Chapter 4 Boundary Value Problems in Spherical and Cylindrical Coordinates

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1 Chapter 4 Boundary Value Problems in Spherical and Cylindrical Coordinates 4. Laplace s Equation in spherical coordinates Laplace s equation in spherical coordinates, (r, θ, φ), has the form where r r Φ (r, θ, φ) r r L LΦ (r, θ, φ). (4.) r L L sin θ sin θ Φ (r, θ, φ) + θ θ sin θ φ (4.) Multiplying Eq. 4. by r,dividing by Φ (r, θ, φ)r (r) Y (θ, φ) and rewriting, r R(r) R r r L L Y (θ, φ). (4.3) Y This equation must be satisfied for all (r, θ, φ) As r, θ, φ are independent variables, Eq. 4.3 can only be satisfied if both sides equal a constant which we choose this constant to be ( +).Then, and L LY (θ, φ) ( +)Y (θ, φ) (4.4) d r R(r) ( +)R(r) (4.5) dr ddr Eq. 4.4 can be further separated using Y (θ, φ) V (θ)w (φ): V (θ) sin θ sin θ V (θ) + θ θ V (θ) sin θ( +)V (θ) W φ W (4.6) Again, since θ,φ are independent variables both sides must equal a constant (the second separation constant) which we take to be m φ W (φ) m W (φ) or (4.7a) and sin θ θ sin θ W m (φ) A m e imφ + B m e imφ (4.7b) θ V (θ) +[sin θ( +) m ]V (θ) (4.8) 73

2 Substituting x cosθ Eq. 4.8 becomes the associated Legendre equation: d ( x ) d dx dx F (x) +[ ( +) m ]F (x).; x cosθ (4.9) x with two linearly independent solutions called the associated Legendre functions of the first, P m (x), and second kind, Q m (x). F, m (x) A, m P m (x)+b, m Q m (x) (4.) One can show that:the P m (cos θ) canbemadetoconvergeforall θ π if : <m m,,. > P m (cos θ) converge.. The Q m (cos θ), howeverdivergeatcos θ for the above conditions on m and. The Q m (cos θ) are possible solutions when cos θ <. In our problem we expect the functions to be well behaved for all x cos θ. A special case is found when m and these solutions, P (x) P (x)are called Legende functions or Legendre polynomials. The P (x) are normalized so that: P (x) (4.) P (x) x P (x) 3x P 3 (x) x 5x 3 P 4 (x) 35x 4 3x +3 8 The associated Legendre polynomials, P m Legendre polynomials by (x) P m (x), of degree and order m. are related to the P m (x) x m/ d m P (x),m ; (4.) dxm P m (x) ( ) m P m (x),m< This particular sign convention is just one possibility. Clearly these are only non-zero when m. FinallywenotethatatP () (by definition), P ( ) ( ), and, if m 6,P m symmetry of these functions for x xis P m ( x) ( ) +m P m (x) (±). The general Legendre s functions of the second kind Q (x), have singularities at x ±. This solution would be one of the set known as Legendre functions of the second kind. The first two Legendre functions of the second kind are, for x, 74

3 Section 4. Spherical harmonics Q (x) µ +x ln tanh (x) (4.3) x Q (x) x µ +x ln x We note the logarithmic divergences for Q n (x) at x ±. In general, the associated Legendre functions can be written in terms of the hypergeometric function, F (a, b, c : z) an infinite series which terminates for a (or b) equal to a negative integer. The termination of the series for a m produces the convergence of the, P m (x): Q m (x) constant m/ x + F (, +, m; x ) (4.4) x P m (x) constant x + x m/ µ x m F (m, + m +,m+; x ) (4.5) where (Handbook of Mathematical Functions, M. Abramowitz and I. Stegun, NBS ) F (a, b, c; z) Γ( c) Γ(a)Γ(b) Σ Γ(a + n)γ(b + n) n z n n!γ(c + n) (4.6) Γ(a) (a )! 4. Spherical harmonics Laplace s and Poisson s equation in spherical coordinates are encountered in a wide range of problems (E&M, thermodynamics, mechanics of continuous media, and quantum mechanics). As might be expected the solutions of these equations have been well studied. The solutions to the angular part of the equation are the spherical harmonics. Theseare the appropriately normalized products of the associated Legendre polynomials in cos θ and the exp (imφ) Y m (θ, φ) ( ) m s +( m)! ( + m)! P m (cos (θ)) e imφ (4.7) The Y m (θ, φ) are solutions to Eq. 4.3: and Some useful properties are: L z Y m L L Y m l ( +)Y m my m ; where L z ẑ L Y m (θ, φ) ( ) m Y m (θ, φ) (4.8a) 75

