ELECTROMAG THEORY 1
ELECTROMAG THEORY 1 PHY 6346
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This 12 page Class Notes was uploaded by Mrs. Linda Wiegand on Friday September 18, 2015. The Class Notes belongs to PHY 6346 at University of Florida taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/206774/phy-6346-university-of-florida in Physics 2 at University of Florida.
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Date Created: 09/18/15
Chapter 6 Complex Variable Methods In two dimensions there are elegant methods for solving electrostatics problems which take advantage of the fact that analytic functions in the complex 2 ziy plane are harmonic7 they are solutions of Laplace7s equation The use of these techniques is almost an art form and will only be brie y touched upon in the notes For further details I can recommend G F Carrier M Krook and C E Pearson Functions of a Complccc Variable 61 Analytic Functions and the CauchyRiemann Equa tions Let f2 be a complex function of the complex variable 2 z iy A complex function f2 is analytic at the point 20 if its derivative dfd2 exists at 2 20 and at each point in some neighborhood of 20 The derivative is de ned similarly to the derivative of a function of a real variable f 20 2 df lim fz 4W 61 d2 1Her 2 7 20 Most common functions are differentiable and therefore analytic for instance 2 61 ln 2 and so on However since the limit in Eq 61 must be the same for any mode of approach differentiability for functions of a complex variable is much more restrictive than for functions of a real variable For instance the function f2 22 22 y2 is not differentiable To see this consider the limit 7 7 1m w hm 2 202 20gt 62 ZHZO 2 7 20 Z720 2 7 20 and let 2 approach 20 from a direction making a xed angle 6 with the real axis so that 2 7 20 rem with r 7 0 Then the limit is 22 7 2023 li H1 2720 Z 7 20 23 20672 63 41 which depends on 0 ie7 the derivative depends on the direction of approach to the point z Analytic functions have an amazing propertyitheir real and imaginary parts satisfy the Cauchy Riemann equations To see how this works7 let7s call the real and imaginary parts of fz u and i ie7 f u ii7 with u and 1 real The derivative of this function with respect to z is fzo lim 64 hm Wowoll tll w l 07yoll memosaye z 7 zo my 7 yo If this derivative exists7 then it should be independent of the mode of approach to the point 20 For instance7 we could take y yo and then take the limit as x approaches zo fZO hm 1495790 Mew0 l 95790 07yol 6395 mama 95 7 950 6 u ia v 7 x an Alternatively7 we could have x 0 and take the limit in y fZO hm luWow o oll t W me U07yol 6396 24410 2y 7 yo i u 31 Z52 6y By equating the real and imaginary parts of these expressions7 we obtain the Cauchy Riemann equations7 3U 31 3U 31 7 3y7 3y i An analytic function will satisfy the Cauchy Riemann equations They can be combined to obtain Laplace7s equation7 67 2 2 2 2 68 3x2 3y 3x2 3y Therefore the real and imaginary parts of an analytic function are quotharmonicquot for an ana lytic function Fz we have VZF 0 so that analytic functions are solutions of Laplace s equation Some terminology o Analytic functions are often referred to as holomorphic 0 An entire function is analytic for all nite z examples would be polynomials in z and 62 o A meromorphic function is one whose only singularities are poles 0 Functions such as 212 are multi valued we can insure a consistent de nition of such a function by placing a branch cut in the complex plane which terminates at a branch point For 212 the branch point is at z 0 and we can place the cut along the negative real axis Such functions are extremely important in applications as the cuts are often associated with conducting surfaces For applications in electrostatics we can introduce the complex potential 102 such that ltIgt Rw and KI Iw with ltIgt the potential and KI the stream function77 terminology borrowed from uid mechanics the lines of constant ltIgt are then the equipo tentials and the lines of constant KI are the eld lines or the streamlines By combining the Cauchy Riemann equations once again we can show that 63 9K1 6ltIgt 9K1 VltIgtVII0 69 6x 6x 6y 6y 7 lt l which shows that the streamlines and the eld lines are mutually orthogonal For example suppose 102 22 then separating into real and imaginary parts we nd w 2 7 y2 Z39sz 610 so that ltIgt 2 7 y2 and KI 2mg Lines of constant ltIgt the equipotentials give hyperbolae lines of constant KI the eld lines also give hyperbolae rotated by 45 with respect to the equipotentials From the complex potential we can calculate the compler electric eld using E Em 7Z39Ey 7dwd2 611 62 Some Simple Examples 1 102 7E02 Then ltIgt Rw 7E0x which is the potential due to a uniform eld in the x direction E0 102 7A27T60 lnz Writing 2 pew we have ltIgt 7A27T60 lnp which is the potential due to an in nite line charge at the origin 9 Take two line charges of opposite charge centered at izog 102 7A27T60lnz 7 20 7 lnz 20 Taking the limit that 20 7 0 we have 102 2A2027T602 which is a two dimensional point dipole with a dipole moment per unit length p 2A20 7 By taking superpositions of these simple solutions we can solve more complicated problems For instance by adding a point dipole of strength p 2760an2 to the potential due to a uniform eld we have 102 