# How is it reasonable/correct to say that the magnetic current $\overline{\mathcal{M}}$ is a "fictitious source"?

Chapter 1.2 Maxwell's Equations of the textbook Microwave Engineering, fourth edition, by David Pozar says the following:

The general form of time-varying Maxwell equations, then, can be written in "point," or differential, form as

$$\nabla \times \overline{\mathcal{E}} = \dfrac{-\partial{\overline{\mathcal{B}}}}{\partial{t}} - \overline{\mathcal{M}}, \tag{1.1a}$$ $$\nabla \times \overline{\mathcal{H}} = \dfrac{\partial{\overline{\mathcal{D}}}}{\partial{t}} + \overline{\mathcal{J}}, \tag{1.1b}$$ $$\nabla \cdot \overline{\mathcal{D}} = \rho, \tag{1.1c}$$ $$\nabla \cdot \overline{\mathcal{B}} = 0 \tag{1.1d}$$ The MKS system of units is used throughout this book. The script quantities represent time-varying vector fields and are real functions of spatial coordinates $$x$$, $$y$$, $$z$$, and the time variable $$t$$. These quantities are defined as follows:

$$\overline{\mathcal{E}}$$ is the electric field, in volts per meter $$(\text{V}/\text{m})$$.
$$\overline{\mathcal{H}}$$ is the magnetic field, in empires per meter $$(\text{A}/\text{m})$$.
$$\overline{\mathcal{D}}$$ is the electric flux density, in coulombs per meter squared ($$\text{Coul}/\text{m}^2$$).
$$\overline{\mathcal{B}}$$ is the magnetic flux density, in webers per meter squared ($$\text{Wb}/\text{m}^2$$).
$$\overline{\mathcal{M}}$$ is the (fictitious) magnetic current density, in volts per meter $$(\text{V}/\text{m}^2)$$.
$$\overline{\mathcal{J}}$$ is the electric current density, in amperes per meter squared ($$\text{A}/\text{m}^2$$).
$$\rho$$ is the electric charge density, in coulombs per meter cubed ($$\text{Coul}/\text{m}^3$$).

The sources of the electromagnetic field are the currents $$\overline{\mathcal{M}}$$ and $$\overline{\mathcal{J}}$$ and the electric charge density $$\rho$$. The magnetic current $$\overline{\mathcal{M}}$$ is a fictitious source in the sense that it is only a mathematical convenience: the real source of a magnetic current is always a loop of electric current or some similar type of magnetic dipole, as opposed to the flow of an actual magnetic charge (magnetic monopole charges are not known to exist).

I'm curious about this part:

The magnetic current $$\overline{\mathcal{M}}$$ is a fictitious source in the sense that it is only a mathematical convenience: the real source of a magnetic current is always a loop of electric current or some similar type of magnetic dipole, as opposed to the flow of an actual magnetic charge (magnetic monopole charges are not known to exist).

Maxwell's equations are mathematics that describe the physical reality of electromagnetism. So how is it reasonable/correct to say that the magnetic current $$\overline{\mathcal{M}}$$ is a "fictitious source"? Since Maxwell's equations describe the physical reality of electromagnetic (that is, they are "correct" in describing electromagnetism), how does this make sense?

I wonder if this is an analogous situation to "quasiparticles," like phonons and plasmons, or other emergent phenomena, like "effective mass." But the way this is written ("the real source of a magnetic current is always a loop of electric current or some similar type of magnetic dipole") makes it sound like something something else – like a mistake/falsity.

## 1 Answer

I have three statements:

1- Maxwell's equations do not originally include the magnetic current term you see in electrical engineering related texts, like Pozar book. And, the theory does not need that term to describe the phenomena.

2- However, adding that term in the equations (as if there were real magnetic currents) is very useful to derive a duality theory that helps to solve electromagnetics problems. In such theory, you will find that the fields of an electric dipole is totally dual to the fields of a current loop, which is the magnetic dipole he mentions.

3- You said "Maxwell's equations are mathematics that describe the physical reality of electromagnetism". In fact, No they are not. They describe "our thoughts and understanding" of that reality, "on the macroscopic scale".

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Oct 18, 2022 at 6:23