# Electromagnetism duality theorem

Concerning Electromagnetism, textbooks often refer to the Duality Theorem. Sometimes it is presented like this:

«Consider the Maxwell's Equations (with phasors) and a known field $\mathbf{E}_1$, $\mathbf{H}_1$:

$\nabla \times \mathbf{E}_1 = - j \omega \mu \mathbf{H}_1$

$\nabla \times \mathbf{H}_1 = j \omega \epsilon \mathbf{E}_1$

If $\mathbf{E}_1$ is replaced with $\mathbf{H}_2$ (the magnetic field of another electromagnetic field: $\mathbf{E}_2$, $\mathbf{H}_2$) and $\mathbf{H}_1$ is replaced with $-\mathbf{E}_2$, $\mu$ is replaced with $\epsilon$ and $\epsilon$ with $\mu$, then the above equations become respectively

$\nabla \times \mathbf{H}_2 = j \omega \epsilon \mathbf{E}_2$

$\nabla \times \mathbf{E}_2 = - j \omega \mu \mathbf{H}_2$»

They are valid Maxwell's Equations too. But now what does follow from this substitution?

1) Should be $\mathbf{E}_1 = \mathbf{H}_2$ and $\mathbf{H}_1 = -\mathbf{E}_2$? It is dimensionally incorrect.

2) I read also $\mathbf{E}_1 = \eta \mathbf{H}_2$ and $\eta \mathbf{H}_1 = -\mathbf{E}_2$.

My question is double: what is the advantage of Duality Theorem and which is the correct form between the two just written?

Thank you anyway!

• regarding the dimensions, there are many different unit conventions you can use, and although E and H have different dimensions in SI units, they are the same in cgs. – George G Mar 25 '14 at 20:11

The idea is that Maxwell's equations are symmetric under the exchange of certain quantities. This does not mean that you identify the electric and the magnetic field, but just say that they are dual to each other. This resolves your problem of dimensionality.

If one extends this principle to the two equations which contain divergences of the fields, one encounters an interesting statement: in order for the duality to hold in this case, the existence of magnetic monopoles is required. One can introduce a so-called monopole-charge which would be the dual of the electric charge. One consequence would be the quantization of electric charge. Unfortunately, magnetic monopoles have not been found in nature.

A well-known application of the duality you refer to occurs in the design of antennas, for example, magnetic vs. electric dipole antennas. A magnetic dipole is a loop antenna, an electric dipole is a linear antenna, still their radiation pattern is similar as one replaces E with H.