Ampère–Maxwell law for a conducting wire: a topological perspective

Over the years the Ampère–Maxwell law, cast in its differential form and/or in its integral form, has been the subject of considerable interest, partly due to its relevance to topics such as whether the displacement current term is independent of the conduction current source, whether such a term can be derived from the Biot–Savart law, and the concept of displacement current; e.g. see [1–9] and references therein. In contrast to such an abundance of literature on the displacement and conduction current terms of Ampère–Maxwell's law, the discussion of a such law in a topological context has been confined to specialized areas such as electromagnetic theory and knots, Ampèrian and current loops interchangeability; e.g. see [10] and references therein. In a recent study, Anacleto et al [11] explored the intrinsic symmetry of Ampère's circuital law from an educational perspective and illustrated the concept of displacement current. Although some concepts from topology were used within the context of the examples treated therein, Anacleto et al fell short of recasting Ampère–Maxwell's law from a topological perspective and, as far as is known to us, this remains a gap in the existing literature. The aim of the present study is to help to fill this gap by revisiting, from a topological perspective, the integral form of Ampère–Maxwell's law for an arbitrarily-shaped conducting wire connecting two opposite charges, in order to answer the question of whether such a law can be recast in a unified form that does not require the separation of the conduction current from the displacement current. We believe that this discussion brings new insights into Ampère–Maxwell's law, being therefore relevant from a scientific and an educational perspective.

Previously, [7, 11] used a simple example of two opposite charges placed at the ends of a straight conducting segment to illustrate Ampère–Maxwell's law (referred to in [11] as the generalized Ampère circuital law). Abandoning the symmetry restrictions of [7, 11], consider an arbitrarily-shaped conducting wire connecting two charges +Q and −Q (Q > 0), as illustrated in figure 1. At any point P on the conducting wire, a solid angle Ω is subtended by an arbitrarily-shaped closed curve C. As explained in figure 1, such a choice of solid angle is not unique. If we set such an angle at +Q to be Ω = Ω⁺ (figure 2), such a value of Ω undergoes a continuous variation along the conducting wire path from +Q to −Q, so that at −Q, Ω = Ω⁻.

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Consider a decreasing current I which flows through the conducting wire as Q decreases, originating a magnetic field $\vec B$ whose circulation along C (figure 2) is given by the Ampère–Maxwell law:

$$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 \left( {\int_{\rm S} {\vec J \cdot {\rm d}\vec a} + \varepsilon _0 \frac{\partial }{{\partial t}}\int_{\rm S} {\vec E \cdot {\rm d}\vec a} } \right){\rm ,} \end{equation} \tag{ 1 }$$

where $\vec J$ and $\vec E$ are the conduction current density and the electric field at each point of any open surface S bounded by C, the second RHS term in equation (1) being the displacement current across S, denoted as

$$\begin{equation} I_{\rm D} {\rm (}{\rm S )} = \varepsilon _0 \frac{\partial }{{\partial t}}\int_{\rm S} {\vec E \cdot {\rm d}\vec a} . \end{equation} \tag{ 2 }$$

Assuming that the decrease in I is sufficiently slow for Coulomb's law to be valid, the electric field flux across S in figure 2 due to charges +Q and −Q is

$$\begin{equation} \int_{\rm S} {\vec E \cdot {\rm d}\vec a} = \frac{Q}{{4\pi \varepsilon _0 }}\left( {\int_{\rm S} {\frac{{\hat r_ + \cdot {\rm d}\vec a}}{{r_ + ^2 }}} - \int_{\rm S} {\frac{{\hat r_ - \cdot {\rm d}\vec a}}{{r_ - ^2 }}} } \right){\rm ,} \end{equation} \tag{ 3 }$$

$\hat r_ +$ and $\hat r_ -$ being the unit vectors ${{\vec r_ + } / {r_ + }}$ and ${{\vec r_ - } / {r_ - }}$, respectively. Since Q decreases with time, we have I = −dQ/dt > 0, i.e. the conducting wire current is always positive. From equations (2) and (3), the displacement current across S is given by

$$\begin{equation} I_{\rm D} {(\rm S)} = - \frac{I}{{4\pi }}\left( {\int_{\rm S} {\frac{{\hat r_ + \cdot {\rm d}\vec a}}{{r_ + ^2 }}} - \int_{\rm S} {\frac{{\hat r_ - \cdot {\rm d}\vec a}}{{r_ - ^2 }}} } \right){\rm ,} \end{equation} \tag{ 4 }$$

which can be related to solid angles Ω⁺ and Ω⁻ in figure 2, it being instructive to express the circulation of the magnetic field, i.e. equation (1), as a function of Ω⁺ and Ω⁻, for which the following four possible situations are considered.