4 Section 4.3 Poisson s equation with point charge in (r, θ, φ) Y m (π θ, π φ) ( ) Y m (θ, φ) (4.8b) The orthonormality condition: Z π Z π Y m (θ, φ) Y m (θ, φ)sinθdθdφ δ δ mm (4.9) (Y m,ym ) δ δ mm and the completeness condition: X m Y θ,φ Y m (θ, φ) δ φ φ δ cos θ cos θ (4.) 3 Y (θ, φ) 3 Y (θ, φ) Y (θ, φ) / 5 Y (θ, φ) / cos θ Y (θ, φ) / sin θe iφ Y (θ, φ) / 3 cos θ / 5 sin θ cos θe iφ 8π / 5 sin θe iφ (4.) Y 3 (θ, φ) Y 3 (θ, φ) 4 Y 3 (θ, φ) 4 Y 3 3 (θ, φ) 4 / 7 5 cos3 θ 3 cos θ / sin θ 5cos θ e iφ / 5 sin θ cos θe iφ π 35 / sin 3 θe i3φ (4.) 4.3 Poisson s equation with point charge in (r, θ, φ) We will now re-examine the solution to Poisson s equation with a point charge q. Let the charge lie at coordinates r,θ,φ and let the observation point have coordinates (r, θ, φ). The solution will be a Green s function and we consider the dependence on r,θ,φ and on (r, θ, φ) The charge density is ρ (r, θ, φ) qδ(r r ) Gaussian units (4.3) q r δ (r r ) δ cos θ cos θ δ φ φ 76

5 Section 4.3 Poisson s equation with point charge in (r, θ, φ) and Poisson s equation is Φ (r, θ, φ) k ρ (r, θ, φ) (4.4) δ(r r ). Gaussian units (4.4) Φ (r, θ, φ) ρ (r, θ, φ) S.I. units We look for a solution of the form Φ (r, θ, φ) X m Inserting this into Poisson s equation we obtain (using Gaussian units) r X m X X m f m r; r,θ,φ Y m (θ, φ) (4.5) f m r; r,θ,φ Y m (θ, φ) q r δ (r r ) δ cos θ cos θ δ φ φ (4.6) µ d r d dr dr f m ( +)f m Y m (θ, φ) q r δ (r r ) δ cos θ cos θ δ φ φ Multiply both sides by Y m (θ, φ) and integrate over RR d(cos θ)dφ Using the orthonormality of the spherical harmonics one finds: ZZ X µ d r r d dr dr f m ( +)f m Y m m (θ, φ) Y (θ, φ) d(cos θ)dφ r X m X m d dr q r δ (r r ) µr ddr f m ( +)f m δ δ m m ZZ δ φ φ δ cos θ cos θ Y m (θ, φ) d(cos θ)dφ d µr ddr r dr f m ( +)f m q r δ (r r ) Y m θ,φ (4.7) Thus the angular dependence of f m r; r,θ,φ is given by Y m θ,φ and we define The functions g (r; r ) satisfy f m r; r,θ,φ g (r; r ) Y m θ,φ. (4.8) µ d r r d dr dr g ( +)g q r δ (r r ) (4.9) These equations are solved by : () obtaining the solutions to the homogeneous equations for r<r and r>r, () making g (r; r ) continuous at r r, and (3) satisfying the condition placed on g (r; r ) obtained by integrating Eq. 4.9 from r r ε to r r + ε. 77

6 Section 4.3 Poisson s equation with point charge in (r, θ, φ) The solution to the homogeneous equation in r has the general form g (r; r )α (r ) r + β (r ) r. (4.3) For r<r g (r; r )α (r ) r and for r>r g (r; r )β (r ) r. Continuity at r r requires that Integrating Eq. 4.9 from r r ε to r r + ε we obtain Z r +ε r ε g (r; r ) a r r +,r<r (4.3) g (r; r r ) a,r>r r+ µ d r r d dr dr g ( +)g dr q r Z r +ε r ε δ (r r ) dr Z r +ε r ε µ d r r d dr dr g ( +)g dr q r The second term on the left is continuous at r r and drops out of the equation. Z r +ε r ε µ d r r d dr dr g dr q r Integrating by parts twice, µ r r d dr g rr +ε Z r +ε rr ε + r ε r d dr g dr q r µ r r d dr g rr +ε rr ε + r g rr +ε Only the first term on the left is non-zero at r r so Z r +ε rr ε + r ε " µdg r (r; r µ ) dg (r; r ) dr rr +ε dr Now use the specificformsforg (r; r ): r a " µ d dr r µ d r + rr +ε dr r g dr rr ε r # r + q rr ε q r # q (4.3) Thus r r r a[ ( +) ]a[ ( +) ] a( +) r+ r + a q + 78 (4.33)

7 Section 4.4 The Coulomb potential in spherical coordinates 4.4 The Coulomb potential in spherical coordinates Using Eq we have the expansion of a point charge coulomb potential given by, X r< Φ (r, θ, φ) q +r + Y m (θ, φ) Y θ,φ. (4.33b) m > Φ (r, θ, φ) qδ(r r ) Gaussian units where the point charge is at r and r < min(r, r ) and r > max(r, r ). Since the position of the point charge is provided in this simple example, one considers r r a constant vector and qδ(r r ) is only a function of.r However, if wehaveachargedensity,ρ(r) q f(r, θ, ϕ) it is clear that ρ(r) is a function of r, andisspecified by f(r, θ, ϕ). The latter includes any dependence on known vectors such as r. 4.5 Green s Function and Expansion of r r Note that the Green s function for Laplace s equation (for V all space) satisfies the following: Thus. r r δ(r r ) G(r, r ) X r< +r + m > G(r, r ) δ(r r ) Gaussian units Recall that the G(r, r ) can also be written as follows: Y m (θ, φ) Y θ,φ. (4.34a) G(r, r ) r r (4.34b) The Green s function differs from the Φ (r) (ineq4.33)asφ(r) describes the electrostatic potential due to a point charge, q at a given point, r r. In contrast, G(r, r ) represents a unit point charge at r, which is as yet unspecified.. Equations 4.34 yield a convenient expansion for r r in spherical harmonics r r X X m + and provides a convenient way to evaluate integrals of the form: r< r> + Y m (θ, φ) Y θ,φ (4.35) ZZZ ρ(r ZZZ ) r r dv 79 ρ(r )G(r, r )dv

