Electrical forces).” Essentially, the measured force is dependent on an observer’s frame of reference.

3. Problems in defining electric field:

Strictly speaking, the electric field is not simply equal to (1/4πϵ₀)q/r². Feynman would discuss problems of defining electric field at a point P as follows: “The electric field, E, is given by 

What do the various terms tell us? Take the first term, E= −qe_r′/4πϵ₀r′². That, of course, is Coulomb’s law, which we already know: q is the charge that is producing the field; e_r′ is the unit vector in the direction from the point P where E is measured, r is the distance from P to q. But, Coulomb’s law is wrong. The discoveries of the 19^th century showed that influences cannot travel faster than a certain fundamental speed c, which we now call the speed of light (Feynman et al., 1963, section 28–1 Electromagnetism).” That is, Coulomb’s law is not exactly correct because the influences (or signals) cannot travel faster than the speed of light and this can result in a delay in action. The time delay (or retarded time) is the time it takes for the influences moving at the speed of light to travel from the charge to the point P.

Moreover, Feynman might add that “[f]or those purists who know more (the professors who happen to be reading this), we should add that when we say that (28.3) is a complete expression of the knowledge of electrodynamics, we are not being entirely accurate. There was a problem that was not quite solved at the end of the 19^th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today (Feynman et al., 1963, section 28–1 Electromagnetism).” In other words, Feynman believes that there is no complete answer to the puzzle of self-force (or self-action).

          To conclude, Feynman would define electric field as the electric force per unit charge due to a distribution of charges (or protons and electrons). Experimentally, the measured force would include magnetic force if the test charge is not stationary. Importantly, he might discuss problems of defining electric field due to a time delay in the speed of influences and self-force of the electric charge.

References:

1. 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.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.