Electromagnetic Waves Staelin Trial Solution Satisfies

What fraction of the incident power P is reflected What is (overlinemathrmH(mathrmt)) at the dielectric surface (z 0).

The general electromagnetic boundary value problem treated in Sections 9.14 involves determining exactly which, if any, combination of waves matches any given set of boundary conditions, which are the relations between the electric and magnetic fields adjacent to both sides of each boundary.

These boundaries can generally be both active and passive, the active boundaries usually being sources.

Boundary conditions generally constrain (overlinemathrmE) andor (overlinemathrmH) for all time on the boundary of the two- or three-dimensional region of interest.

Therefore we may solve boundary value problems simply by hypothesizing the correct combination of waves and testing it against Maxwells equations.

That is, we leave undetermined the numerical constants that characterize the chosen combination of waves, and then determine which values of those constraints satisfy Maxwells equations.

This strategy eases the challenge of hypothesizing the final answer directly.

Moreover, symmetry and other considerations often suggest the nature of the wave combination required by the problem, thus reducing the numbers of unknown constants that must be determined.

Here our basic building blocks are usually uniform plane waves, but other more compact expansions are typically used if the symmetry of the problem permits, as illustrated in Section 4.5.2 for cylindrical and spherical geometries, Section 7.2.2 for TEM transmission lines, and Section 9.3.1 for waveguide modes.

This more complete problem definition is sufficient to yield a unique solution.

Often the first step in solving a problem is to ensure its definition is complete.

In particular, we need to match the boundary condition (overlinemathrmE 0)at z 0.

This can be done by adding a single uniform plane wave propagating in the -z direction with an electric field that cancels the incident electric field at z 0 for all time t.

Inside the conductor (overlinemathrmEoverlinemathrmH0), and boundary conditions (2.6.17) require a surface current.

Electromagnetic Waves Staelin Trial Solution Satisfies

We know that our trial solution satisfies the wave equation in our source-free region because our solution is the superposition of waves that do; it therefore also satisfies Faradays and Amperes laws in a source-free region, as well as Gausss laws.

At the perfectly conducting boundary we require ( overlinemathrmE 0) and (overlinemathrmHperp0); these constraints are also satisfied by our trial solution, and therefore the problem is solved for the vacuum.

Zero-value fields inside the conductor satisfy all Maxwells equations, and the surface current ( overlinemathrmJmathrmS) (9.1.10) satisfies the final boundary condition.

Note that the total electric field is zero not only at the surface of the conductor, but also at a series of null planes parallel to the conductor and spaced at intervals along the z axis such that kz nulls -n(pi), where n 0, 1, 2.

On the other hand, the magnetic field is maximum at those planes where E is zero (the null planes of E), and has nulls where E is maximum.

Since the time average power flow and the Poynting vector are clearly zero at each of these planes, there is no net power flow to the right.

Except at the field nulls, however, there is reactive power, as discussed in Section 2.7.3.

Because no average power is flowing via these waves and the energy and waves are approximately stationary in space, the solution is called a standing wave, as illustrated in Figures 7.2.3 for VSWR and 7.4.1 for resonance on perfectly reflecting TEM transmission lines.

For example, assume the same uniform plane wave (9.1.12) in vacuum is incident upon the same planar interface, where a medium having, o, o for z 0 has replaced the conductor.

We have no reason to suspect that the fields beyond the interface are zero, so we might try a trial solution with both a reflected wave E r (z,t) and a transmitted wave E t (z,t) having unknown amplitudes (E r and E t ) and phases ( and ) for which we can solve.

Electromagnetic Waves Staelin Pdf