8 Section 4.4 The Coulomb potential in spherical coordinates Example: Determine the electrostatic potential due to the following distribution of charge (a spherical shell of charge with a cos θ angular dependence); ρ(r) Q cos θδ(r a) a assuming that the potential like as r r Solution: We use the Green s function for all of space (Eq. 4.34a) as follows in S.I. units: φ (r) ε ZZZ all space ZZ [ ρ (r )]G (r, r ) d 3 r + φ (r) ε r ZZZ ZZ φ (r s) G (r, r s) ds G (r, r s) φ (r s) ds r r >a since the surface integrals vanish like r as r. (seepage59). φ (r) ε ε ZZZ ZZZ Q ε a all space all space m ρ (r ) X ρ (r ) G (r, r ) d 3 r r r d3 r (note r dependence of ρ) SI units Q a cos θ δ(r a) Z + Y m (θ, φ) X m r< r> + + r< r> + Y m r δ(r a)dr ZZ Next, we use the expansion of cos θ in spherical harmonics: r cos θ 3 Y (θ,φ ) φ (r) φ (r) φ (r) ε Q ε a Q ε a X m X m Q a + Q 3εa r cos θ Q a 3 3εa r cos θ Z + Y m (θ, φ) Z + Y m (θ, φ) r 3 Y (θ, φ) if r<a if r>a r< r> + Note that the final angular dependence of φ (r) is r< r> + r< r> + r ZZ r δ(r a)dr 3 r δ(r a)dr (θ, φ) Y θ,φ d 3 r r 3 δ δ m r δ(r a)dr Q 3εa cos θ r < of radius a and falls off like outside the sphere. Continuity at r a is preserved r cos θ Y θ,φ d(cos θ )dϕ Y (θ,φ )Y θ,φ d(cos θ )dϕ r> + a r 3 Y (θ, φ) cosθ.and the r dependence is linear inside the sphere 8

9 Section 4.6 Addition Theorem for Spherical Harmonics 4.6 Addition Theorem for Spherical Harmonics Suppose in the expansion of r r X X m + r< r> + Y m (θ, φ) Y θ,φ (see 4.33) we let r r ẑ.sothat θ. Then only the m terms will appear in the su Y m,φ if m 6.Inthiscase, using the expression for Y m (θ, φ) in Eq. 4.7 and P () r r ẑ r r ẑ [r r r r cos θ+r ] / r< +r> + r< +r> + r< P (cos θ) r + > r< r + > P (cos θ) Y (θ, φ) Y,φ r r + + P (cos θ) P () The angular dependence of this expression depends only on the angle, γ, between the vector locating the charge, r,and the vector locating the observation point, r. That is, r r rr cos γ. So [r r r r cos γ+r ] / r< r + > X m P (cos γ) + Thus, one obtains the addition theorem for spherical harmonics: r< r> + Y m (θ, φ) Y θ,φ P (cos γ) X + Y m (θ, φ) Y θ,φ m ˆr ˆr cos γ sinθ sin θ [cos φ cos φ +sinφsin φ ]+cosθcos θ sinθ sin θ cos φ φ +cosθ cos θ (4.36) 4.7 Problems with azimuthal symmetry The angular dependence of systems in which the boundary conditions and the charge densities have axial symmetry, i.e., 8

10 Section 4.6 Addition Theorem for Spherical Harmonics they are independent of φ,is given by the Legendre functions P ν (cos θ). The potential will have the general form Φ (r, θ) f n (r) P νn (cos θ) (4.37) n Problems with θ π If the range of θ is from zero to π then the condition that Φ (r, θ) be non-singular requires that ν n n (an integer). In this case the sum will be over Legendre polynomials. The potential is then given by Φ (r, θ) αn r n + β n r n P n (cos θ). (4.38) n To evaluate the coefficients we need the orthogonality condition for the Legendre polynomials δ mn n + Z π Z π P m (cos θ) P n (cos θ)sinθdθ (4.39) P m (cos θ) P n (cos θ) d(cos θ). Specifying the potential at r a and at r b>athe coefficients satisfy the equations Φ (a, θ) Φ (b, θ) αn a n + β n a n P n (cos θ) I n (a) P n (cos θ) n n αn b n + β n b n P n (cos θ) I n (b) P n (cos θ) n n Mulitply each side byp n (cos θ) and integrate over θ: αn a n + β n a n µ n + Z π P n (cos θ) Φ (a, θ)sinθdθ I n (a) (4.4) αn b n + β n b n µ n + Z π P n (cos θ) Φ (b, θ)sinθdθ I n (b) which have solutions α n an+ I n (a) b n+ I n (b) a n+ b n+ (4.4) β n (ab) n+ b n I n (a) a n I n (b) a n+ b n+ In the limit that a we have that α n b n I n (b) and β n, while if b we have α n and β n a n+ I n (a). Problems with θ θ m <π If we require that the solution of Legendre s equation be zero at θ θ m andthatitbefinite for θ θ m the most general solution will be a Legendre function of the first kind, P ν (cos θ) where ν 6 integer (See Abramowitz & Stegun, pp33, 556). Legendre functions of the second kind, which diverges at cos θ,are excluded. The P ν (cos θ) are generally expressed in terms of the hypergeometric functions, F (a, b, c; w) (See Eqs. 4.5 and 4.6) where we set a ν, b +ν, 8

