# Why is the curl of electric field vector inside a hollow waveguide zero for TEM waves?

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?

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.

• Thank you for your response. I think, I now understand what is meant by the curl being zero as stated in the book. Thanks. Commented Mar 3, 2022 at 14:35
• @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. Commented Mar 3, 2022 at 14:37
• 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. Commented Mar 4, 2022 at 6:38
• 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. Commented Mar 4, 2022 at 7:50