Hassani S. Mathematical Methods: For Students of Physics and Related Fields

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26.8 Problems 637

26.19. Suppose that the sphere of Example 26.7.3 is held at potential V

(a) Find the potential Φ(r, θ) and the electrostatic ﬁeld at all points in space.

(b) Calculate the surface charge density on the sphere.

entire sphere.

26.20. Using the inﬁnite series expansion, ﬁnd the electrostatic potential

both inside and outside a conducting sphere of radius a held at the constant

potential V

26.21. Find the electrostatic potential inside a sphere of radius a with an

insulating small gap at the equator if the bottom hemisphere is grounded and

the top hemisphere is maintained at a constant potential V

26.22. A sphere of radius a is maintained at a temperature T

. The sphere is

inside a large heat-conducting mass. Find the expressions for the steady-state

temperature distribution both inside and outside the sphere.

26.23. A ring of total charge q and radius a in the xy-plane with its center

at the origin constitutes an azimuthally symmetric charge distribution whose

potential is also azimuthally symmetric.

(a) Write the most general potential function valid for r>a.

(b) By direct integration show that

Φ(r, θ =0)=

4π

dq(r



)

|r − r





θ=0

4π

√

+ a

the series in (a) to ﬁnd the coeﬃcients of Legendre expansion and show that

Φ(r, θ)=

4π

∞



k=0

(−1)

(2k)!

(k!)





(cos θ).

(d) Find a similar expression for Φ(r, θ)forr<a.

26.24. A conducting sphere of radius a is inside another conducting sphere

of radius b. The inner sphere is held at potential V

; the outer sphere at

. Find the potential inside the inner sphere, between the two spheres, and

outside the outer sphere.

26.25. A conducting sphere of radius a is inside another conducting sphere

of radius b which is composed of two hemispheres with an inﬁnitesimal gap

between them. The inner sphere is held at potential V

. The upper half of the

outer sphere is at potential +V

and its lower half at −V

. Find the potential

inside the inner sphere, between the two spheres, and outside the outer sphere.

26.26. A heat conducting sphere of radius a is composed of two hemispheres

with an inﬁnitesimal gap between them. The upper and lower halves of the

sphere are in contact with heat baths of temperatures +T

and −T

, respec-

tively. The sphere is inside a second heat conducting sphere of radius b held

at temperature T

. Find the temperature inside the inner sphere, between

the two spheres, and outside the outer sphere.

Chapter 27

Laplace’s Equation:

Cylindrical Coordinates

Before working speciﬁc examples of cylindrical geometry, let us consider a

question that has more general implications. We saw in Chapter 22 that

separation of variables led to ODEs in which certain constants appeared,

and that diﬀerent choices of signs for these constants can lead to a diﬀerent

functional form of the general solution. For example, an equation such as

x/dt

− kx = 0 can have exponential solutions if k>0 and trigonometric

solutions if k<0. One cannot a priori assign a speciﬁc sign to k.Thus,the

general form of the solution is indeterminate. However, once the boundary

conditions are imposed, the unique solutions will emerge regardless of the

initial functional form of the solutions. The following argument illustrates this

point on the angular DE resulting from the separation of Laplace’s equation

in cylindrical coordinates.

27.1 The ODEs

The separation of variables Φ(ρ, ϕ, z)=R(ρ)S(ϕ)Z(z) for Laplace’s equation

∇

Φ = 0 yields the following three ODEs [see Equation (22.14) noting that

λ = 0]. In what follows, we shall use λ for λ

dρ



dρ





λρ +



R =0,

dϕ

− μS =0,

− λZ =0. (27.1)

640 Laplace’s Equation: Cylindrical Coordinates

Let us concentrate on the second equation whose most general solution we

can write as

The dependence

of the solution on

ϕ is dictated by

physical

conditions.

S(ϕ)=

√

μϕ

+ Be

−

√

μϕ

if μ =0,

Cϕ + D if μ =0.

(27.2)

No matter what type of boundary conditions are imposed on the potential Φ,

it must give the same value at ϕ and at ϕ +2π while keeping the other two

variables ﬁxed.

This is because (ρ, ϕ, z)and(ρ, ϕ +2π, z)representthesame

physical point in space. It follows that

R(ρ)S(ϕ)Z(z)=R(ρ)S(ϕ +2π)Z(z) ⇒ S(ϕ +2π)=S(ϕ)

because the identity holds for all values of ρ and z. If the last relation is to

be true for the case of μ =0,wemusthaveC =0andS(ϕ)=D.Forμ =0,

Equation (27.2) yields

√

μ (ϕ+2π)

+ Be

−

√

μ (ϕ+2π)

= Ae

√

μϕ

+ Be

−

√

μϕ

√

μϕ

√

μ 2π

− 1) + Be

−

√

μϕ

−

√

μ 2π

− 1) = 0.

