13.半空间的场+感应定理(11-0).docVIP

5. Fields in Half Space 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. To illustrate, consider a problem consisting of matter and sources in region , and the free space in region as shown below. 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. 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. 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. As discussed previously, the E-field in whole space produced by above magnetic current is Example Consider 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. The boundary value problem for the electric potential inside the coax is The solution to the boundary value problem is determined as follows. and the boundary conditions namely yield The final solution of potential becomes The electric field inside the coax is According 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. 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. where the magnetic current is 6