7E0z7a2z 612 43 which is zero on a7 and therefore is the solution to the problem of a grounded conducting cylinder in a uniform electric eld The real part of the potential is ltIgt 7E0pia2p cost97 613 and the stream function which gives the eld lines is 1 7E0pa2p sin0 614 102 AzlZ7 with A a real constant lt7s best to invert this one7 so that w2 lt1gt2 i 112 2lt1gt1 z y Tz A27 615 so that gt2 i 112 2lt1gtK11 If we want to nd the equipotentials7 then eliminate 1 between these two equations lt1gt2 A2 2 x i y 617 These curves are parabolas tipped on their sides For ltIgt 07 the parabola becomes the negative real axis7 so our potential is the the solution to Laplace7s equation in the presence of a semi in nite charged conducting strip along x lt 0 We can also nd the eld lines7 112 A2 2 7 my These are also parabolas7 orthogonal to the equipotentials 618 More complicated problems can be solved using conformal mapping methods7 which I wont discuss in any detail However7 to illustrate the power of these techniques7 I will discuss the solution of one problem7 the charged conducting strip 63 The Charged Grounded Conducting Strip Assume that the strip is very thin7 so that it can be taken as a line let7s scale the lengths so that the strip is between z 71 and m 1 The problem is to nd the potential 631 Solution using complex variable methods The complex potential is 102 Ci sinquotlz7 619 44 with C an as yet undetermined constant This is an analytic function with suitably de ned cuts in the complex z plane so it solves Laplace7s equation To see that it satis es the boundary conditions its probably best to invert the function to obtain 2w 2w 5m sin1O coshltIgtO ficos1OsinhltIgtO 620 where we7ve used w ltIgt Z39KI and have written 2 in terms of its real and imaginary parts This in turn yields the two equations which determine the eld lines and equipotentials z sinIlC coshltIgtC y 7 cosqlC sinhltIgtC 621 Notice that when ltIgt 0 x sinIlC and y 0 as we vary 11 z varies between 1 and 1 Therefore the strip 71 S m S 1 and y 0 is an equipotential and can be thought of as a charged conducting strip What about the other equipotentials From Eq 621 we have sin1O W cosTlC whim 622 Adding and squaring we have x y 2 W 1 623 so we see that the equipotentials are ellipses Similarly we nd for the eld lines z y 2 17 63924 so the eld lines are hyperbolae From the potential we can calculate the complex electric eld E E 7 2E idwdz 7L 625 m so that the y component of the eld on the strip is E Lsigny 626 m This is related to the surface charge density through 039 260Ey integrating the surface charge density over the width of the strip we obtain the charge per unit length A Therefore we nd that C A27r60 Notice that the solution is obtained in a relatively compact formithis is characteristic of solutions obtained using complex variable methods 632 Solution using separation of variables in elliptical coordi nates From the form of the equipotentials and eld lines obtained above7 we might guess that the solution could also have been obtained using separation of variables in elliptical coordinates We introduce the coordinates 51752 through m cosh 1 sin 27 y sinh L cos 2 then we see that 2 y 2 cosh 1 sinh 1 17 627 z 2 y 2 7 003 62 17 628 so that 1 labels ellipses and 2 labels hyperbolae7 which are orthogonal Using these new variables in the Laplacian7 after a little algebra we nd that Laplace7s equation becomes 62 62 6 6 6 6 7 629 The boundary condition on the grounded conducting strip is ltlgt 1 0752 0 Using separation of variables7 51752 F 1G 2 using the boundary conditions at 007 we nd that F 17 G constant7 so that the solution is ltlgt 0amp1 The nal step is to invert the equations which de ne the elliptical coordinates to nd 1 in terms of z and y The end result is7 of course7 the same as the result obtained using complex variable methods 64 Edges and Corners As a nal application of complex variable methods7 let7s consider the behavior of the elec trostatic potential near a corner7 The appropriate complex potential is apart from a multiplicative constant 102 22quot 630 By introducing radial coordinates7 z pew7 and separating real and imaginary parts7 we have p 7p sinm9 631 Now if we want ltlgt 0 on 6 0 and t9 67 then we need to choose V mg with n a positive integer Taking n 17 we see that V 7r 7 and therefore ltlgt 7N sin7n9 7 632 and the electric eld near the edge is E pm 633 For B gt 7T7 the electric eld strength diverges near the edge7 and for B 27139 a knife edge7 it diverges as p lZ Now for a real knife edge this divergence will be cut off by the thickness of the edge7 but the fact remains that sharp edges and points can produce very large electric elds This is the idea behind the lightning rodiit produces a large eld in its vicinity7 which may encourage dielectric breakdown of the air7 and therefore produce a conducting path through the air to the rod rather than to your house Chapter 13 Magnetism in matter Magnetization in matter varies on a microscopic scale7 because of electron currents in atomic and molecular orbitals and because of the magnetization density associated with intrinsic magnetic moment of the electrons and also of nuclei7 but the effective nuclear magneton is smaller than lelh2MB by a factor of about 2000 The separation of currents into micro scopic77 