Electromagnetic Waves

12.1.1  EM Waves - Qualitatively

One of the hallmarks of scientific thought of the last 20 centuries is the derivation of the fundamental connection between electric and magnetic fields that leads to the notion of the electromagnetic wave.

This derivation, first done by James Clerk Maxwell around 1865, laid the foundation for the fundamental discoveries of radio that have lead to the basis for remote communication throughout the world. While the mathematics needed to rigorously derive the results of Maxwell starting from Maxwell's equations are generally beyond what would normally be expected for someone taking a freshman course in electricity and magnetism, Maxwell's approach can be done in a "physics" sense that more or less mimics the actual derivation of the wave equation for electricity and magnetism. This approach begins by first thinking about the characteristics of electric and magnetic fields for cases where both must always be produced. Essentially any situation which causes static charges to move will create a situation in which both electric and magnetic fields are present. Using Maxwell's equations, we can then look at the constraints on the form these fields must take.

We begin with the description of an oscillating electric dipole. The picture here is one of a dipole in which we allow the electric charges to change their relative positions. For our purposes, we wish to have both the charges move such that their positions with respect to time are defined by sinusoidal functions (let's say a cosine to be definite although sine would do just as well). In this case, the charges oscillate back and forth as though each were connected to a (separate) spring which had the middle of the dipole as its equilibrium position. The charges therefore go from one end of the dipole to the other as shown in the Java animation (see figure 12.1) if you click the "Start the animation" button. Their positions are independently specified by cos(t/T + f) where T is the period of the motion and f is, of course, the phase specifying the initial position of the charges.

The animation shows two views, one from the side and one from the top, above the dipole. These two views allow us to see both the electric field (in the side view) and the magnetic field (top view) where, in both views, the other field is perpendicular to the screen. Note that the magnetic field exists because, as the charges oscillate, they constitute a current and every current creates a magnetic field. According to Ampere's Law, this field is perpendicular to the plane containing the current and is seen to have equal magnitude for a circular path of given radius around the current as shown in the top view of the animation.

Let's focus on the electric field for now. A static picture of the dipole at any given time would appear as shown in figure 12.2a.)

Figure 12.2: Two views of an electric dipole showing
a.) the positive charge at the top
b.) the negative charge at the top.

When the charges are in their reversed position, we see the field as shown in figure 12.2b.) These two views come about with period T. Therefore, if we try to find what the field looks like at intermediate points in time, that is, between the two views, we are faced with the situation shown in figure  12.3.

Figure 12.3: A transition region exists between the two view shown in figure 12.2 in which we must connect the field lines so that they are continuous between the two views in time.

Since we don't assume the change can propagate instantly once the new position is reached, the "blank" area represents what has to happen to the fields in the meantime. Since Faraday has explained that E field lines can't cross and that we need to have them be continuous (except when they end on charges), then a plausible guess for what the fields look like is shown in figure  12.4.

Figure 12.4: A plausible guess for how the fields lines between the two views shown in figure 12.2 must connect. The second picture shows the view a little further out and the third picture shows the pattern far away from the oscillating current source. The last picture shows an animated view.

The base conclusions we can take from this simple, qualitative analysis are that

1. The E and B fields are always at right angles to each other. This is difficult to see as "true" for any given position in space, but certainly it is obvious for many points that we can readily see.
2. The propagation of the fields, i.e. their direction of travel away from the oscillating dipole, is perpendicular to the direction in which the fields point at any given position in space.
3. Go far enough away from the dipole and the electric field appears to form closed loops which are not connected to either charge. This is, of course, always true for any B field and this case is no exception. Thus, far from the dipole, we find that the E and B fields are traveling independent of the charges! They propagate away from the dipole and spread out through space.

Believe it or not, with just these three observations in hand, plus some calculus, we can come up with a startling conclusion as to how the fields interact with each other.

If we extrapolate these fields out far enough, they look like plane waves, i.e. the E and B fields stretch out enough that, for a sufficiently small section of space, they look approximately straight rather than curved as shown in figure 12.5. In this case, we can identify a small section of space as a wave front, a plane which is perpendicular to the direction of propagation of the fields. It is this wave front which we use for our quantitative analysis.

Figure 12.5: Far from the source, the E and B fields look approximately linear rather than curved. In this case, we can define a flat plane that contains the E and B field and is perpendicular to the direction of travel.

