# Close electric field lines in wave guides

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?

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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.