The Traveling Electromagnetic Wave, Qualitatively — Question 9

Question 9

A static field added to a wave can destroy the single-wave field identities

At one event a +x vacuum wave has $\mathbf E_w=E_*\hat y$ and $\mathbf B_w=E_*\hat z/c$. Add a uniform static field $B_s\hat y$, with Bs nonzero, and no added electric field.

1. Find total E dot B and the quantity E squared minus c squared B squared.

2. Can the total pair be a single vacuum plane wave at that event? Explain whether this conclusion forbids the superposition itself.

Question 9 - Solution

A static field added to a wave can destroy the single-wave field identities

(a) Method and calculation

$$\mathbf E=E_*\hat y,\quad\mathbf B=B_s\hat y+(E_*/c)\hat z.$$$$\boxed{\mathbf E\cdot\mathbf B=E_*B_s}.$$

(b) Interpretation and further calculation

$$E^2-c^2B^2=E_*^2-c^2[B_s^2+E_*^2/c^2]=\boxed{-c^2B_s^2\ne0}.$$$$\boxed{\text{Not a single vacuum plane wave; linear superposition is still allowed.}}$$

Physical check

These identities are necessary for one vacuum plane wave; their failure does not imply a violation of Maxwell equations by a sum of valid fields.