Relationship between B, E and k of a wave.

[renebenthien]

Homework Statement

In dielectric, homogeneous media show that:

$$i\textbf{k} \times \tilde{\textbf{E}} = i\omega\tilde{\textbf{B}}$$ (1)

k is the wave vector, and E-tilda is complex constant vector.

Homework Equations

$$\textbf{E} = \textbf{\tilde{E}}e^{i(k.r - \omega t)} + c.c.$$ (2)
where c.c. refers to the complex conjugate.

B field also varies like above. (3)

$$\nabla \times \textbf{E} = - \partial \textbf{B} / \partial t$$ (4)

The Attempt at a Solution

How does the curl operate on this? When I ignore the c.c. and simply substitute the field equations (2) and (3) into the maxwell equation (4) I get (1). But I'm not sure why i can just ignore it.

 

[gabbagabbahey]

$$\textbf{E} = \tilde{\textbf{E}}e^{i(k.r - \omega t)} + c.c.$$ (2)
where c.c. refers to the complex conjugate.

B field also varies like above. (3)

Usually, one writes that $\textbf{E}$ is the real part of the complex plane wave:

$$\textbf{E}=\text{Re}\left[\tilde{\textbf{E}}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}\right]=\frac{\tilde{\textbf{E}}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}+\tilde{\textbf{E}}^* e^{-i(\textbf{k}\cdot\textbf{r}-\omega t)}}{2}=||\tilde{\textbf{E}}||\cos(\textbf{k}\cdot\textbf{r}-\omega t+\delta)$$

where $\delta$ is the complex phase of $$\tilde{\textbf{E}}$$...and likewise for $\textbf{B}$

$$\nabla \times \textbf{E} = - \partial \textbf{B} / \partial t$$ (4)

The Attempt at a Solution

How does the curl operate on this? When I ignore the c.c. and simply substitute the field equations (2) and (3) into the maxwell equation (4) I get (1). But I'm not sure why i can just ignore it.

When you define $\textbf{E}$ and $\textbf{B}$ in the manner above, they differ from the corresponding complex plane waves $$\tilde{\textbf{E}}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}$$ and $$\tilde{\textbf{B}}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}$$ only by the addition of some imaginary components. The imaginary components of these plane waves will differ from the real components only by the replacement of all the cosines by sines. (!)Hence, if the real components satisfy Maxwell's laws (and they must!) so too do the imaginary components, and hence the entire complex fields will as well:

$$\implies \nabla \times \left(\tilde{\textbf{E}}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}\right)=- \frac{\partial}{\partial t}\left( \tilde{\textbf{B}}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}\right)$$

Most introductory/intermediate EM texts give a similar argument, or even a proof of this when they introduce the concept of complex fields, and from then on the authors expect you to be familiar with this property.

 

[turin]

Alternatively, you can obtain real-valued fields by adding E to E^* and B to B^* (and dividing by some normalization if you want to). If E and B satisfy Maxwell's Equations, then so must E^* and B^* (assuming that ρ and J are replaced by ρ^* and J^* if there are sources). You can easily verify this by taking the complex conjugate of Maxwell's Equations.