Close electric field lines in wave guides

Question

In a wave guide, graphics of propagation of Transversal Magnetic modes show closed field lines for the electric field.

For example, for a rectangular guide:

$E_x (x,y,z) = \frac {-j\beta m \pi}{a k^2_c} B_{mn}\cos\frac{m\pi x}{a}\sin\frac{n\pi y}{b}e^{-j(\beta z + \omega t)}$

$E_y (x,y,z) = \frac {-j\beta n \pi}{b k^2_c} B_{mn}\sin\frac{m\pi x}{a}\cos\frac{n\pi y}{b}e^{-j(\beta z + \omega t)}$

$E_z (x,y,z) = B_{mn}\sin \frac{m\pi x}{a}\sin\frac{n\pi y}{b}e^{-j(\beta z + \omega t)}$

Is it possible to have closed lines for the electric field?

Answer 1

Yes, it is possible. Maxwell's equations say $$ \oint_l \vec{E}\; d\vec{l} = -\frac{1}{c}\int_{S(l)}\frac{\partial \vec{B}}{\partial t} d\vec{S}. $$ The electric field of this closed line is proportional to the rate of the change of the magnetic flux.

There is no problem with energy conservation. An electron moving along this closed line will be accelerated but it consumes the energy we waste to keep the field changing. This effect is used in some particle accelerators while the reversal effect is used in most microwave sources where the EMW consumes the kinetic energy of moving electrons.