11 c and w cos θ series. Section 4.6 Addition Theorem for Spherical Harmonics sin θ.whenν 6 integer ( a is not a negative integer) the F ( ν,+ν;,w) is an infinite P ν (cos θ) P υ (x) F µ µ θ ν,+ν;, sin with the function (z) n given by [sin (θ/)] n ( ν) n (ν +) n n! n (z) n z (z +)(z +)... (z + n ). (z + n )! (z )! For a given angle, θ β, the Legendre functions can be plotted as a function of ν to determine the values of ν which yield P ν (cos β). The figure below illustrates the approach with β π/6, π/3, and π/. In the case that β π/ we note that the appropriate functions will be the Legendre polynomials of odd order. Plot versus order.63 Legendre function order 9 deg. 3 deg. 6 deg.fig. 4. Example: the fields near the point of a cone The results of the last section can be applied to the problem of determining the fields and surface charge density near the tip of a conical conductor. In the region of interest the surface of the cone makes an angle β with the z axis. For β<π/ the region of interest is inside the cone while for β>π/ the region is outside the conical surface. If we take the cone to be at zero potential then the leading term in the potential will be where ν is determined by the condition that cos β is the first zero of P ν (cos θ). 83 Φ (r, θ, φ) Φ r ν P ν (cos θ) (4.43)

12 Section 4.8 Green s function in a spherical shell For example if β π/6 then ν 4 (see Fig. 4.). Near the tip of the cone the electric field will have components E r (r, θ) Φ ν r ν P ν (cos θ) E θ (r, θ) r θ Φ r ν P ν (cos θ) Φ r ν sin θpν (cos θ) and the surface charge density will be given by ˆn E σ (r) where ˆn is normal to the conical surface defined by θ π/6. Thatis, ˆn ˆθ. σ (r) () Φ r ν sin βp ν (cos β). (4.45) We first consider the case with β<π/. For this system ν > and the fields and surface charge vanish as r. The geometry of this system would correspond to a conical hole in a conductor. As expected the field does not readily penetrate a hole in a conductor. The alternate case has β>π/. From Fig. we deduce that ν <. Clearly the fields and surface charge diverge as one approaches the point of the cone. This is confirmation of the folk-lore that the fields are strong at pointed regions of a charged conductor. 4.8 Green s function in a spherical shell To illustrate the Green s function techniques we consider systems which lie in a spherical shell a r b bounded by conducting (segmented?) spherical shells. In this case the potentials on the conductors are generally specified and the Dirichlet green s function is required. Excluding the value r r gives two homogeneous equations, one for a r<r and the other for r <r b. The solutions for the r and r dependence are of the form in Eq To make the g (r; r ) at the surfaces of the conductor we set " # g (r; r ) α (r r ) r + a+ (rr ) +, a r<r (4.46) at r " a # g (r; r ) β (r ) (rr ) b + + r r +,r <r b at r b Continuity of g (r; r ) at r r requires that " # " # α (r ) r a+ (r ) + β (r ) (r ) b + + r (4.47) while to satisfy the differential equation, Eq. 4.9, one needs to carry out the procedure in Eq 4.3. (That is one needs to integrate the differential equation, O Φ(r, r ) δ(r r ) from r r to r r + ). The procedure (in Gaussian units) results in the following equation: α (r ( +)a+ r ) + r r + + β (r ( +) ) + b+ r r (4.48) 84

13 Section 4.8 Green s function in a spherical shell The solution to these equations is α (r ) β (r ) (r /b) + + (a/b) + (3-49) (a/r ) + + (a/b) + giving the result Φ (r; r ) G (r; r ) b + R> + R + < a + ( +)(b + a + )(rr ) + X+ m b + R> + R + < a + ( +)(b + a + )(rr ) + R < min(r, r ), R > max(r, r ) Y m X+ m (θ, φ) Y θ,φ, (4.5) Y m (θ, φ) Y θ,φ If a this gives the Green s function for the system confined to lie inside a spherical shell of radius b while if b this is the Green s function for the system confined to lie outside a spherical shell of radius a. Example : Uniformly charged annulus in θ π/ plane As an application for the Green s function of the last section we consider the system consisting of grounded concentric spheres, radii a<b,separated by a uniformly charged annulus (ring) with total charge Q. The problem is to determine the total charges on the spherical surfaces r a and r b. The surface charge density on the ring between the spheres is (4.6) the charge density is σ Q π (b a ), ρ (r) σ δ (z) σ δ (cos θ), a r b r and the boundary conditions are Φ (a, θ, φ) Φ (b, θ, φ). Solution: With the given boundary conditions and the azimuthal symmetry The charge on the inner sphere is ZZZ Φ (r, θ, φ) G (r; r )[ ρ (r )]d 3 r Gaussian units a r b X Z P (cos θ) P () b b πσ (b + a + r + R + > R + < a + ) a (rr ) + dr Q a ZZ ra πσ a b a ZZ σ(r)r dω Z b a ra E r r dω a (b r ) dr πσ a (b a) 85 ZZ Φ (r, θ, φ) r dω ra

