I have read, in many places, statements like this:

Heaviside was able to greatly simplify Maxwell's 20 equations in 20 variables, replacing them by four equations in two variables. Today we call these 'Maxwell's equations' forgetting that they are in fact 'Heaviside's equations'.

... but nowhere can I find exactly what these equations look like in the form Heaviside produced. There are many versions of the equations, most with more than just four.

(From this biographical article)

Please can anyone tell me exactly what are the original "four equations in two variables"?


1 Answer 1


This gives the four equations in the form Heaviside came up with: $$\varepsilon E = \rho$$ $$\nabla \times E = - \mu \frac{\partial H}{\partial t}$$ $$\nabla \cdot \mu H = 0$$ $$\nabla \times H = k E + \varepsilon \frac{\partial E}{\partial t}$$

where $E$ represents the electric field, $H$ represents the magnetic field, $\varepsilon$ is the permittivity, $\mu$ is the permeability, $\rho$ is the charge density, and $k$ is the conductivity.

Here, $\times$ and $\cdot$ denote the cross product and dot product. Also, $H$ is not the same as the perhaps more commonly used $B$.

Note: The first equation should properly be $\varepsilon \nabla \cdot E = \rho$, though the source leaves it out, probably due to a typo.


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