• (1)  
  S intersects the conducting wire, and wire current I exits S (figure 2): from figure 2, the solid angles through which charges +Q and −Q view S are, respectively, $\int_{\rm S} {\frac{{\hat r_ + \cdot {\rm d}\vec a}}{{r_ + ^2 }}} = - 4\pi + \Omega ^ +$ and $\int_{\rm S} {\frac{{\hat r_ - \cdot {\rm d}\vec a}}{{r_ - ^2 }}} = \Omega ^ -$. Inserting these angles into equation (4), gives the displacement current across S as

  $$\begin{equation} I_{\rm D} {\rm (} {\rm S )} = - I\left( {\frac{{\Omega ^ + - \Omega ^ - }}{{4\pi }} - 1} \right). \end{equation} \tag{ 5 }$$

  Since the current I exits S, $\int_{\rm S} {\vec J \cdot {\rm d}\vec a}$ is negative, and I = −dQ/dt being a positive quantity we have that $\int_{\rm S} {\vec J \cdot {\rm d}\vec a} = - I$, which together with equation (5) can be inserted into equation (1), enabling the Ampère–Maxwell law to be written as

  $$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 I\frac{{\Omega ^ - - \Omega ^ + }}{{4\pi }}. \end{equation} \tag{ 6 }$$
• (2)  
  S intersects the conducting wire, and wire current I enters S (figure 3): from figure 3, the solid angles through which charges +Q and −Q view S are, respectively, $\int_{\rm S} {\frac{{\hat r_ + \cdot {\rm d}\vec a}}{{r_ + ^2 }}} = \Omega ^ +$ and $\int_{\rm S} {\frac{{\hat r_ - \cdot {\rm d}\vec a}}{{r_ - ^2 }}} = - 4\pi + \Omega ^ -$, and inserting these angles into equation (4), gives the displacement current across S as

  $$\begin{equation} I_{\rm D} {\rm (S)} = - I\left( {1 + \frac{{\Omega ^ + - \Omega ^ - }}{{4\pi }}} \right). \end{equation} \tag{ 7 }$$

  Carefully note that, since for this situation the current I enters S, $\int_{\rm S} {\vec J \cdot {\rm d}\vec a}$ is positive, and I = −dQ/dt being a positive quantity we have that $\int_{\rm S} {\vec J \cdot {\rm d}\vec a} = I$ which, together with equation (7), can be inserted into equation (1) giving a magnetic field circulation identical to that in equation (6) of the previous situation, an expected result since the magnetic field circulation around C has to be independent of the choice of surface bounded by C (see figure 2 with 3).
• (3)  
  S does not intersect the conducting wire, the latter viewing S from its concave side (figure 4): from figure 4, the solid angles through which charges +Q and −Q view S are, respectively, $\int_{\rm S} {\frac{{\hat r_ + \cdot {\rm d}\vec a}}{{r_ + ^2 }}} = - 4\pi + \Omega ^ +$ and $\int_{\rm S} {\frac{{\hat r_ - \cdot {\rm d}\vec a}}{{r_ - ^2 }}} = - 4\pi + \Omega ^ -$, and inserting these angles into equation (4), gives the displacement current across S as

  $$\begin{equation} I_{\rm D} {(\rm S)} = - I\left( {\frac{{\Omega ^ + - \Omega ^ - }}{{4\pi }}} \right). \end{equation} \tag{ 8 }$$

  Since for this situation the wire current I does not cross S, $\int_{\rm S} {\vec J \cdot {\rm d}\vec a} = 0$, which together with equation (8) can be inserted into equation (1) giving again, as expected, a magnetic field circulation identical to that in equation (6).
• (4)  
  S does not intersect the conducting wire, the latter viewing S from its convex side (figure 5): from figure 5, the solid angles through which charges +Q and −Q view S are, respectively, $\int_{\rm S} {\frac{{\hat r_ + \cdot {\rm d}\vec a}}{{r_ + ^2 }}} = \Omega ^ +$ and $\int_{\rm S} {\frac{{\hat r_ - \cdot {\rm d}\vec a}}{{r_ - ^2 }}} = \Omega ^ -$, and inserting these angles into equation (4) gives a displacement current which, as expected, is identical to that in equation (8) and, since as in situation 3 we have $\int_{\rm S} {\vec J \cdot {\rm d}\vec a} = 0$, inserting these values into equation (1) again gives the expected result for the magnetic field circulation, i.e. equation (6).