14 Section 4.9 Laplace s equation in cylindrical coordinates and on the inner surface of the outer sphere ZZ Φ Q b b (r, θ, φ) r πσ Z b b b a a where we have used the orthogonality of the Legendre polynomials Z dω rb (r a) dr πσ b (b a) P (x) dx δ. (4.5) As a variation on this problem we relax the condition that the two spheres are at the same potential. Instead we require that the inner sphere be neutral and determine the potential of the inner sphere minus the potential of the outer sphere. The result can be readily obtained using the superposition principle. The inner sphere is made neutral by distributing a charge πσ a (b a) uniformly over its surface. The inner surface of the outer sphere will then have a total charge of π b a σ. Applying the superposition principle yields that Φ (a) Φ (b) πσ (b a) b Q (b a) b (b + a) Example : Inner sphere potential is Φ P (cos θ), outer sphere is grounded To demonstrate another application of the Green s function we consider a boundary value problem. Again the system consists of concentric spheres, radii a<b.the boundary conditions are Φ (a, θ, φ) Φ cos θ, Φ (b, θ, φ). The green s function solution to this problem is given by ZZ Φ (r, θ, φ) a Φ cos θ r G (r; r ) dω r a Using the orthogonality of the Legendre polynomials this reduces to Ã Φ (r, θ, φ) a 3 Φ cos θ b 3 r 3 r 3 a 3! r (b 3 a 3 )(rr ) a Φ cos θ b 3 r 3 r (b 3 a 3 ) This result illustrates the general result that the boundary value problems. The boundary conditions are expanded in spherical harmonics and the boundary value problem is solved for each coefficient in the expansion. The solution of the problem is then given by the superposition principle r a 4.9 Laplace s equation in cylindrical coordinates Problems in which the system is confined to cylindrical volumes are best analyzed in cylindrical coordinates. In cylindrical coordinates Laplace s equation is ρ ρ z) ρ ρ Φ (ρ, ϕ, + ρ Φ (ρ, ϕ, z)+ Φ (ρ, ϕ, z). (4.5) ϕ z 86

15 Section 4.9 Laplace s equation in cylindrical coordinates As in the case of spherical coordinates, this equation is solved by a series expansion in terms of products of functions of the individual cylindrical coordinates. That is, we use separation of variables. Φ (ρ, ϕ, z) X α lmn U l (ρ) V m (ϕ) W k (z). (4.53) l,m,n Substituting Φ lmk (ρ, ϕ, z) U l (ρ) V m (ϕ) W k (z).into Eq 4.5 and dividing by Φ lmk (ρ, ϕ, z), we can sequentially separate all the variable.dependence: d U l ρ dρ ρ d dρ U l + d V m ρ dϕ V m d W n dz W k ±k (4.54) ρ d ρ d U l dρ dρ U l ρ k d V m dϕ V m ±m (4.55) These functions satisfy the Eqs. 4.54, 4.55 and 4.56: d W k (z) dz λ z W k (z) ; λ z k (4.54a) W k (z) A k e kz + B k e kz (4.54b) Note that k can be complex, pure imaginary or real. d dϕ V m (ϕ) λ ϕ V m (ϕ) ; λ ϕ m (4.55) V m (ϕ) C m e imϕ + D m e imϕ (4.55b) Note that m could be non-integer and/or complex ρ d dρ U (ρ)+ρ d dρ U (ρ)+ k ρ m U (ρ) (4.56) (kρ) d d(kρ) U d m (kρ)+kρ d(kρ) U m (kρ)+ (kρ) m U m (kρ) U m (kρ) a m J m (kρ)+b m N m (kρ) (4.56b) wherenowthelabelonu m (kρ) becomes m and the J m and.n m are called Bessel functions of the first and second kind, respectively. [Note that one could also use Hankel functions, H m(kρ) ± J m (kρ) ± in m (kρ), rather than J m (kρ) and N m (kρ).] The boundary conditions on the system determine the allowed values of k and m. Typically the system fills the region from ϕ to ϕ π. In this case the continuity of V m (ϕ) and Vm (ϕ) requires that m be an integer. We consider these systems first. The values of k are determined by boundary conditions requiring solutions to be periodic in z ( k must be pure imaginary, k ±i k ) or exponential in z ( k ± k ). 87

16 Section 4.9 Laplace s equation in cylindrical coordinates 4.9. Potential is zero at ρ a, ρ b In the case that the potential on the cylindrical surfaces are zero, Eq determines the allowed values for k. The appropriate solutions, U (ρ), will be the Bessel functions of the first and second kind, J m (kρ) and N m (kρ) (see Abramowitz & Stegun, Chapts. 9-) which can be made equal to zero at.ka and kb. We can obtain the small ρ dependence of the solutions by neglecting the term k ρ in the differential equation. In this case, if m 6, U (ρ) J m (kρ) (kρ/)m Γ (m +) U (ρ) N m (kρ) Γ (p)(kρ/) m /π. (4.57) If m,we find that U (ρ) or U (ρ) (/π)lnρ. For real m the non-singular solutions are Bessel functions of the first kind, J m (kρ), and the singular solutions are Bessel functions of the second kind, N p (kρ). For large values of kρ the asymptotic behavior of these functions is J m (kρ) N m (kρ) r ³ πkρ cos kρ mπ π 4 r ³ πkρ sin kρ mπ π 4, (4.58). Note: Either of these approximations (4.57 and 4.58) can be used to determine the normalization of the functions for the Green s function. Also, the N m (kρ) has already been made linearly independent of J m (kρ) for integer m. The n th root, in order of magnitude, of the equation J m (x) is given by (see Hildebrand, Advance Calculus for Engineers, Prentice-Hall 949, p ) where µ x n nπ + m c 4n k 4c 3k 3 3c 3... (4.59) 5k5 k π (m +4n ), q 4m, (4.59b) c q, c (q ) (7q 3), c 3 (q ) 83q 98q n m m m m 3 m 4 m Some values for x n are It is often useful to have the zeros of Jm (x). The n th root, in order of magnitude, of the equation Jm (x) is given by µ m + x n nπ + c 4n k 4c 3k 3 3c k5 (4.6) where k π (m +4n +), q 4m, (4.6b) c q +3, c 7q +8q 9, c 3 83q q 339q

