# Plane Waves and the Poynting Vector

• ninevolt
In summary: However, for plane waves, the Poynting vector is constant since the energy is evenly distributed over an infinite plane. This is why we use plane waves as approximations, since they have constant energy density and direction of propagation, making calculations simpler. The decrease in intensity with distance is still captured in the Poynting vector, just in a different way.
ninevolt
Why is it that that poynting vector is independent of distance from the source?
Is it because EM waves are plane waves?
Furthermore I do not fully understand why EM waves have to be plane waves. I understand that changing magnetic fields give rise to electric fields and vice versa, but does that constitute a constant E and B field over an infinite plane?

My book states that the intensity of the wave decreases as 1/r^2, but I don't see this anywhere in the poynting vector, and doesn't that violate the premise of a plane wave?

ninevolt said:
Why is it that that poynting vector is independent of distance from the source?
Is it because EM waves are plane waves?
Furthermore I do not fully understand why EM waves have to be plane waves. I understand that changing magnetic fields give rise to electric fields and vice versa, but does that constitute a constant E and B field over an infinite plane?

My book states that the intensity of the wave decreases as 1/r^2, but I don't see this anywhere in the poynting vector, and doesn't that violate the premise of a plane wave?

There's a lot in your post I don't understand: the Poynting vector is defined locally, in terms of the local E and H fields- it has no information about the source.

EM waves do not *have* to be plane waves; plane waves are a solution to Maxwell's equations, and so arbitrary solutions can be decomposed into a summation of plane waves.

The intensity of the electromagnetic field, for a plane wave, does not have 1/r^2 dependence: the intensity of spherical waves do.

So if the poynting vector has no information about the source, does that mean the distance from the source doesn't matter when describing Power/Area

When I spoke of plane waves I just wanted to know why is it that EM waves can be described as plane waves. I want to build up some physical inituition of plane waves, but as of now I cannot imagine how EM waves have constant values over their infinite plane. My intuition tells me that it should decrease. Or is this where the "changing E field give rise to B field and visa versa?"

I believe when the wave propagate out from a source far away, even though it start out in spherical shape, but when the radius getting very large, the surface approx a straight plane rather than a spherical shape.

From the solution of Maxwell's equation of:

$$\nabla X \vec E \;=\; -\frac{\partial \vec B}{\partial t} \;\hbox { and }\; \nabla X \vec H \;=\; \vec J \;+\; \frac{\partial \vec D}{\partial t}$$

E and B are perpendicular to each other and the direction of propagation is perpendicular to both of them. Both E and B are in a plane that propagate in direction of the normal of the plane.

Poynting vector:

$$\vec P = \vec E X \vec H$$

at a point. It is calculated from E and B at that point. It say nothing about where the E and B come from. It tell you the EM energy density (W/unit area) and the direction of the energy flow.

These are my understanding, please correct me if I am wrong. I am studying Poynting vector also.

Last edited:
ninevolt said:
So if the poynting vector has no information about the source, does that mean the distance from the source doesn't matter when describing Power/Area

When I spoke of plane waves I just wanted to know why is it that EM waves can be described as plane waves. I want to build up some physical inituition of plane waves, but as of now I cannot imagine how EM waves have constant values over their infinite plane. My intuition tells me that it should decrease. Or is this where the "changing E field give rise to B field and visa versa?"

Plane waves are indeed unphysical (as are Gaussian, Bessel, spherical...). These solutions approximate reality, and you have the freedom to choose which approximate solution you choose (although some approximate better than others in specific situations). Plane waves, in particular, are good approximations to use when then surface of constant phase (the wavefront) is very nearly flat- this is a good approximation to starlight incident on Earth, a collimated beam of light, etc.

This is one way to understand diffraction by edges- say an initially flat wavefront is truncated by an aperture or an edge. The wavefront is then given by a summation of multiple plane waves, all traveling is slightly different directions- the wavefront spreads out during propagation.

The distance from the source does matter. Any localized source of radiation will create to a good approximation outwardly traveling spherical waves. At larger distances from the source, the same total energy is spread over a larger sphere and must be weaker. That is why we need so many cell phone towers spread all over the map, instead of on huge tower in the ocean, the signal gets too weak if you are too far away. Mathematically this means that the Poynting vector is proportional to 1/r^2 for spherical waves, i.e. radiation caused by small sources.

## 1. What is a plane wave?

A plane wave is a type of electromagnetic wave that has a constant amplitude and wavelength, and travels in a straight line. It is characterized by its electric and magnetic fields being perpendicular to each other and to the direction of wave propagation.

## 2. What is the Poynting vector?

The Poynting vector is a mathematical tool used to describe the direction and magnitude of energy flow in an electromagnetic wave. It is defined as the cross product of the electric and magnetic fields, and represents the energy per unit time per unit area that is carried by the wave.

## 3. How is the Poynting vector related to plane waves?

In the case of a plane wave, the Poynting vector is always perpendicular to the direction of wave propagation. This means that the energy flow is in the same direction as the wave, and the magnitude of the Poynting vector represents the intensity of the wave.

## 4. What is the significance of the Poynting vector in practical applications?

The Poynting vector is important in understanding how electromagnetic waves transfer energy and how they interact with matter. It is used in various fields, such as telecommunications, optics, and radar, to calculate the power and intensity of waves and to design devices that can manipulate them.

## 5. How is the Poynting vector related to the speed of light?

The magnitude of the Poynting vector is proportional to the square of the speed of light. This means that as the speed of light increases, the energy carried by the wave also increases. Additionally, the direction of the Poynting vector is always perpendicular to the electric and magnetic fields, which are also perpendicular to the direction of wave propagation, as described by the laws of electromagnetism.

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