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 chapter 4 electricfield boundary value problems 258 Electric Field Boundary Value Problems The electric field distribution due to external sources is disturbed by the addition of a conducting or dielectric body because the resulting induced charges also contribute to the field The complete solution must now also satisfy boundary conditions imposed by the materials 4 1 THE UNIQUENESS THEOREM Consider a linear dielectric material where the permittivity may vary with position D e r E e r VV 1 The special case of different constant permittivity media separated by an interface has e r as a step function Using 1 in Gauss s law yields V r VV pf 2 which reduces to Poisson s equation in regions where E r is a constant Let us call V a solution to 2 The solution VL to the homogeneous equation V e r V VI 0 3 which reduces to Laplace s equation when e r is constant can be added to Vp and still satisfy 2 because 2 is linear in the potential V e r V Vp VL V e r V VP V e r V VL Pf 0 4 Any linear physical problem must only have one solution yet 3 and thus 2 have many solutions We need to find what boundary conditions are necessary to uniquely specify this solution Our method is to consider two different solutions V1 and V2 for the same charge distribution V eV Vi P V eV V 2 Pf 5 so that we can determine what boundary conditions force these solutions to be identical V V 2 259 Boundary Value Problems in CartesianGeometries The difference of these two solutions VT V V 2 obeys the homogeneous equation 6 V eV Vr 0 We examine the vector expansion V eVTVVT VTV EVVT eVVT VVT eVVTI 2 7 0 noting that the first term in the expansion is zero from 6 and that the second term is never negative We now integrate 7 over the volume of interest V which may be of infinite extent and thus include all space V eVTVVT dV V eVTVVT dS I EIVVTI dV 8 The volume integral is converted to a surface integral over the surface bounding the region using the divergence theorem Since the integrand in the last volume integral of 8 is never negative the integral itself can only be zero if VT is zero at every point in the volume making the solution unique VT O0 V 1 V2 To force the volume integral to be zero the surface integral term in 8 must be zero This requires that on the surface S the two solutions must have the same value VI V2 or their normal derivatives must be equal V V 1 n V V2 n This last condition is equivalent to requiring that the normal components of the electric fields be equal E V V Thus a problem is uniquely posed when in addition to giving the charge distribution the potential or the normal component of the electric field on the bounding surface surrounding the volume is specified The bounding surface can be taken in sections with some sections having the potential specified and other sections having the normal field component specified If a particular solution satisfies 2 but it does not satisfy the boundary conditions additional homogeneous solutions where pf 0 must be added so that the boundary conditions are met No matter how a solution is obtained even if guessed if it satisfies 2 and all the boundary conditions it is the only solution 4 2 BOUNDARY VALUE PROBLEMS IN CARTESIAN GEOMETRIES For most of the problems treated in Chapters 2 and 3 we restricted ourselves to one dimensional problems where the electric field points in a single direction and only depends on that coordinate For many cases the volume is free of charge so that the system is described by Laplace s equation Surface 260 Electric FieldBoundary Value Problems charge is present only on interfacial boundaries separating dissimilar conducting materials We now consider such volume charge free problems with two and three dimensional variations 4 2 1 Separation of Variables Let us assume that within a region of space of constant permittivity with no volume charge that solutions do not depend on the z coordinate Then Laplace s equation reduces to 82V O2V ax2 y2 1 0 We try a solution that is a product of a function only of the x coordinate and a function only of y 2 V x y X x Y y This assumed solution is often convenient to use if the system boundaries lay in constant x or constant y planes Then along a boundary one of the functions in 2 is constant When 2 is substituted into 1 we have d 2X d2Y Y X 0 dx dy 1 d2X X dx2 1 d2 Y Y dy 3 where the partial derivatives become total derivatives because each function only depends on a single coordinate The second relation is obtained by dividing through by XY so that the first term is only a function of x while the second is only a function of y The only way the sum of these two terms can be zero for all values of x and y is if each term is separately equal to a constant so that 3 separates into two equations 1 d2X X k2 1 d 2 Y k d 4 where k2 is called the separation constant and in general can be a complex number These equations can then be rewritten as the ordinary differential equations d 2X Sk 2X O dx d2 dy k Y O2 Boundary Value Problems in CartesianGeometries 4 2 2 261 Zero Separation Constant Solutions When the separation constant is zero A2 0 the solutions to 5 are X arx bl Y cry dl where a b1 cl and dl are constants The potential is given by the product of these terms which is of the form V a 2 b2 x C2 y d 2xy The linear and constant terms we have seen before as the potential distribution within a parallel plate capacitor with no fringing so that the electric field is uniform The last term we have not seen previously a Hyperbolic Electrodes A hyperbolically shaped electrode whose surface shape obeys the equation xy ab is at potential Vo and is placed above a grounded right angle corner as in Figure 4 1 The Vo 0 5 25 125 Equipotential lines Vo ab Field lines y2 X2 const Figure 4 1 The equipotential and field lines for a hyperbolically shaped electrode at potential Vo above a right angle conducting corner are orthogonal hyperbolas 262 Electric FieldBoundary Value Problems boundary conditions are …
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