17 Section 4. Expansion of the Green s function in cylindrical coordinates The reference also gives formulae for the solutions of J m (x) N m (ax) J m (ax) N m (x) (a>) (4.6) Jm (x) Nm (ax) Jm (ax) Nm (x) (a>) If the system is contained in a cylinder of radius a and length L, a boundary value problem has the potential given on the end caps. In this case the solutions are Bessel functions of the first kind which satisfy the condition J m (ka). This requirement determines values of k, k n with n,,...these functions satisfy the orthogonality relationship Z a ρj m (k n ρ) J m (k n ρ) dρ.5 δ n n [aj m+ (k n a)]. (4.6) For a given value of m we can assume that these functions form a complete set of orthogonal functions on the interval ρ a. Finally, if a the orthogonality condition is ρj m (kρ) J m (k ρ) dρ k δ (k k ). (4.63) 4.9. Potential is zero at z z,z z With the potential equal to zero on the end-caps of the cylindrical region Eq provides the eigenvalue problem for λ z. The required solutions would be a superposition of cos ( k z) and sin ( k z) and λ z (ik) k. Theρ dependence is given by the modified Bessel functions, I m ( k ρ) i m J m (i k ρ) and K m ( k ρ) π im+ H + m(i k ρ) The solutions to Bessel s equation would be the associated Bessel functions I m ( k z) and K m ( k z). For k z these functions behave as µ m k z I m ( k z) (4.64) Γ (m +) K m ( k z) Γ (m) µ m, m 6 k z µ k z ln, m and for large kz these functions have the asymptotic form I m (kz) (π k z) / exp ( k z) (4.65) µ / π K m (kz) exp ( k z). k z A general cylindrical problem would expected to involve both types of solutions. 4. Expansion of the Green s function in cylindrical coordinates The following steps illustrate the general procedure for finding a Green s function expansion...in cylindrical coordinates the Green s function satisfies G (r; r )δ (r r ) ρ δ (ρ ρ ) δ (z z ) δ (ϕ ϕ ) (4.66). Note that there are three representations of delta functions, useful for cylindrical problems: δ (z z ) π e ik(z z ) dk π 89 cos [k (z z )] dk,, (4.67a)

18 Section 4. Expansion of the Green s function in cylindrical coordinates δ (ϕ ϕ ) π m e im(ϕ ϕ ) π {+cos[m (ϕ ϕ )]} (4.67b) δ (ρ ρ )ρ kj p (kρ) J p (kρ ) dk. (4.67c) 3.Making use of the orthogonal functions appearing in the delta function expansions, write a general form for G (r; r ) leaving one set of variables (in this case z and z ) in terms of an unknown function, g m (z; z ): G (r; r ) π m e im(ϕ ϕ ) kg m (z; z ) J m (kρ) J m (kρ ) dk. (4.68) 4. Apply to the general form forg (r; r ) and set the result equal to δ (r r ): G (r; r ) [ ρ ρ ρ + ρ ρ ϕ + z ] π kg m (z; z ) J m (kρ) J m (kρ ) dk m e im(ϕ ϕ ) m e im(ϕ ϕ ) kj m (kρ ) J m (kρ) π [g m (z; z ) ρj m (kρ) ρ + m g m (z; z ) ρ + g m (z; z ) z ] ρ J m (kρ) ρ m e im(ϕ ϕ ) kj m (kρ ) J m (kρ) π [(m ρ k + m ρ )g m (z; z )+ g m (z; z ) z ]dk e im(ϕ ϕ ) kj m (kρ ) J m (kρ) π [ k g m (z; z )+ g m (z; z ) z ]dk m δ (ρ ρ ) δ (z z ) δ (ϕ ϕ ) ρ 9

