b the value of the normal derivative grad 57498 is specified on the whole boundary Neumann condition c the is specified on part of the boundary and grad 57498 on the rest 3 If satisfies Laplaces equation in a region bounded by the surface can at ID: 25998 Download Pdf

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b the value of the normal derivative grad 57498 is specified on the whole boundary Neumann condition c the is specified on part of the boundary and grad 57498 on the rest 3 If satisfies Laplaces equation in a region bounded by the surface can at

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LAPLACE'S AND POISSON'S EQUATIONS UW Poisson's equation In regions of no charges the equation turns into: Laplace's equation Solutions to Laplace's equation are called Harmonic Functions Properties of harmonic functions 1) Principle of superposition holds 2) A function ) that satisfies Laplace's equation in an enclosed volume and satisfies one of the following type of boundary conditions on the enclosing boundary is unique (a) the value of the f unction is specified on the whole boundary (Dirichlet condition). (b) the value of the normal derivative, grad + , is

specified on the whole boundary (Neumann condition). (c) the is specified on part of the boundary and grad + on the rest. 3) If ) satisfies Laplace's equation in a region , bounded by the surface , can attain neither a maximum nor a minimum within . Extreme values occur only at the surface Discussion: If the potential is constant on a boundary of a volume not containing any charges the potential has the same constant value within the whole volume.

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Bo E. Sernelius 4:2 SOLUTION OF LAPLACE'S EQUATION WITH SEPARATION OF VARIABLES. There are eleven different coordinate

systems in which the Laplace equation is separable. We will here treat the most important ones: the rectangular or cartesian; the spherical; the cylindrical. The geometry of a typical electrostatic problem is a region free of charges bounded by a surface of a given geometry. It can be of rectangular box type, spherical, cylindrical or of some other type. The standard method then is to choose a coordinate system in which the boundary surface coincides with the surface where one of the coordinates is constant. In the special case of a 2D configuration, where the bounding surface and the boundary

conditions on the surface only depend on two variables, one may use conformal mapping to go from one geometrical shape to another. Other situations are too complicated to solve by these methods. Then one has to rely on purely numerical methods, like solving finite-difference versions of the Laplace's equation, finite element methods (FEM) or some other method.

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Bo E. Sernelius 4:3 RECTANGULAR COORDINATES +++ xyz Assume we may write xyz XxYyZz ,, YZ dX dx XZ dY dy XY dZ dz Note that the derivatives are no longer partial. 111 dX dx dY dy dZ dz The first term depends on only, the

second on only and the third on only. The equation can only be valid if each of the terms is a constant: 222 dX dx dY dy dZ dz FGL ''' Since we are considering the electrostatic potential it is real valued. This means that all these squares are real valued, but the last relation shows that the constants themselves cannot all be real valued, neither can they all be imaginary.

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Bo E. Sernelius 4:4 We can only have the following cases a) two real, one imaginary b) one real, two imaginary c) one real, one imaginary, one zero d) th ree zero An imaginary separation constant leads to

an oscillatory solution while a real valued leads to an exponential. Let us arbitrarily let ' and ' be imaginary: FF GG LL 22 22 22 } } , and are all real valued. dX dx dY dy dZ dz 222 22 LFGLFG ; Xx Ae Be Yy Ce De Zz Ee Fe ix ix iy iy zz FF GG LL

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Bo E. Sernelius 4:5 The complete solution is xyz XxYyZz Ae Be Ce De Ee Fe ix ix rs iy iy rs rs rrss rs rs ,, FFGG LL Short hand notation: xyz e e e ix iy z ,, ~ FGL All the constants will be determined from the boundary conditions of the problem.

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Bo E. Sernelius 4:6 SPHERICAL COORDINATES 222 11 1 +++ rr yV yV yO sin

sin sin rRrPQ ,, VOVO 11 1 222 rR dr dR dr rP dP rQ dQ sin sin sin VO multiply with 22 sin sin sin sin VV dr dR dr P dP dQ dQ The left-hand side depends only on and , while the right-hand side depends only on . Thus the two sides must be a constant, dQ mQ Q e m im 0012 ; ~ ; , , Note: If the physical problem limits to a restricted range can be a non- integer. Now we return to the left-hand side and rearrange the terms: 11 dr dR dr P dP sin sin sin VV The new left-hand side depends only on and the right-hand side on only Thus, they must be a constant, +1).

