We can clearly see in page $6$ of Pozar's Microwave Engineering Faraday's law written under this form

$$\nabla \times \bar{\mathcal{E}}= \frac{\partial \bar{\mathcal{B}}}{\partial t}-\bar{\mathcal{M}}$$ $\bar{\mathcal{E}}$ is the electric field, in volts per meter (V/m).

$\bar{\mathcal{B}}$ is the magnetic flux density, in webers per meter squared (Wb/m²).

$\bar{\mathcal{M}}$ is the (fictitious) magnetic current density, in volts per meter (V/m²).

I can't find in any other reference (Jackson, Griffiths,...) a form of Faraday's law in which $\bar{\mathcal{M}}$ appears. Is the form above simply a mistake?

  • 2
    $\begingroup$ See Griffiths eqn. 7.44 $\endgroup$
    – d_b
    Commented Jun 2, 2020 at 16:00

1 Answer 1


It is not a mistake. In your quote Pozar explicitly states that this magnetic current density $\mathcal{M}$ is a fictitious one. There are no magnetic currents for there are no magnetic charges. RF engineers, more specifically antenna engineers do like to introduce a practical analogy to the real current $\mathcal{J}$ of true electrical charges. The underlying reason for this is the very real analogy between a true $+e,-e$ electric dipole and its radiation pattern and a magnetic dipole moment of a loop current and its radiation pattern. The static field of an electric dipole and that of a steady state loop current is the same far away from their sources. Prompted by this similarity that carries on into the radiation pattern of their corresponding far fields one can represent the behavior of a loop antenna as resulting from a pair of fictitious oscillating magnetic charges $+m,-m$ that form a radiating dipole, a magnetic Hertzian dipole. A somewhat more sophisticated example would be to treat a radiating slot antenna as a magnetic analogue of a conventional electric wire antenna (see, also Babinet's principle). You can even go a step further a treat an array of radiating slots as being of a sheet of magnetic surface currents, etc.

  • $\begingroup$ Very cool. Shouldnt it be possible to derive this 'new' equation from the fundamental maxwell equations? $\endgroup$
    – lalala
    Commented Jan 31 at 7:15
  • $\begingroup$ @lalala there is no "derivation", you just postulate the existence of such an $\vec {\mathcal M$ and then you can derive the equivalent Helmholtz equation from the modified Maxwell as in the question with this magnetic current as source. $\endgroup$
    – hyportnex
    Commented Jan 31 at 12:52
  • $\begingroup$ If it cannot be derived from maxwell, wouldnt it mean, that maxwell is incomplete and that would be a rather bigger thing? if it is just a computational tool, than it shd be derivable, no? $\endgroup$
    – lalala
    Commented Feb 1 at 14:20
  • $\begingroup$ @lalala There are no magnetic charges or magnetic currents, certainly not in classical physics (Maxwell) and by all accounts not even in HEP physics. It is an engineering trick to notice the analogy between the radiation field $E,H$ of a small oscillating electric dipole (Hertz) and the radiation field $H, E$ of a small oscillating current loop. Antenna engineers use it as a convenient analogy, nothing deeper than most of linear circuit theory has analogies between "Thevenin" and "Norton", impedance and admittance, thinking of inductors as current sources in a power supply, etc., $\endgroup$
    – hyportnex
    Commented Feb 1 at 14:34
  • $\begingroup$ thanks. How would you calculate that without the 'trick'? $\endgroup$
    – lalala
    Commented Feb 2 at 8:57

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