0
$\begingroup$

I'm reading the book "The Quantum Theory of Light", by Rodney Loudon, and I came across the following passage describing the general form of the electromagnetic field in a cubical conductive cavity with side length $L$:

The electric field in empty space must satisfy...the Maxwell equation $\nabla\cdot\overrightarrow{\mathbf E}=0$. The solution that satisfies the boundary conditions has components:$$E_x(\mathbf{rt})=E_x(t)\cos(k_xx)\sin(k_yy)\sin(k_zz),$$ $$E_y(\mathbf{rt})=E_y(t)\sin(k_xx)\cos(k_yy)\sin(k_zz),$$ $$E_z(\mathbf{rt})=E_z(t)\sin(k_xx)\sin(k_yy)\cos(k_zz).$$

The problem is, I don't see how this equation satisfies $\nabla\cdot\overrightarrow{\mathbf E}=0$ at all. Specifically, it seems like the value of the lhs of that equation is$$-(E_x(t)+E_y(t)+E_z(t))(\sin(k_xx)\sin(k_yy)\sin(k_zz)).$$There's no reason that $(E_x+E_y+E_z)$ needs to be zero, but it also can't be true that $\sin(k_xx)\sin(k_yy)\sin(k_zz)=0$ for every point in the cavity. Can anyone tell me what I am missing here?

$\endgroup$

1 Answer 1

2
$\begingroup$

Your expression given for $\, \vec{\nabla} \cdot \vec{E}(\vec{r}, t)\,$ is wrong. It should instead read $$-\left(k_x E_x(t)+k_y E_y(t)+k_z E_z(t)\right)\left(\sin(k_x x) \sin(k_y y) \sin(k_z z)\right).$$ Thus the Maxwell equation $\, \vec{\nabla} \cdot \vec{E}(\vec{r},t) =0 \, $ implies $\, k_x E_x(t)+k_y E_y(t)+k_z E_z(t)=\vec{k} \cdot \vec{E}(t) =0 \, $ in conformity with eq. (1.1.6) of your textbook.

$\endgroup$

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.