# Electromagnetic plane waves in vacuo

In summary, the conversation discusses the application of Maxwell's equation to a plane wave, resulting in the conclusion that the wave vector k must equal 0. However, this oversimplification is addressed by considering the more general case where the wave vector is perpendicular to the field. This highlights the conceptual issue of choosing coordinates that align with the wave vector, but not considering the direction of the field.
Consider a plane wave f = cos(kz - wt). Applying Maxwell's equation (divE=0 in vacuo) gives
kcos(kz - wt) = 0 which means that k = 0. This surely doesn't make sense?

Well, mathematically,

$$\bigtriangledown \cdot E_0 \cos(kz - wt) = E_0 \sin(kz - wt) = 0$$

$$\Longrightarrow k = 0$$ or $$sin(kz - wt) = 0$$ $$\forall{ z, t }$$

EDIT: But yes, addressed below is the conceptual issue here...

Last edited:
Well the reason you get that is that EM waves are transverse waves, and you have oversimplified the situation. The more general result is:
Consider a plane wave
$$\vec{E} = \vec{E_0}cos(\vec{k}\cdot\vec{r} - \omega t})$$
Then,
$$\nabla\cdot\vec{E} = \vec{k}\cdot\vec{E} = 0$$

Which shows that the wave vector is perpendicular to the field.

I think I see my mistake. I thought all i was doing was choosing my coordinates such that my z axis coincided with the wave vector k. The way i did it missed out the fact that k and E point in different directions.

## 1. What are electromagnetic plane waves in vacuo?

Electromagnetic plane waves in vacuo are a type of electromagnetic radiation that propagate through empty space. They consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation.

## 2. How are electromagnetic plane waves in vacuo produced?

Electromagnetic plane waves in vacuo are produced when an electrically charged particle accelerates. This acceleration creates a changing electric field, which in turn generates a changing magnetic field. The resulting combination of electric and magnetic fields propagates as a plane wave.

## 3. What is the speed of electromagnetic plane waves in vacuo?

The speed of electromagnetic plane waves in vacuo is approximately 299,792,458 meters per second, which is equivalent to the speed of light in a vacuum. This is a fundamental constant that is the same for all electromagnetic radiation.

## 4. How do electromagnetic plane waves in vacuo interact with matter?

Electromagnetic plane waves in vacuo can interact with matter in various ways. They can be absorbed, transmitted, or reflected depending on the properties of the material they encounter. Additionally, they can induce electrical currents in conductive materials.

## 5. What are some real-world applications of electromagnetic plane waves in vacuo?

Electromagnetic plane waves in vacuo have numerous applications in our daily lives. Some common examples include radio and television broadcasting, wireless communication technologies, and medical imaging techniques such as MRI. They are also used in scientific research for studying the properties of materials and the universe.

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