19 Section 4. Expansion of the Green s function in cylindrical coordinates 5. Multiply both sides by R π dϕe im ϕ R Z π π m + g m (z; z ) z ]dk Z π dϕ e i(m m)ϕ e imϕ dϕ e im (ϕ ) Jm (k ρ ) δ (z z ) ρj m (k ρ) dρ and integrate over dϕ and dρ : kj m (kρ )[ dρδ (ρ ρ ) δ (z z ) δ (ϕ ϕ ) e im (ϕ) J m (k ρ) ρj m (kρ) J m (k ρ) dρ][ k g m (z; z ) R π 6. On the left hand side, use the orthogonality condition, π e imϕ e imϕ dϕ δ mm to reduce the sum over m to a single term with m m, and the integral, R ρj m (kρ) J m (k ρ) dρ k δ (k k ) to carry out the integration over k. e im (ϕ ) kj m (kρ ) k δ (k k )[ k g m (z; z )+ g m (z; z ) z ]dk e im (ϕ ) Jm (k ρ ) [ k g m (z; z )+ g m (z; z ) z ]e im (ϕ ) Jm (k ρ ) δ (z z ) 7. Finally, extract the differential equation for, g m (z; z ), which in this case must satisfy The Dirac delta function on the right hand side indicates d dz g m (z; z ) k g m (z; z )δ (z z ). (4.69) that dg m dz has a discontinuity at zz. 8.To determine the overall normalization constant for g m integrate both sides of 4.69 over R z + z dz Z z + z [ d dz g m (z; z ) k g m (z; z )]dz Z z + z δ (z z ).dz Z z + z d dz g m (z; z ) dz Z z + z k g m (z; z ) dz Z z + z δ (z z ).dz Z z + z d dz [ d dz g m (z; z )]dz d dz g m (z; z ) z + z 9. Assume a general form for g m (z; z ) in terms of the functions given in step.and evaluate the limit in step 8. g m (z; z )C m exp(kz < )exp( kz > ) d dz g m (z; z ) z + z C m[exp(kz ) d dz exp( k(z + )) d dz exp(kz)exp( kz ) C m [exp(kz )( k)exp( k(z + )) k exp(k(z )exp( kz ) 9

20 Section 4. Expansion of r r and the Coulomb potential: As, C m [ k exp(kz kz ) C m k. Put the function, g m (z; z ) k exp(kz < kz >) into the expansion for the Greens function: G (r; r ) π m G (r; r ) e im(ϕ ϕ ) m r r z > max(z,z ), z < min(z, z ) k k exp(kz <)exp( kz > )J m (kρ) J m (kρ ) dk. e im(ϕ ϕ ) exp(kz < )exp( kz > )J m (kρ) J m (kρ ) dk. (4.7) 4. Expansion of r r and the Coulomb potential: Thus by comparing G (r; r ) with its compact form, r r, the expansion of r r is obtained r r X e im(ϕ ϕ ) exp [ k (z > z < )] J m (kρ) J m (kρ ) dk (4.7) m and the Coulomb potential (with unit charge) in cylindrical coordinates can be written q r r q P ) R m eim(ϕ ϕ exp [ k (z > z < )]J m (kρ)j m (kρ )dk (4.7b) If we set ρ and z we can obtain the Laplace transform of J (t) from p ρ + z +s e kz J (kρ) dk or (4.7) e st J (t) dt You should refer to Jackson to review the expansion of the Green s function in terms of the modified Bessel functions. 4. Eigenfunction expansions for the Green s function We consider a system in a volume Ω. A property of the system satisfies the differential equation ψ (r)+[f (r)+λ] ψ (r) (4.73) 9

21 Section 4.3 Green s function in orthogonal functions of spherical coordinates which with appropriate boundary conditions on ψ (r) becomes an eigenvalue equation for λ. Let λ n and ψ n (r) be the eigenvalues and corresponding eigenfunctions. The {ψ n (r)} can be assumed to provide a complete set of functions for describing the property of the system and, since the ψ n satisfy an eigenvalue problem in a finite volume, they form an orthogonal set and Z ψ n (r) ψ m (r) d 3 r δ mn. (4.74) Completeness implies that for r, r in Ω. We now use the functions {ψ n (r)} to expand the Green s function solution to Ω X ψ n (r) ψ n (r )δ(r r ) (4.75) n G (r; r )+[f (r)+λ] G (r; r ) δ (r r ) (4.76) X ψ n (r) ψ n (r ) n The solution is In the case that λ we have G (r; r ) X n G (r; r ) X n ψ n (r) ψ n (r ) λ λ n ψ n (r) ψ n (r ) λ n (4.77a) (4.77b) This is a common form for the Green s function. 4.3 Green s function in orthogonal functions of spherical coordinates We consider an infinite system, the eigenfunctions are taken to be zero on a sphere of radius R and then R istakentoinfinity. The eigenvalue problem is from which we immediately extract the angular dependence obtaining The radial dependence satisfies ψ (r) ψ (r)+k ψ (r) (4.78) lx l m l R l (r) Y m l (θ, φ). (4.79) r d dr R l (r)+r d dr R l (r) l (l +)R l (r)+k r R l (r). (4.8) For small r the solutions behave as r l and r l. Requiring that the solution is regular at the origin we pick the solution which varies as r l for small r. These solutions are the spherical Bessel functions. r π j l (kr) kr J l+ (kr) (4.8) 93

22 Section 4.4 Simple application: point charge at origin The first few spherical Bessel functions are l l j l (z) z sin z z sin z z cos z 3z 3 z sin z 3z cos z The eigenfunctions satisfy the orthogonality condition ( Cohen-Tannoudji, Du, and Laloë, Qunatum Mechanics, Vol. II, p. 95) ZZ r j l (kr) j l (k r) dr Yl m (θ, φ) Yl m (θ, φ) dω π kk δ (k k ) δ ll δ mm. (4.8) The (assumed) completeness relationship is given by lx k j l (kr) j l (kr ) dk Yl m (θ, φ) Yl θ,φ π δ (r r ) (4.83) m l For reference a plane wave can be written as exp (ik r) X lm i l j l (kr) Yl m (θ r,φ r ) Yl m (θ k,φ k ) (4.84) Assuming that the Green s function has the form G (r, r ) X lm we find that g l (k) 8k. It follows that k g l (k) j l (kr) j l (kr ) Y m l (θ, φ) Yl θ,φ (4.85) r r 8X lm j l (kr) j l (kr ) dk Y m l (θ, φ) Yl θ,φ (4.86) This can be compared to the expansion in terms of plane waves r r ZZZ exp [ik (r r )] π k d 3 k (4.87) Comparing Eq.?? with Eq.?? (taking a and b ) we find that j l (kr) j l (kr π ) dk (l +) r< l r> l+ (4.88) 4.4 Simple application: point charge at origin To illustrate the application of this Green s function expansion we consider a point charge q locatedonthezaxisatz a. The potential will be evaluated for a. The charge density for this system is ρ (r) q a δ (r a) δ cos θ δ φ φ. (4.89) 94

