13.半空间的场+感应定理(11-0).doc

5. Fields in Half Spacel A combination of the equivalence principle and image theory can be used to obtain solutions to boundary value problems for which the field in half space is to be determined from its tangential components over the boundary plane.l To illustrate, consider a problem consisting of matter and sources in region , and the free space in region as shown below.Sources & Matterl According to the equivalence principle for electric conductors, a problem equivalent to the original problem in the region can be composed of an infinite conducting plane and a magnetic current at , as shown below.Zero FieldElectric Conductor l According to the image theory, the magnetic current is imaged to the conducting plane , which will result in the magnetic current radiating into unbounded space, and producing the same field in the region as do the original sources. Image Fieldl At the same time, this magnetic current will also radiate an image field in region , but it is different from the field of the original problem, and is therefore not of interest to us.l As discussed previously, the E-field in whole space produced by above magnetic current isl ExampleConsider a coaxial line of inner radius a and outer radius b opening into a ground plane, and the voltage between the outer and inner conductors is V, find the magnetic current in its equivalent problem.2a2bxyz1) The boundary value problem for the electric potential inside the coax is2) The solution to the boundary value problem is determined as follows. and the boundary conditions namely yield3) The final solution of potential becomes 4) The electric field inside the coax isAccording to the equivalence principle, the field in region must be the same to that produced by the magnetic current on the aperture (coax opening) with the aperture covered by a conductor.5) By employing the method of image, the actual problem can be equivalently pictured as another problem in which a magnetic current on the aperture radiates into unbounded space.xyz where the magnetic current is6. Induction TheoremA. The original probleml Consider a problem in which some sources radiate in presence of an obstacle as shown below.SourcesObstacleSl The field is termed the impressed field or incident field which is produced by the sources in absence of the obstacle.The field is called the scattered field defined as the difference between the field with the obstacle present and the field with the obstacle absent, namelyl The scattered field is thought of as the field produced by the current (either conduction or polarization current) on the obstacle.B. The induction theoreml Another problem is constructed by retaining the obstacle and postulating that the original field exists internal to it and that the scattered field exists external to it.ObstacleSl To support these fields, there must be surface currents on S,l It follows from the uniqueness theorem that these currents, and , radiating in presence of the obstacle, produce the postulated field internal to S and the scattered field external to S. l The above statement is termed the induction theorem which is closely related to, but somewhat different from, the concept of the equivalence theorem.C. The equivalence theoreml According to the equivalence principle, an equivalent problem can be given in a way that the field internal to S is the same to that of the original field, , and zero field external to S, as shown below.Zero fieldObstacleSl To support the field in the equivalent problem, the surface currents on the obstacle must bel These currents, and , radiating into an unbounded medium having constitutive parameters equal to those of the obstacle, produce the postulated field inside the obstacle. This problem is equivalent to the original problem only within the obstacle where both problems possess the same field .D. Comparisonl For the equivalence theorem, the currents, and , are thought of as radiating into an unbounded medium, hence the field internal to the obstacle can be calculated by using the following previously-mentioned formulas:wherel For the induction theorem, the currents, and , radiate in presence of the obstacle, hence the field either inside or outside the obstacle can not be calculated by using above potential-involved formulas.A determination of this field is a boundary value problem of the same order of complexity as the original problem.l For the equivalence theorem, in order to calculate the field inside the obstacle, we have to know the currents and first, namely we have to know the field on the obstacle, which can be obtained by solving the original problem, namely it is a problem of the same order of complexity as the original problem.l For the induction theorem, in order to calculate the field both inside and outside the obstacle, we only need to know the currents and first, and the field on the obstacle is usually given. E. Conducting obstaclel If the obstacle in the original problem is perfectly conducting shown below,SourcesConductorSthen the E-field boundary condition is on Snamely on Sl On the electric conductor S, the tangential E-field component is zero (just behind ) and equal to the scattered E-field component (just in front of ), the magnetic current supporting this discontinuity should bel At the same time, the tangential H-field component on the electric conductor (just behind ) is equal to the scattered H-field component (just in front of ), the electric current supporting this continuity should be zero,l The induction representation for the problem can be constructed as follows:ConductorSl In other words, the magnetic current (the tangential impressed E-field) radiating in presence of the conductor will produce a scattered field outside this conductor. This conclusion is often applied for analysis of antennas and scatterers.l Example 1Assuming that a plane wave defined byis normally incident to an infinite conducting plane as shown below,Incident wavez0xl According to the induction theorem, the scattered field in region is produced by a magnetic current on ,l This current , radiating in presence of the conducting plane , produces a scattered field in region , shown below, z0xl According to the image theory, it means radiates in free space,zxl Example 2Consider a relatively large rectangular conducting plate of width a in the y direction and b in z direction as shown below