currents that give rise to M and macroscopic77 currents that are caused by batteries or by varying magnetic elds is not always clear cut We will assume that such a separation is possible7 and consider only simple cases 1 31 Macroscopic equations Associated with the magnetization density M there is a vector potential M0 MX X XX 3 M0 1 3 A d M d 131 X 47139 x7 x lg z 47139 X X W lxix l z lntegrating by parts7 this becomes WMMw Ax 7 47T X 7 KW d 95 132 showing that there is an effective magnetization current JM V gtlt M 133 Then the B eld obeys the equations V B 07 134 V gtlt B u0J V X M 135 The macroscopic equations are obtained by a suitable average of these7 and have the same form From now on in this lecture7 M and B are averaged quantities It is also convenient to introduce the eld H BMo 7 M 136 so that the macroscopic equations become V B 07 137 V gtlt H J 138 in analogy with V D p 139 V gtlt E 0 1310 H is called the magnetic eld and B is called the magnetic flare density The trouble with these names is that they suggest that the fundamental eld is H7 in analogy to electrostatics7 where the fundamental eld is E7 while actually the fundamental eld is B As in the electrical case7 the boundary conditions at the interface of two media are that Bn is continuous and HH is continuous in the absence of a surface current density A major difference from the electric case is that many materials exhibit a spontaneous magnetization and do not have a simple local relation between M and H that is analogous to P X5E If there is a relation M XltmgtH 1311 where X quot is the magnetic susceptibility7 then one de nes the magnetic permeability a 01 Xm and obtains B pH 1312 in analogy with D 6E Because this relation makes life simple7 one tries to use it as much as possible7 allowing a to depend on H and even on the history of the sample 1 32 Magnetic susceptibility There are many types of magnetic behavior in matter Strictly speaking7 none of them can be discussed classically7 because one can show that classically there is no magnetism at thermal equilibrium This theorem comes from the basic fact that the magnetic force on a moving particle of charge q and velocity V7 the Lorentz force F qv gtlt B 1313 is such that F V 0 Thus the energy of any particle is not changed when a magnetic eld is switched on and the thermal distribution7 which depends only on energy7 is unchanged More formally7 the Hamiltonian of a particle of mass M7 classically or quantally7 is 1 H m p i qAx2 Vx 1314 All thermodynamic results are derivable from the partition function Z Classically 2d d3pd3z expiHkBT 1315 The integral over momenta can be performed by changing variable from p to p 7 qAX7 leaving a 20 that does not depend on A z 27erBT32 dSz expiVxkBT 1316 Quantum theory is essential to describe magnetism in the real world To see how things work out7 we take a constant eld in the z direction The vector potential depends on the gauge A popular gauge choice in quantum theory is the Landau gauge7 where Am7By Ay0 2110 1317 but in our case it is more elegant to use the symmetrical gauge7 where A B gtlt x 1318 Then 2 2B2 1 0 q q 2 2 7 B 1319 2M 2M0 XXP8M02 WHWX Let us consider separately the two eld dependent terms in H7 rst for atoms7 then for molecules and solids 1321 Diamagnetism and paramagnetism of atoms In what follows l7m going to revert to using cgs units rather than SI units in keeping with the literature The diamagnetz39c term QZBZ W x 12 1320 always raises the energy For atoms or ions 2 2 2 7 2 ltz y 7 3 ltrgt 1321 is almost temperature independent The energy change for an atom or ion containing Z electrons is then 122975436 lt 2 1322 99 if we understand that ltT2gt fr2px ds fr2px dsx 13 23 f px d3z Z 39 Equating the energy change to i B mm we obtain the induced dipole moment Z62 r2 dw 7 1324 m 6Mec2 We should use Blow instead of B7 but the difference turns out to be insigni cant With N atoms per unit volume7 the diamagnetic susceptibility is then iNZeZ r2 dia X 6MB 02 1325 For atomic hydrogen r2 3a and recalling XM 92a3 we nd ixwmlXM 1962M56210 19 62h02 2 592 10 6 The ratio ezMeczao is also equal to 12 027 in agreement with the earlier remark that magnetism is a relativistic effect In general7 the magnitude of X01 is Z 10 6 per mole The paramagnetic term can be written in various ways 7 q 2M0 B 2M0 2M0 Bxxp7 g L7 g Bpyypz 1326 Classically7 we compute 20 by integrating over the direction and the magnitude of L This is equivalent to integrating over all values of p and gives a result that does not depend on B7 as one can double check In the real world7 both the direction and the magnitude of L are quantized7 and there is also the spin The full paramagnetic term is e 2Mec B L 90 S 1327 for each electron7 where go 2 100116 The individual electron7s L and S combine to form a total L and S7 and these in turn add up to a total angular momentum J LS not to be confused with the current In an atom or ion only one value of J contributes signi cantly to the quantum partition function Z7 and to rst order in B the paramagnetic interaction is not obviously equivalent to 59 7 1328 2Mec where7 taking go 27 the Lande 9 factor is given by 3 1 SS1 7LL1 1329 g 2 2 JJ 1 ln effect7 we are back to a magnetic moment 69 J 1330 M 2MBC 100
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