23 Section 4.5 Multipole expansion of charge distributions In terms of the Green s function, expanded in the spherical Bessel functions and spherical harmonics, the potential is Φ (r) 4q ZZZ µ r dr d cos θ dφ δ (r a) δ cos θ (4.9) π a X j l (kr) j l (kr ) dk Yl m (θ, φ) Y l θ,φ lm The spatial integration over the delta functions can be performed to obtain Φ (r) q π (l +)P l (cos θ) l Since j l () for l 6 and j (), if a Φ (r) q π j (kr) dk q πr j l (kr) j l (ka) dk (4.9) sin (kr) dk q k r It is obvious that the expansion of the Green s function for Poisson s equation in terms of spherical Bessel function is not as convenient as the expansion in terms of powers of r. Expansions of Green s functions in terms of orthogonal functions is more commonly used when differential equations are converted into integral equations. In that case the solutions are often expanded in a set of orthogonal functions automatically leading to the expansion of the Green s function in orthogonal functions. (4.9) 4.5 Multipole expansion of charge distributions In the case that the charge is restricted to a finite volume, say a sphere of radius R, then the potential outside of this volume is obtained using the form of the Green s function given in Eq.?? Φ (r) lx l m l l + Yl m ZZZ (θ, φ) r l+ ρ (r ) r l Yl θ,φ d 3 r (4.93) The integral in this expression defines the multipole moments of the charge distribution because of the relationship Y m l (θ, φ) ( ) m Yl m (θ, φ) it follows that q l m ( ) m qlm. In terms of the multipole moments of the charge distribution the potential for r>ris Φ (r) lx l m l i q lm l + Y m l (θ, φ) r l+. (4.95) The set of elements {q lm ; m l,..., l,l} form a spherical tensor of rank l. There are two common representations of tensors, the familiar rectilinear representation and the spherical representation. For comparison we consider the charge dipole moment. In rectilinear representation 3X ZZZ p p i e i r ρ (r) d 3 r p i ZZZ or {p i ; i,, 3}. In the spherical representation we find that r i ρ (r) d 3 r q ± q r 3 8π [±p ip ] (4.96) r 3 p 3. 95

24 Section 4.6 Multipole expansion of the energy of a charge distribution in an external field 4.6 Multipole expansion of the energy of a charge distribution in an external field Consider a charge distribution contained in a volume of radius R. Let there be a potential Φ (r) in this region due to charges external to the volume. The potential energy of the charge distribution is ZZZ W ρ (r) Φ (r) d 3 r (4.97) For convenience we can chose the origin of the coordinate system to be at the middle of the charge distribution. Since the sources of the potential are external to the volume containing the charge, the electric potential can be expanded in a Taylor series. 3X Φ (r ) Φ (r) Φ () + r i ri (4.98) + The potential energy has the expansion ZZZ W Φ () " + 3X i, j 3X i, j i r i r j " Φ (r ) r i r j r # ZZZ ρ (r) d 3 r E () # Φ (r ) r i r j r ZZZ r +... rρ (r) d 3 r (4.99) r i r j ρ (r) d 3 r +... The first term is a kind of average potential energy, the second is the interaction of the electric dipole distribution with the applied electric field, and the third term gives the interaction of the quadrupole moment of the charge distribution with the gradient of the electric field. The quadrupole moment of the charge distribution, which transforms as a second rank tensor under rotations, is Q ij ZZZ 3ri r j δ ij r ρ (r) d 3 r. (4.) The additional diagonal term, which is invariant under rotations, does not contribute to the potential energy. Since the quadrupole moment of the charge distribution is a symmetric, traceless second rank tensor. It is therefore specified by five parameters. These parameters for this rectilinear tensor are related to the five parameters of the spherical tensor q m, The simplest relationship occurs for q and Q 33 This simplicity is obtained because the spherical tensors we are using have taken the z or 3 axis as a preferred axis. 4.7 Boundary value problems, conclusions We have developed the solutions of Laplace s and Poisson s equations in rectilinear, cylindrical, and spherical coordinates. Most problems which we encounter are most readily solved in one of these coordinate systems. 3 i, j δ ij Φ (r) r i r j r Φ (r) r 96

25 Section 4.6 Multipole expansion of the energy of a charge distribution in an external field The solutions of Laplace s equation were obtained by taking a superposition of potentials. Each solution matched the given boundary conditions on a different segment of the bounding surface and was zero on the rest of the surfaces. The solutions to Poisson s equation, and to Laplace s equation, were obtained in terms of Green s functions. We found that given the Green s function for a given boundary geometry the solution to a potential problem with this boundary was reduced to integration. A number of Green s functions, in various forms, for rectangular, cylindrical, and spherical volumes were obtained. Finally, we have found that the potential generated by a charge distribution is a superposition of potentials due to the multipole moments of the charge distribution. 97