Problem — E-B symmetry and asymmetries

Problem

The four Maxwell equations show a partial symmetry between the electric field $\vv E$ and the magnetic field $\vv B$. Identify at least three structural asymmetries between the pairs of analogous laws.

Solution

The four laws (compact form):

$\nabla\cdot\vv E = \frac{\rho}{\varepsilon_0}\qquad \nabla\cdot\vv B = 0$

$\nabla\times\vv E = -\frac{d\vv B}{dt}\qquad \nabla\times\vv B = \mu_0\vv j + \mu_0\varepsilon_0\frac{d\vv E}{dt}$

First asymmetry — the sources (Gauss’s laws). Gauss’s law for the electric field has a source term, the charge density: $\nabla\cdot\vv E = \rho/\varepsilon_0$. Gauss’s law for the magnetic field has a zero right-hand side: $\nabla\cdot\vv B = 0$. This reflects the existence of isolated electric charges but the absence of magnetic monopoles ($\vv B$ field lines are always closed).

Second asymmetry — current and varying flux (curl laws). In the Ampère-Maxwell law, $\vv B$ has two sources: the conduction current $\vv j$ and the varying electric flux $d\vv E/dt$. In Faraday’s law, however, $\vv E$ is generated only by the varying magnetic flux $-d\vv B/dt$: there is no analogous “magnetic current” term (again due to the absence of monopoles).

Third asymmetry — signs and constants. A minus sign appears in Faraday’s law ($-d\vv B/dt$, Lenz’s law) with no symmetric positive counterpart in the other law; furthermore, the roles are weighted by different constants, $\varepsilon_0 \neq \mu_0$, so that $\vv E$ and $\vv B$ do not enter the equations in a perfectly interchangeable way.

$\ev{\text{(1) sources: } \rho \text{ vs } 0;\ \text{(2) double source for } \vv B \text{ vs single source for } \vv E;\ \text{(3) Lenz's sign and } \varepsilon_0\neq\mu_0}$

The symmetry becomes almost perfect only in a source-free vacuum ($\rho = 0$, $\vv j = 0$): it is precisely there that the equations take a symmetric form and admit electromagnetic waves as a solution.

Connections

Topics: Electromagnetic waves Concepts: Maxwell’s equations