Electromagnetic waves: How broad is a plane wave?

[frustrationboltzmann]

Hello all again,
I was just thinking again about another aspect of electromagnetic waves: Assume we have a planar wave. How "broad" is it or how far does the electric field of it reach? For instance if we have a single planar wave, assume the k-vector in the direction of propagation and then the electric field-vector is normal to it in vacuum. now if the planar wave hits a plane perpendicular to the k-vector, let's say a plane consisting of infinitely many little dipoles how far would the electric field of the planar wave affect the dipoles? only at the "point" where it hits or somehow radially decreasing with 1/r^2?

 

[Nugatory]

A plane wave extends out to infinity. That is, for all infinite planes perpendicular to the direction of propagation, at any given moment the E field is the same at all points on that plane.

Obviously this particular solution of Maxwell's equations is unphysical. We spend a lot of time on it for several reasons:
1) It is far and away the simplest solution to Maxwell's wave equation, and the more complicated solutions can be written as sums of different plane waves moving in different directions. Thus, we can analyze the complicated ones by rewriting them as a sum of plane waves, solving the simpler problem of how each plane wave interacts whatever system we're analyzing, and then adding these results back together.
2) Within a small area, the plane wave is a very good approximation for several common situations. For example, waves expanding spherically out from a central source obviously aren't a plane wave... But if you consider a small region on the surface of the sphere far from the source, across that region a plane wave is a very good approximation for the much more complicated spherical wave (just as "earth is flat" is a good approximation for small regions on the curved surface of the earth). Likewise, if you're illuminating a screen with beam of light, the plane wave will be a very good approximation to what's going on inside the illuminated area and away from the edges.

So to answer your question about a plane wave encountering a perpendicular plane: If it really were a true plane wave, every point on that plane would exerience the same oscillating E and B fields, all the way out to infinity, no weakening with distance. In practice here are no plane waves, but that doesn't matter because there also are no infinite plane screens for the infinite plane wave to hit. Instead you have a finite-sized flat screen, and within that region you don;t really have a plane wave, but you do have something like a beam of light that can be approximated as a plane wave across the screen: the E field is the same everywhere on the screen.

 

[sophiecentaur]

The bigger the separation between source an detector, the less the wave front will depart from a plane. It has to depend on what you actually need for a particular experiment. The physical size of any detector is also relevant to this. You can look upon it in terms of diffraction at work at both ends of the link.

 

[tech99]

Hello all again,
I was just thinking again about another aspect of electromagnetic waves: Assume we have a planar wave. How "broad" is it or how far does the electric field of it reach? For instance if we have a single planar wave, assume the k-vector in the direction of propagation and then the electric field-vector is normal to it in vacuum. now if the planar wave hits a plane perpendicular to the k-vector, let's say a plane consisting of infinitely many little dipoles how far would the electric field of the planar wave affect the dipoles? only at the "point" where it hits or somehow radially decreasing with 1/r^2?

You seem to have an idea of a single wave hitting a point, which is not correct. There is always a wave front involved, occupying a finite area.
We can restrict the original infinite wave front to a small area of perhaps a few square wavelengths by using a conducting sheet with a hole in it. This process will cause the energy to taper towards the edges of the beam and may create side lobes. The beam will also diverge with distance.