MIT OpenCourseWare http ocw mit edu Electromagnetic Field Theory A Problem Solving Approach For any use or distribution of this textbook please cite as follows Markus Zahn Electromagnetic Field Theory A Problem Solving Approach Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms ProbLnus 231 PROBLEMS Section 3 1 1 A two dimensional dipole is formed by two infinitely long parallel line charges of opposite polarity X a small distance di apart d x a What is the potential at any coordinate r 46 z b What are the potential and electric field far from the dipole r d What is the dipole moment per unit length c What is the equation of the field lines 2 Find the dipole moment for each of the following charge distributions I 2 II jL L L o L Xo X d a t o c L d e a Two uniform colinear opposite polarity line charges Ao each a small distance L along the z axis b Same as a with the line charge distribution as AAo 1 z L O z L A Ao l z L L z O c Two uniform opposite polarity line charges Ao each of length L but at right angles d Two parallel uniform opposite polarity line charges Ao each of length L a distance di apart 232 Polarization and Conduction e A spherical shell with total uniformly distributed surface charge Q on the upper half and Q on the lower half Hint i sin 0 cos i sin 0 sin 4 i cos Oi f A spherical volume with total uniformly distributed volume charge of Q in the upper half and Q on the lower half Hint Integrate the results of e 3 The linear quadrapole consists of two superposed dipoles along the z axis Find the potential and electric field for distances far away from the charges r d rl 1 r2 11 A r 1 1 r r cos 0 s 2 rT 2 2 r 0 1 3 0 cos2 0 di Linear quadrapole 4 Model an atom as a fixed positive nucleus of charge Q with a surrounding spherical negative electron cloud of nonuniform charge density P po 1 r Ro r Ro a If the atom is neutral what is po b An electric field is applied with local field ELo causing a slight shift d between the center of the spherical cloud and the positive nucleus What is the equilibrium dipole spacing c What is the approximate polarizability a if 9eoELoE poRo 1 5 Two colinear dipoles with polarizability a are a distance a apart along the z axis A uniform field Eoi is applied p aEL a a What is the total local field seen by each dipole b Repeat a if we have an infinite array of dipoles with constant spacing a Hint 1 11 n s 1 2 c If we assume that we have one such dipole within each volume of a s what is the permittivity of the medium 6 A dipole is modeled as a point charge Q surrounded by a spherical cloud of electrons with radius Ro Then the local Problnm 283 field EL differs from the applied field E by the field due to the dipole itself Since Edip varies within the spherical cloud we use the average field within the sphere P4 Q rR 0 3 a sin th etro h lu a h rgn hwta a Using the center of the cloud as the origin show that the dipole electric field within the cloud is Q ri di Qri Edp 4ireoRo 4vreo d r 2 2rd cos S b Show that the average x and y field components are zero Hint i sin 0 cos 0i sin 0 sin Oi cos Oi c What is the average z component of the field Hint Change variables to u r d 2rdcos and remember r Ir dj d If we have one dipole within every volume of 31rR3 how is the polarization P related to the applied field E 7 Assume that in the dipole model of Figure 3 5a the mass of the positive charge is so large that only the election cloud moves as a solid mass m a The local electric field is E 0 What is the dipole spacing b At t 0 the local field is turned off Eo 0 What is the subsequent motion of the electron cloud c What is the oscillation frequency if Q has the charge and mass of an electron with Ro 10 m d In a real system there is always some damping that we take to be proportional to the velocity fdampin nv What is the equation of motion of the electron cloud for a sinusoidal electric field Re Eoe e Writing the driven displacement of the dipole as d Re de i what is the complex polarizability d where Q Q Eo f What is the complex dielectric constant i e je 6 of the system Hint Define o Q2 N meo g Such a dielectric is placed between parallel plate electrodes Show that the equivalent circuit is a series R L C shunted by a capacitor What are C1 C 2 L and R h Consider the limit where the electron cloud has no mass m 0 With the frequency w as a parameter show that 234 Polarizationand Conduction Re fe j i Area A C1 Re vej t I I 2 L R a plot of er versus e is a circle Where is the center of the circle and what is its radius Such a diagram is called a Cole Cole plot i What is the maximum value of ei and at what frequency does it occur 8 Two point charges of opposite sign Q are a distance L above and below the center of a grounded conducting sphere of radius R Q a What is the electric field everywhere along the z axis and in the 0 v 2 plane Hint Use the method of images b We would like this problem to model the case of a conducting sphere in a uniform electric field by bringing the point charges Q out to infinity L o What must the ratio Q L 2 be such that the field far from the sphere in the 0 wr 2 plane is Eoi c In this limit what is the electric field everywhere 9 A dipole with moment p is placed in a nonuniform electric field a Show that the force on a dipole is f p V E I Problems 235 Sn J 2 v I V P tP T P1 b c b Find the force on dipole 1 due to dipole 2 when the two dipoles are colinear or are adjacent a distance a apart c Find the force on dipole 1 if it is the last dipole in an infinite array of identical colinear or adjacent dipoles with spacing a Hint E …
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