12.1.2  EM Waves - Quantitatively

First, let's draw a portion of the plane wave. The wave propagates along the x axis which means the E and B fields lie along the y and z axes, respectively. We then pick two plane wavefronts which are of sizes Dx by Dy and Dx by Dz as shown in figure 12.6. We keep the sizes of Dx, Dy, and Dz smaller than the range of distance over which the curvature of E and B are non-negligible.

Figure 12.6: A plane wave front showing the E and B fields far away from the oscillating dipole that produced them.

To find the quantitative description of the E and B fields as functions of time and x position, we make use of Ampere-Maxwell's Law and Faraday's Law. First consider the case of the B field's effect on the E field. We assume that B field moves along the x  axis. It is also changing along the z axis in figure 12.6 above, but we do not yet have a description of how it is changing. For the moment, we take the average value of B_z along the x axis as being somewhere between B_z, 1 and B_z, 2. For simplicity, we assume that the average is the field all along the Dx distance between the two plane fronts. Since B_z is, in general, not zero, we have a flux through the area bounded by E_y, 1  and E_y, 2. That flux is

FB, avg. = BzDxDy

(12.1.2.1)

By Faraday's Law, the electric field around the area bounded by Dx and Dy must be related to the time rate of change of this magnetic flux as follows

ó õ ® E   ·d ® l   = - ¶ ¶t (BzDxDy)

(12.1.2.2)

Why should the magnetic flux change with time? Remember that the B field is presumed to be propagating out away from the dipole. Just the motion of this field along the x axis is enough to create a time-varying flux if the field in fact changes as a function of x and time. Note that the integral around the Dx, Dy box is easy to do. The parts of the path that are parallel to x  have no E field since, according to our oscillating dipole model, E has no components along x, only along y. Evaluating the integral according to the right-hand rule (i.e. up along E_y, 2, then along -x, then down along E_y, 1 and back along +x) for the direction of B_z, we get

ó (ç) õ ® E   ·d ® l   = - ¶ ¶t (BzDxDy) Ey, 2Dy - Ey, 1Dy = - ¶Bz ¶t DxDy Ey, 2 - Ey, 1 Dx = - ¶Bz ¶t ¶Ey ¶x = - ¶Bz ¶t (12.1.2.3)

Let's make sure we have done justice to Faraday's Law. If an observer sits at point P in figure 12.6, she will see that the B_z field decreases with time, hence, to oppose the change, the E field must be as we show it in the figure (E_y, 2 > E_y, 1), i.e. the E field magnitude also decreases with time as it goes past point P.

We can perform the same analysis for the effect of the E field on B. If we note that the E field is not constant over the Dx, Dz area bounded by B_z, 1 and B_z, 2 in figure 12.6, but use the average value E_y over this interval, then we have a non-zero electric flux (usually) over this interval. Since the electric flux will also not be constant in time or space across this interval (the E field is moving along x and probably has time variation in magnitude as a function of x), the Ampere-Maxwell relation will come into play. Again, in doing the line integral of B around the Dx, Dz boundary, remember that the B field is parallel to the z axis and has no components along x. In this case, we evaluate the line integral by going along B_z, 1, parallel to +x, down along B_z, 2, and along -x.

ó (ç) õ ® B   ·d ® l   = m0e0(EyDxDz) -Bz, 2Dz + Bz, 1Dz = m0e0 ¶Ey ¶t DxDz Þ - Bz, 2 - Bz, 1 Dx = m0e0 ¶Ey ¶t ¶Bz ¶x = -m0e0 ¶Ey ¶t (12.1.2.4)

We now only need one more step to derive equations of motion for both the E and fields: we need to know the second derivative with respect to time for both in terms of their spatial distribution. Remember that the equation of motion represents something like an acceleration (2nd derivative w.r.t. time) to the position or velocity or spatial extent of something. We can perform the second derivative for both fields with respect to time, then play around with the equations we just derived relating first order derivatives to get a surprising result.

-m0e0 ¶ ¶t ¶Ey ¶t = ¶ ¶t ¶Bz ¶x -m0e0 ¶2Ey ¶t2 = ¶ ¶x ¶Bz ¶t Þ -m0e0 ¶2Ey ¶t2 = - ¶ ¶x ¶Ey ¶x Þ m0e0 ¶2Ey ¶t2 = ¶2Ez ¶x2 (12.1.2.5)

We can do exactly the same procedure for B.