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Inspecting equation (6), it is clear that the circulation of $\vec B$ along C depends only on the conducting wire current I and on the difference in solid angles Ω⁻ − Ω⁺. Carefully note that, from the previous discussion in the context of figure 1, such a difference must be understood as the variation in solid angle along the conducting wire path from +Q to −Q. In other words, for a set value of Ω⁺, the angle Ω⁻ is determined not only by the position of charge −Q, but also by the path connecting +Q to −Q. For instance, consider the arrangement in figure 6. In going from z = z₁ to z = z₂ through C (path 1) the solid angle Ω, being initially Ω⁺, gradually increases towards 2π as one approaches C, and continues to increase as one goes through C until it reaches the final value Ω⁻(1). In this case, equation (6) gives the magnetic field circulation around C as

$$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 I\frac{{\Omega ^ - {\rm (}1{\rm )} - \Omega ^ + }}{{4\pi }}. \end{equation} \tag{ 9 }$$

If z₁ = +∞ and z₂ = −∞, then Ω⁺ = 0 and Ω⁻(1) = 4π, which inserted into equation (9) gives the well-known result for an infinite conducting wire that goes through C, i.e. $\oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 I$. On the other hand, if the conducting wire from charge +Q to charge −Q does not go through C (path 2 in figure 6), the solid angle Ω, being initially Ω⁺, gradually collapses towards zero, overlaps, changes sign to negative, and continues to decrease until it reaches a value whose magnitude is |Ω⁻(2)| = 4π − Ω⁻(1), i.e. Ω⁻(2) = Ω⁻(1) − 4π. In this case, equation (6) gives the magnetic field circulation as

$$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 I\frac{{\Omega ^ - {\rm (}2{\rm )} - \Omega ^ + }}{{4\pi }} \end{equation} \tag{ 10a }$$

$$\begin{equation} \hphantom{\oint_{\rm C} {\vec B \cdot {\rm d}\vec l}}= \mu _0 I\left[ {\frac{{\Omega ^ - {\rm (}1{\rm )} - \Omega ^ + }}{{4\pi }} - 1} \right]. \end{equation} \tag{ 10b }$$

If z₁ = +∞ and z₂ = −∞, then Ω⁺ = 0 and Ω⁻(2) = 0, which inserted into equation (10a) gives the well-known result for an infinite conducting wire that does not go through C, i.e. $\oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = 0$. Using the superposition principle, subtracting equation (10b) from equation (9) we obtain the current loop $z_1 \mathop \to \limits_{{\rm (}1{\rm )}} z_2 \mathop \to \limits_{{\rm (}2{\rm )}} z_1$, giving

$$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 I\left( {\frac{{\Omega ^ - {\rm (}1{\rm )} - \Omega ^ + }}{{4\pi }}} \right) - \mu _0 I\left( {\frac{{\Omega ^ - {\rm (}1{\rm )} - \Omega ^ + }}{{4\pi }} - 1} \right) = \mu _0 I{\rm ,} \end{equation} \tag{ 11 }$$

which is identical to the well-known result for magnetic field circulation produced by a current loop that interlocks with C. However it is more instructive to model the interlocking closed loop system using figure 7, due to its relevance to the subsequent interpretation of equation (6).

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In figure 7, the solid angle Ω, being initially Ω⁺, increases towards 2π as one approaches C, continues to increase as one goes through C, on overlap reaches the value of 4π, and continues to increase towards the final value of Ω⁻ = 4π + Ω⁺ if charge −Q is infinitely close to +Q. The important feature here is that, unlike in the previous examples considered, the solid angle Ω⁻ exceeds 4π. Inserting the aforementioned value of Ω⁻ into equation (6) again gives the expected result of μ₀I for the circulation around C of the magnetic field produced by a closed loop of current interlocking C. Evidently, if one considers the surface S shown in figure 7, the displacement current obtained by inserting Ω⁻ into equation (7) is zero.

