Interchanging $c\mathbf{B}$ with $\mathbf{E}$ in Ampere's law

Question

I am reading Purcell's Electricity and Magnetism in the section about the displacement current in Maxwell's equations. He states that because the Lorentz-transformations of the electric and magnetic field are symmetric in $\mathbf{E}$ and $c\mathbf{B}$, we can interchange the roles of $\mathbf{E}$ and $c\mathbf{B}$ in Ampere's law so that $$ \nabla \times \mathbf{B} = -\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial{t}} $$ I don't understand why this is true. Why can the roles of $\mathbf{E}$ and $c\mathbf{B}$ be interchanged?

Answer 1

In Purcell Ch 6, he says "The equations in the box confront us with an astonishing fact, their symmetry with respect to E and B. If the printer had mistakenly interchanged E's with B's, and y's with z's, the equations would come out exactly the same!"--- so, you have swap y and z as well.

In Griffith's, "Problem 7.64 (a) Show that Maxwell's equations with magnetic charge (Eq. 7.44) are invariant under the duality transformation (7.68) $$\vec E' = \vec E \cos \alpha + c\vec B\sin\alpha$$ $$c\vec B' = c\vec B \cos \alpha — \vec E \sin \alpha,$$ $$cq'_e = cq_e \cos\alpha + q_m \sin \alpha $$ $$q'_m = q_m \cos \alpha — cq_e \sin \alpha,$$ where $c = 1/\sqrt{\epsilon_0 \mu_0}$ and $\alpha$ is an arbitrary rotation angle in "$\vec E/\vec B$-space." Charge and current densities transform in the same way as $q_e$ and $q_m$. [This means, in Chapter 7 Electrodynamics particular, that if you know the fields produced by a configuration of electric charge, you can immediately (using $\alpha=90^\circ$) write down the fields produced by the corresponding arrangement of magnetic charge.]

Try using $\alpha=90^\circ$.

The fancier term is "Hodge duality". https://en.wikipedia.org/wiki/Hodge_star_operator

In 3-dimensions, this duality allows one to associate an axial vector with a bivector (as in the cross-product). Note: 3-2=1... a sort of complimentary dimension.

In 4-dimensions, this duality allows a 2-form (like the electromagnetic field tensor) to be mapped to its dual (4-2)-form.