¶ ¶t ¶Bz ¶t = - ¶ ¶t ¶Ey ¶x ¶2Bz ¶t2 = - ¶ ¶x ¶Ey ¶t ¶Bz ¶t2 = 1 m0e0 ¶ ¶x ¶Bz ¶x Þ m0e0 ¶2Bz ¶t2 = ¶Bz ¶x2 (12.1.2.6)

What we find is that both E and B can be described by wave equations. These are normally encountered when considering what happens to an elastic medium subjected to a force that deforms it and then goes away. The disturbance to the medium (i.e. the displacement of parts of the material from its equilibrium position) propagates according to an equation of the form

¶2y ¶x2 = 1 v2 ¶2y ¶t2

(12.1.2.7)

where y represents the height of the displacement from equilibrium, x is the direction of propagation of the disturbance, and v is the velocity at which the disturbance travels. The general solution of any such equation is

y(x, t) = Asin(wt - kx + f)

(12.1.2.8)

where k is the wave number, w is the angular frequency, f is the phase angle, and A is the amplitude. Thus, the solutions for equations 5 and  6 are

E(x, t) = Emaxsin(wt - kx + f) B(x, t) = Bmaxsin(wt - kx + f) (12.1.2.9)

The characteristic speed of both the E and B field waves is

v = 1 Ö m0e0 = c = 2.998×108 m/s

(12.1.2.10)

This is the same as the experimentally determined value for the speed of light! Maxwell concluded that light is an electromagnetic wave! Furthermore, although it wasn't understood at the time, these wave equations do not need a medium to propagate. In a sense, the E and B fields oscillate on each other, i.e. their interaction forms the basis for the oscillation rather than disturbance of a medium. The electromagnetic wave is shown in this Java applet.

Finally, we note that if we consider this plane wave as propagating for a very short period of time dt, then we can assume that the electric and magnetic fields are essentially constant over the distance dx = c dt traveled. Considering equation 2, we can integrate the nearly constant E field over an infinitesimal distance dy. We find that there must be a relationship between the magnitude of the electric and magnetic fields as follows.

ó (ç) õ ® E   ·d ® l   = - ¶ ¶t (Bz dx dy) E dy = B ¶ ¶t (c dt)dy Þ E = cB (12.1.2.11)

Therefore, the electric and magnetic field magnitudes have a constant relationship in that E/B = c!

12.1.3  Energy and Momentum in EM Waves

We expect that electromagnetic (EM) waves should carry energy just as mechanical waves do. This is, in fact, true. Any region of empty space that the wave travels through will have an energy density, u, associated with the electric and magnetic fields so that

u = 1 2 e0E2 + 1 2m0 B2

(12.1.3.12)

Since B = E/c at every point,

u = 1 2 e0E2 + 1 2m0 m0e0E2 = e0E2 Þ uE = uB = 1 2 e0E2 = 1 2m0 B2. (12.1.3.13)

So the energy density in the electric and magnetic fields in a vacuum is the same at any instant in time and position in space but the value of the energy density changes as a function of time and position. We define the rate of energy transport per unit area as the Poynting vector, S, where

® S   º 1 m0 ® E   × ® B

(12.1.3.14)

Since the values of E and B vary with position and time, we generally use the time-averaged value of the magnitude of S at a point, which we call the intensity, I.

I º Savg. = é ê ë 1 m0 EmaxBmaxsin2(wt - kx + f) ù ú û avg.  I = EmaxBmax 2m0 = Emax2 2m0c = 1 2 e0cEmax2 (12.1.3.15)

Although we need more physics or more math than we've introduced to prove it, electromagnetic waves, since they carry energy, must also carry momentum and can therefore exert a pressure on matter with which the EM wave interacts. This pressure, termed radiation pressure, can be shown to have one of two values.

pradiation = Savg. c = I c    wave totally absorbed pradiation = 2Savg. c = 2I c     wave totally reflected (12.1.3.16)

This pressure is nominally very, very small, but it can be important for the navigation of spacecraft and can be very large near the cores of stars.

Send comments to larryg@upenn5.hep.upenn.edu.
This page was last modified on 04/05/2003 at 18:45:52 (EST).
Current date/time is Saturday, 11-Oct-2008 14:34:30 EDT
16696 hits since