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 231d-x_PROBLEMSSection 3-11. A two-dimensional dipole is formed by two infinitely longparallel line charges of opposite polarity ±X a small distance di,apart.(a) What is the potential at any coordinate (r, 46, z)?(b) What are the potential and electric field far from thedipole (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 chargedistributions:2 III tjL + XL + o L Xo dL -oL(a) (c) (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 asAAo(1-z/L), O<z<LA-Ao(l+z/L), -L<z<O(c) Two uniform opposite polarity line charges *Ao eachof 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 sur-face charge Q on the upper half and -Q on the lowerhalf. (Hint: i, = sin 0 cos i,. +sin 0 sin 4$i, +cos Oi,.)(f) A spherical volume with total uniformly distributedvolume charge of Q in the upper half and -Q on thelower half. (Hint: Integrate the results of (e).)3. The linear quadrapole consists of two superposeddipoles along the z axis. Find the potential and electric fieldfor distances far away from the charges (r >d).' 1 1 + A) . -0 --s' 0)rl r 2 rT1 _1 1 _ + cos 0 _ ()2 (1 -3 cos2 0)r2 r ( r 2 rLinear quadrapole4. Model an atom as a fixed positive nucleus of charge Qwith a surrounding spherical negative electron cloud ofnonuniform 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 aslight shift d between the center of the spherical cloud andthe positive nucleus. What is the equilibrium dipole spacing?(c) What is the approximate polarizability a if9eoELoE(poRo)<< 1?5. Two colinear dipoles with polarizability a are a distance aapart 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 withconstant spacing a. (Hint: :111/ns • 1.2.)(c) If we assume that we have one such dipole within eachvolume of as, what is the permittivity of the medium?6. A dipole is modeled as a point charge Q surrounded by aspherical cloud of electrons with radius Ro. Then the local__diProblnm 283field EL, differs from the applied field E by the field due tothe dipole itself. Since Edip varies within the spherical cloud,we use the average field within the sphere.QP4 3~-rR0(a sin th etro h lu a h rgn hwta(a) Using the center of the cloud as the origin, show thatthe dipole electric field within the cloud isQri, Q(ri, -di)Edp= -4ireoRo + 4vreo[d +r2-2rd cos ]S(b) Show that the average x and y field components arezero. (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 andremember (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 massof the positive charge is so large that only the election cloudmoves as a solid mass m.(a) The local electric field is E0.What is the dipole spacing?(b) At t = 0, the local field is turned off (Eo = 0). What is thesubsequent motion of the electron cloud?(c) What is the oscillation frequency if Q has the chargeand mass of an electron with Ro= 10-' m?(d) In a real system there is always some damping that wetake to be proportional to the velocity (fdampin,, = -nv). Whatis the equation of motion of the electron cloud for a sinusoi-dal electric field Re(Eoe")?(e) Writing the driven displacement of the dipole asd = Re(de-i).what is the complex polarizability d, where Q= Q= Eo?(f) What is the complex dielectric constant i = e,+je6 ofthe system? (Hint: Define o = Q2N/(meo).)(g) Such a dielectric is placed between parallel plate elec-trodes. Show that the equivalent circuit is a series R, L, Cshunted by a capacitor. What are C1, C2, L, and R?(h) Consider the limit where the electron cloud has nomass (m = 0). With the frequency w as a parameter show thatRe(fe j~iArea AC12 L Ra plot of er versus e, is a circle. Where is the center of thecircle and what is its radius? Such a diagram is called aCole-Cole plot.(i) What is the maximum value of ei and at what frequencydoes it occur?8. Two point charges of opposite sign Q are a distance Labove and below the center of a grounded conducting sphereof radius R._-Q(a) What is the electric field everywhere along the z axisand in the 0 = v/2 plane? (Hint: Use the method of images.)(b) We would like this problem to model the case of aconducting sphere in a uniform electric field by bringing thepoint charges ± Q out to infinity (L -* o). What must the ratioQ/L2 be such that the field far from the sphere in the 0 = wr/2plane is Eoi,?(c) In this limit, what is the electric field everywhere?9. A dipole with moment p is placed in a nonuniform electricfield.(a) Show that the force on a dipole isf = (p-V)E234 Polarizationand ConductionRe(vej t)I IIT PlProblems 235Sn2 P tPv ITV *P1(b), (c)(b) Find the force on dipole 1 due to dipole 2 when the twodipoles are colinear, or are adjacent a distance a apart.(c) Find the force on dipole 1 if it is the last dipole in aninfinite array of identical colinear or adjacent dipoles withspacing a. (Hint: E:-,I 1/n4 = r4/90.)10. A point dipole with moment pi, is a distance D from thecenter of a grounded sphere of radius R.(Hint: d<<D.)(a) What is the induced dipole moment of the sphere?(b) What is the electric field everywhere along the z axis?(c) What is the force on the sphere? (Hint: See Problem9a.)Section 3-211. Find the potential, electric field, and charge density dis-tributions for each of the following charges placed within amedium
View Full Document
Unlocking...