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Chapter 12. Superposition of Electric Fields and Causality

Chapter 12. Superposition of Electric Fields and Causality

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Chapter 12

Superposition of electric fields and causality


Knowledge of the field makes it possible to predict what will happen to

any charge which happens to be placed at the point considered, as a result of

the positions and values of all the other charges.

In short, the superposition principle may be summed up thus1: every unit

charge present at a given time in a quasistatic situation (see chapter 5)

contributes in the same way to the electric field at any point M in space,

according to Coulomb’s law.

This principle may seem unsurprising and obvious, but is it easily

accepted? Is it correctly applied to all the situations that the statement

encompasses, or are the difficulties associated with some situations?

Through preliminary interviews with second year university students

specialising in science, two lines of reasoning were identified as constituting

obstacles;relevantproblematic situations were then proposed.

1. Students often reason as though a cause existed only when there is an

obvious effect (chapter 4). This way of thinking leads many to refuse the

possibility that a field may exist at a point where chargescannot move.

Hence the first series of questionnaires presented below, which involve

insulators: the aim is to call into question the idea that all charge contributes

to the field, even in a context that is mistakenly thought to be unfavourable

to the electric field.

2. The other aspect of reasoning studied here is the attribution of a causal

status to “formulae”: a quantity, X, mentioned in the algebraic expression of

another quantity G=f(X), is interpreted as an exclusive “cause” of the

phenomenon associated with that quantity G. Therefore, the contribution to

the total electric field of charges whose value is not explicitly included in the

expression of that field is not taken into account. Hence the second series of

questionnaires asks about the total field E near a conductor, which is

expressed by a formula which contains the surface charge density present

see box 2). It

on the conductor near the point being considered

may seem strange that it should be possible to express a field whose sources

are all the charges in the universe through such a local variable: in fact, that

variable itself reflects the locations of all the charges present in the situation,

since the distribution of charges on the conductor is determined by the

position of all the other charges.

The students questioned (1300 in all) have all completed their secondary

schooling and are in preparatory classes or at university in France, or at the

Stockholm Polytechnic. They are grouped in four categories in the analysis

of the results, according to the instruction they have received on the subject:


The idea underlying this statement is not limited to the particular context of electric fields, it

is highly important in physics.


Chapter 12

from a scientific baccalaureate level (G1) to a complete course on

electromagnetism (G4).2


The students questioned had received instruction on the following notions, respectively:

(G1): electric circuits and an introduction to electrostatics; the electric field E defined by the

relationship F=qE; the case of a uniform field, related to the capacitor, and the relationship

E=U/d (Where U is the voltage and d the distance between plates);

(G2): (in addition to G1:) Coulomb's law and Gauss’s theorem; the potential (V) defined by

the relationship E=-grad V; the conductivity defined by the relationship j=yE (where j is

the current density); a few rudiments of vector calculus, including the definition of a


(G3): (in addition to G2:) conductors in electrostatic equilibrium, the fact that the electric field

is zero inside such conductors; field near a conductor;

(G4): (in addition to G3:) dielectrics (or “insulators”) and Maxwell’s equations.

Superposition of electric fields and causality





Two questionnaires were used for this study: the “insulator”

questionnaire and the “Neapolitan ice cream” questionnaire.


The “insulator” questionnaire (INS)

The two items of this questionnaire raise the question of the action, inside

an insulator, of an external charge, and of the action, outside the insulator, of

an internal charge. In both cases, by virtue of the superposition principle, the

correct answer is that the charge creates a non-zero electric field at the point


INS 1:

A point charge is outside an insulating body. Does this charge create an

electric field at a point M inside this insulating body (see diagram


INS 2:

A point charge is inside an insulating body. Does this charge create an

electric field at a point M outside this insulating body (see diagram


Chapter 12



Main findings for the INS questionnaire

Table 1 presents the rate of answers (“YES”, “NO”, “I do not know,” or

“?” when no answer is provided). The category “E=0” was introduced

because this indirect answer is not as categorical as a “NO”: it might mean

that the total field is zero, but that the contribution of the charge q is not

negated for all that.

The rate of wrong answers is relatively high for each group, considering

the apparent simplicity of the question. It changes predictably with the level

of education.

But the students’ comments are good indicators of the reasoning that they


In addition to the justifications of correct answers (“YES, the charge

creates a field”), the comments fall into two main categories:

The first category consists of answers alluding to the “blocking role of

the insulator”:

“The insulating property of the body prevents the field from penetrating.”

(INS 1, G1)

“The insulator is globally subjected to the electric field created by q. But

M is not subjected to the electric field because it is protected by the

insulator.” (INS 1, G1)

“The insulator blocks the field inside the body.” (INS 2, G1).

The second category of comments all specify that the charges cannot


“As this body is an insulator, the charges inside it are immovable. If q

created a field, the charges inside would be subjected to an electric field.”

(INS 1, G2)

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