The Traveling Electromagnetic Wave, Quantitatively — Question 6

Question 6

A proposed quarter-cycle magnetic lag fails Faraday law

A proposed vacuum field pair is $E_y=E_0\cos\phi$ and $B_z=(E_0/c)\sin\phi$, where $\phi=kx-\omega t$ and omega=ck. There are no other field components.

1. Calculate both sides of the one-dimensional Faraday equation and show the mismatch.

2. Can adding an ordinary electric current density repair that mismatch without changing E or B? Give the correct traveling Bz with no static offset.

Question 6 - Solution

A proposed quarter-cycle magnetic lag fails Faraday law

(a) Method and calculation

$$\partial_xE_y=-kE_0\sin\phi,\quad-\partial_tB_z=(\omega E_0/c)\cos\phi=kE_0\cos\phi.$$$$\boxed{-kE_0\sin\phi\ne kE_0\cos\phi\text{ in general}}.$$

(b) Interpretation and further calculation

$$\partial_xE_y=-\partial_tB_z\text{ contains no ordinary }\mathbf J\Rightarrow\boxed{\text{current cannot repair it}}.$$$$\boxed{B_z=(E_0/c)\cos\phi}\quad\text{satisfies Faraday law}.$$

Physical check

A field amplitude ratio alone is insufficient; relative phase and propagation signs must also satisfy the field equations.