It is important to stress at this juncture that equation (6) was derived herein for an arbitrarily-shaped conducting wire and that, since in this context equation (6) is equivalent to equation (1), it implicitly contains both the conduction current and displacement current contributions. On rearranging equation (6) as

$$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 \frac{{\rm d}}{{{\rm d}t}}\left[ {( { + Q} )\frac{{\Omega ^ + }}{{4\pi }} + ( { - Q} )\frac{{\Omega ^ - }}{{4\pi }}} \right]{\rm ,} \end{equation} \tag{ 12 }$$

it becomes apparent that the square bracketed term in equation (12) represents the sum of the fraction of charge ( + Q) contained within Ω⁺ with the fraction of the charge ( − Q) contained within Ω⁻; see figures 2–7. Note carefully, however, that such individual fractions must be understood in a generalized sense, since either Ω⁺ or Ω⁻ may individually be larger than 4π, e.g. see figure 7, even though the magnitude of the sum within square brackets does not exceed Q. Equivalently, writing equation (12) as

$$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \mu _0 \varepsilon _0 \frac{{\rm d}}{{{\rm d}t}}\left[ {\frac{{( { + Q} )}}{{4\pi \varepsilon _0 }}\Omega ^ + + \frac{{( { - Q})}}{{4\pi \varepsilon _0 }}\Omega ^ - } \right]{\rm ,} \end{equation} \tag{ 13 }$$

the bracketed term in equation (13) represents the sum of the flux through Ω⁺ of the electric field produced by charge ( + Q), with the flux through Ω⁻ of the electric field produced by charge ( − Q); see figures 2–7. Evidently, such electric field flux must again be understood in a generalized sense. For instance, in figure 7 we have Ω⁻ = 4π + Ω⁺, and therefore the flux through Ω⁻ of the electric field produced by individual charge −Q covers a region of space larger than 4π, even though the magnitude of the sum within square brackets in equation (13) does not exceed Q/ε₀; in fact for this case it equals Q/ε₀. The aforementioned result, that for Ω⁻ − Ω⁺ = 4π we get $\frac{Q}{{4\pi \varepsilon _0 }}( {\Omega ^ - - \Omega ^ + } ) = \frac{Q}{{\varepsilon _0 }}$, parallels Gauss's law for the electric field flux produced by a charge Q through a closed surface that encloses Q. On the other hand, if the conducting wire in figure 7 did not go through C, then Ω⁻ − Ω⁺ = 0 and we would get $\frac{Q}{{4\pi \varepsilon _0 }}( {\Omega ^ - - \Omega ^ + } ) = 0$, which again parallels Gauss's law for the electric field flux produced by a charge Q through a closed surface that does not enclose Q. Designating the generalized electric field flux as

$$\begin{equation} \phi _E^* = \frac{{( { + Q})}}{{4\pi \varepsilon _0 }}\Omega ^ + + \frac{{( { - Q} )}}{{4\pi \varepsilon _0 }}\Omega ^ - {\rm ,} \end{equation} \tag{ 14 }$$

equation (13) can be written as

$$\begin{equation} \oint_{\rm C} {\vec B \cdot {\rm d}\vec l} = \frac{1}{{c^2 }}\frac{{{\rm d}\phi _E^* }}{{{\rm d}t}}{\rm ,} \end{equation} \tag{ 15 }$$

showing that, for the conducting wire, the definition of a generalized flux of the electric field which intrinsically contains both conduction current and displacement current contributions enabled Ampère–Maxwell's equation for magnetic field circulation to be expressed in a form which parallels that in the absence of conduction current, i.e. equation (1) without the first RHS term. Moreover, equation (6) removed the need both to select a supporting surface for C and to consider individual conduction and displacement current density flux contributions across such a surface, with only the conducting wire current I and the solid angles Ω⁺ and Ω⁻ being needed to determine the magnetic field circulation.

The integral form of Ampère–Maxwell's law for an arbitrarily-shaped conducting wire connecting two charges (+Q and −Q) was recast using a topological perspective. The circulation of the conducting wire's magnetic field around an arbitrarily-shaped loop C was found to be proportional to the wire current and to the difference in solid angles subtended at +Q and −Q by C, the value of such a difference being dependent on the path of the conducting wire connecting +Q to −Q. Such an approach removed the need both to define a supporting surface for C and to consider separate conduction and displacement current density contributions across such a surface. For the conducting wire, the definition of a generalized flux of the electric field which implicitly contained both conduction and displacement current terms enabled Ampère–Maxwell's equation for magnetic field circulation to be expressed in a form which paralleled that in the absence of conduction current. It is hoped that the topological perspective presented herein will prove useful both in a scientific and educational context.