1School of Electrical and Computer Engineering, Cornell University ECE 303: Electromagnetic Fields and Waves Fall 2007 Homework 3 Due on Sep. 14, 2007 by 5:00 PM Reading Assignments: i) Review the lecture notes. ii) Relevant sections of the online Haus and Melcher book for this week are 5.0-5.3, 5.9, 6.0-6.2. Note that the book contains more material than you are responsible for in this course. Determine relevance by what is covered in the lectures and the recitations. The book is meant for those of you who are looking for more depth and details. ii) This homework is long – start early. Table of Solutions of Laplace’s Equation Spherical Coordinate System Cylindrical Coordinate System ()Ar =rφ Constant potential ()Ar =rφ Constant potential ()rAr =rφ Spherically symmetric potential ()()rAr ln=rφ Cylindrically symmetric potential ()()θφcosrAr =r Potential for uniform z-directed E-Field ()()φφcosrAr =r Potential for uniform x-directed E-Field ()()φφsinrAr =r Potential for uniform y-directed E-Field ()()2cosrArθφ=r Potential for point-charge-dipole-like solution oriented along the z-axis ()()rArφφcos=r Potential for line-charge-dipole-like solution oriented along the x-axis ()()rArφφsin=r Potential for line-charge-dipole-like solution oriented along the y-axis2Problem 3.1: (A nano-structured dielectric medium) These days nano-technology is being used to “design” materials (as opposed to relying on nature) that have some desired characteristics. In this problem you will explore one such material made out of “nano-dots”. Consider a material made up of nano-sized dielectric spheres (or nano-dots) of dielectric permittivity 1ε embedded in another material that has the same permittivity as that of free-space (oε), as shown below. We will consider this problem in different steps. Consider first a single dielectric sphere of radius a and of dielectric permittivity 1ε embedded in a medium of permittivity oε, as shown below. A constant and uniform E-field has been applied in the +z-direction from outside. a) Find trial solutions, ()rinrφ and ()routrφ, for the potentials inside and outside the dielectric sphere. (Hint: ()routrφ must have a term that has the same form as a dipole potential). b) Write down all the boundary conditions (at least as many as the number of unknown constants in your answer to part (a) above) relevant to solving for the potentials, ()rinrφ and ()routrφ. c) Find all the unknown constants in your solutions in part (a) above by using all the boundary conditions in part (b) above. oε 1ε aθzzEEoˆ=r xoε 1ε3d) Compare the dipole-like term in your solution ()routrφ to that of a point-charge dipole potential (see your homework problem 2.1 solutions) and from that comparison figure out the dipole-moment pr (small “p”) of the polarized dielectric sphere. Make sure you get the correct units. (Hint: the dipole moment must be proportional to the applied E-field magnitude). This dipole moment has been “induced” in the dielectric sphere due to the external E-field. Now come back to the medium made up of nano-sized dielectric spheres of dielectric permittivity 1ε embedded in another medium of permittivity oε, as shown in an earlier picture. Suppose the dots are spaced reasonably far apart and so the field from one dot does not interact with the field of the other dots. Suppose that there are approximately N dots per unit volume. e) What is the polarization vector Pr (capital “P”) of the nano-dot medium given that you know the dipole-moment of each dot? (Hint: the polarization vector must be proportional to the applied E-field magnitude). f) From your answer to part (e), find the electrical susceptibility eχ of the nano-dot medium. g) From your answer in part (f), find the dielectric permittivity ε of the nano-dot medium. Problem 3.2: (Dielectric image charges) Consider a point charge Q+ sitting in free space at a distance d above a dielectric medium of permittivity ε, as shown below. The electric field from the charge will get partially screened by the surface polarization charge density (paired surface charge density) that will exist at the surface of the dielectric medium. But unlike the perfect metal case, the electric field will not get fully screened out of the dielectric material. In order to solve this problem, one needs to realize that the actual field solution, both inside and outside the dielectric medium, must be a superposition of the field due to the point charge and the field due to the surface polarization charge density (i.e. the paired charge density at the surface of the dielectric medium). A priori, we don’t know what this surface charge density looks like so we will try to construct a guess solution. Q+ zε oε d4We will assume that OUTSIDE the dielectric, the potential looks like the superposition of the potential of the original charge Q+ and the potential due to an image charge of strength aQ− sitting a distance d below the dielectric interface and that the whole space is filled with free space. The image charge has been assumed to have a different strength then the original charge because dielectric screening, unlike perfect metal screening, is not expected to be perfect. For the potential INSIDE the dielectric we will assume that it looks like that of a charge of strength bQ+ sitting outside the dielectric at a distance daway from the interface and that the whole space is filled with material of permittivity ε. This is because the actual field from the charge Q+ will get partially screened by the polarization (or paired) surface charged density at the surface of the dielectric. a) Write an expression for the guess potential ()routrφ outside the dielectric in terms of the distances +r and −r from the charges Q+ and aQ− , respectively, and for the guess potential ()rinrφ inside the dielectric in terms of the distance +r from the charge bQ+. b) You have two unknowns in your solution (the strength of the charges aQ− and bQ+ ) and you need two boundary conditions. What are these two boundary conditions? c) Using the boundary conditions from part (b) find the strength of the charges aQ− and bQ+ in terms of the charge strength Q+ and the permittivities ε and oε. d) Show from your result in part (c) that if oεε= then 0=aQ and QQb= which is what one would expect on physical grounds. e) Show from your