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Bo E. Sernelius 4:7

We get dr dR dr ll R 10 and 10 sin sin sin VV dP ll To solve the first, we make the ansatz: RAr and obtain the two solutions and -(l+1) . The general solution is then Rr Ar B ll For the polar-angle function ) it is customary to make the substitution cos ; sin VV qq dx This gives dx dP dx ll 11 We will first limit ourselves to axial or azimuthal symmetry.

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Bo E. Sernelius 4:8 Axial symmetry 1210 dP dx dP dx ll Legendre's equation Note that if =±1 are excluded from the problem may be non-integer. The solution is the Legendre polynomial of order : cos Thus we have the

general solution to Laplace's equation in spherical coordinates for the special case of axial symmetry as: rArB cos VV The Legendre polynomials can be obtained from Px dx Rodrigues' formula or from the generating function Fx Px 12 12 or from recursion relations such as: lPx lxPxlPx lll 121 11 or dP dx lxP x lP x ll

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Bo E. Sernelius 4:9 The polynomials form a complete, orthogonal set of functions in the domain -1 1 (0 fx APx fxPxdx ll 21

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Bo E. Sernelius 4:10 General case, no axial symmetry. In this case we have in general a non-zero value and the differential

equation for is more elaborate. The Legendre polynomials are replaced by the associated Legendre polynomials , cos . For a given -value there are +1 possible -values: m = 0, ±1, ±2,, ±3, ... There is a more general Rodrigues' formula for these functions: Px dx xlml lm lm ff 11 ; For any given the functions cos and cos are orthogonal and the associated Legendre polynomials for a fixed form a complete set of functions in the variable . The product of Px and im forms a complete set for the expansion of an arbitrary function on the surface of a sphere. These functions are called spherical

harmonics llm lm Pe mim VO cos 21 They are orthonormal YY d ddYY ll mm VOVO OVVVOVOII UU ,*, sin , * , ''

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Bo E. Sernelius 4:11 fCY ml VOVO ,, and CfY d VOVO ,*, The general solution to Laplace's equation in terms of spherical harmonics is rArB ml ml ,, , VOVO

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Bo E. Sernelius 4:12 CYLINDRICAL COORDINATES 11 +++ rr rz yV rz RrQZz ,, VV 111 rR r dr dR r dr rQ dQ Zz dZz dz Rr dr dR r dr Zz dZz dz dQ 22 dQ nQ Qe n in ~ ; ,,, 012 ( may sometimes be non-integer) 11 rR dr dR dr dZ dz dZ dz kZ Zz e kz dr dR dr kr n R 22 2

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Bo E. Sernelius 4:13

Cylindrical symmetry and Cylindrical Harmonics Then we may let vanish and dr dR dr nR The n = 0 term has to be treated separately Rr ABrn Ar B 00 123 ln , , , , CD n CnDnn nn VV 00 123 , cos sin , , , General solution in cylindrical coordinates with no -dependence. rABr ArB CD ,ln cos sin VVV 00 The terms are called cylindrical harmonics

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Bo E. Sernelius 4:14 No cylindrical symmetry and Bessel functions. Now, we have to keep the constant in the differential equation for dr dR dr kr n R 22 2 To solve this one usually makes the substitution ukr dr du ; This leads to Bessel's

equation dR du dR du unR 22 The solution to this equation is the so-called Bessel function of order n , (u) -n (u) is also a solution. These are linearly dependent for integer orders but not for non-integer orders. One usually introduces another function instead of -n (u), the so-called Neumann function or Bessel function of the second kind , (u). Nu Ju n J u nn cos sin General solution to Bessel's equation may be written as Rkr AJkr BNkr nnnnn (u) is regular at origin and at infinity. (u) is not regular at origin but at infinity.

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Bo E. Sernelius 4:15 The general solution to

Laplace's equation in cylindrical coordinates can be written as the Fourier-Bessel expansion rz AJkr BNkre e mn n m mn n m in k z mn ,, ~ Other useful properties of the Bessel function Let be the th root of kr ), i.e., ) = 0. Then ) form a complete orthogonal set for the expansion of a function of in the interval 0 fr D J k r n mn nm for any Jk frJ k rdr mn nm nm WW Fourier-Bessel series analogous to the Fourier transform.

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Bo E. Sernelius 4:16 Discussion: If we had chosen + instead of - 11 rR dr dR dr dZ dz The - dependence had been plane waves instead of exponentials and the

dependence had been found as solutions to the modified Bessel equation: dR du dR du unR 22 with the modified Bessel functions (u) and (u) as solutions. The first is bounded for small arguments and the second for large. Thus, an alternative expression for the general solution is rz AIkr BKkre e mn n m mn n m in ik z mn ,, ~

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