I have been studying David Griffiths 'Introduction to Electrodynamics', Electromagnetic Waves chapter. For hollow wave guide, the Maxwell's equations

$$\begin{align} \nabla\cdot\mathbf E &= 0 \\ \nabla\times\mathbf E &= - \frac{\partial \mathbf B}{\partial t} \end{align}$$

From these we can obtain,

$$\begin{align} \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} &= i\omega B_z \\ \frac{\partial E_z}{\partial y} - ikE_y &= i\omega B_x \\ ikE_x - \frac{\partial E_z}{\partial x} &= i\omega B_y \end{align}$$

The left hand side of these equations are the components of the curl of the electric field vector. To show that TEM waves (where the z component of both the electric and magnetic fields are zero) cannot exist in a hollow wave-guide, it's stated that, the divergence and curl of the electric field vector are both zero. My question is, how can we show that the curl is zero for this electric field?


1 Answer 1


I assume that Griffiths does not mean the curl of the total electric field, just the transverse part of the electric field.

If we assume a wave of the following kind $$\mathbf{E} = \mathbf{E}_T(x,y)e^{j(\omega t - kz)},$$

and we define a tranverse del operator as $\nabla_T = \hat{x}(\frac{\partial}{\partial x}) + \hat{y}(\frac{\partial}{\partial y}),$ then we can state that $\nabla_T \cdot \mathbf{E}_T = 0 $ and $\nabla_T \times \mathbf{E}_T = 0$. The tranverse field can thus be written in terms of a potential function, similar to the electrostatics case. It is solved by formulating Laplace's equation with the appropriate boundary equations.

  • $\begingroup$ Thank you for your response. I think, I now understand what is meant by the curl being zero as stated in the book. Thanks. $\endgroup$ Mar 3, 2022 at 14:35
  • $\begingroup$ @Mobin_Haque It is worth mentioning that if the curl of the total electric field were zero, then there wouldn't be a wave possible at all since the magnetic field would then also be zero, per faraday's law. $\endgroup$ Mar 3, 2022 at 14:37
  • $\begingroup$ Yes, that's right. This is what has been done eventually to prove that TEM wave cannot exist in a hollow waveguide. Ultimately solving the Laplace equation with appropriate boundary condition along with the fact that Ez=Bz=0, it has been shown that Ex and Ey are zero. So, the total electric field is zero and hence there cannot be any wave. I just thought that, if I input the information Ez=Bz=0 in the three curl equations I mentioned in my question, I would arrive at the conclusion that the curl is zero, which was my mistake. We simply arrive at free space E and B wave relation. $\endgroup$ Mar 4, 2022 at 6:38
  • $\begingroup$ Yes, that is exactly right. It is the fact that Laplace's equation has no non-trivial solutions for hollow waveguides that shows that there is no TEM wave possible. $\endgroup$ Mar 4, 2022 at 7:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.