This must hold for all ϕ. The only way that can happen (we want to keep A

and B nonzero) is to have

√

μ 2π

− 1=0 and e

−

√

μ 2π

− 1=0

both of which are equivalent to e

√

μ 2π

=1.

If we conﬁne ourselves to real

μ, we get only trivial solutions. To avoid this, we have to have

√

μ = im for

m =0, ±1, ±2,... or μ = −m

for m =0, ±1, ±2,.... With this choice of μ,

the DE for S(ϕ) becomes S



+ m

S = 0 whose general solution is a sum of

trigonometric functions. We summarize this ﬁnding:

Theorem 27.1.1. For all physical problems for which the azimuthal angle

varies between 0 and 2π, one is forced to restrict the value of μ to the negative

of the square of an integer. The solution for the angular part then becomes

S(ϕ)=A

cos mϕ + B

sin mϕ, m =0, 1, 2,..., (27.3)

where A

and B

are constants that may diﬀer for diﬀerent m’s.

The negative values of m will not give rise to any new solutions, so they

are not included in the range of m.Thecaseofμ = 0 need not be treated

separately, because the acceptable solution for this case is S = D =const.,

which is what is obtained in (27.3) when m =0.

This argument is valid only for physical situations deﬁned for the entire range of ϕ.

If the region of interest restricts ϕ to a subset of the interval [0, 2π], the argument breaks

down.

The second equation can be obtained by multiplying the ﬁrst equation by e

−

√

μ 2π

27.1 The ODEs 641

The DE for Z(z) is independent of m and has an exponential solution

if λ>0 and a trigonometric solution if λ<0. Assuming the former, and

writing λ ≡ l

,wehave

Z(z)=Ae

+ Be

−lz

. (27.4)

Least familiar is the radial DE which, in terms of l =

√

λ, can be rewritten

dρ



−



R =0. (27.5)

Furthermore, if we deﬁne the variable v = lρ, we can cast (27.5) in the form

Bessel diﬀerential

equation



1 −



R =0. (27.6)

Equation (27.5), or (27.6), is one of the most famous DEs of mathematical

physics called the Bessel diﬀerential equation. Our task for the remainder

of this chapter is to ﬁnd solutions of this DE and list some of their properties

and examples of their usage.

Historical Notes

Friedrich Wilhelm Bessel showed no signs of unusual academic ability in school,

although he did show a liking for mathematics and physics. He left school intending

to become a merchant’s apprentice, a desire that soon materialized with a seven-year

unpaid apprenticeship with a large mercantile ﬁrm in Bremen. The young Bessel

proved so adept at accounting and calculation that he was granted a small salary,

with raises, after only the ﬁrst year. An interest in foreign trade led Bessel to study

geography and languages at night, astonishingly learning to read and write English

in only three months. He also studied navigation in order to qualify as a cargo oﬃcer

aboard ship, but his innate curiosity soon compelled him to investigate astronomy

at a more fundamental level. Still serving his apprenticeship, Bessel learned to

observe the positions of stars with suﬃcient accuracy to determine the longitude

of Bremen, checking his results against professional astronomical journals. He then

tackled the more formidable problem of determining the orbit of Halley’s comet

from published observations. After seeing the close agreement between Bessel’s

calculations and those of Halley, the German astronomer Olbers encouraged Bessel

to improve his already impressive work with more observations. The improved

calculations, an achievement tantamount to a modern doctoral dissertation, were

published with Olbers’s recommendation. Bessel later received appointments with

increasing authority at observatories near Bremen and in K¨onigsberg, the latter

position being accompanied by a professorship. (The title of doctor, required for the

professorship, was granted by the University of G¨ottingen on the recommendation

of Gauss.)

Friedrich Wilhelm

Bessel 1784–1846

Bessel proved himself an excellent observational astronomer. His careful mea-

surements coupled with his mathematical aptitude allowed him to produce accurate

positions for a number of previously mapped stars, taking account of instrumental

eﬀects, atmospheric refraction, and the position and motion of the observation site.

In 1820 he determined the position of the vernal equinox accurate to 0.01 second, in

agreement with modern values. His observation of the variation of the proper motion

642 Laplace’s Equation: Cylindrical Coordinates

of the stars Sirius and Procyon led him to posit the existence of nearby, large, low-

luminosity stars called dark companions. Between 1821 and 1833 he catalogued the

positions of about 75,000 stars, publishing his measurements in detail. One of his

most important contributions to astronomy was the determination of the distance

to a star using parallax. This method uses triangulation, or the determination of

the apparent positions of a distant object viewed from two points a known distance

apart, in this case two diametrically opposed points of the Earth’s orbit. The angle

subtended by the baseline of the Earth’s orbit, viewed from the star’s perspective,

is known as the star’s parallax. Before Bessel’s measurement, stars were assumed

to be so distant that their parallaxes were too small to measure, and it was further

assumed that bright stars (thought to be nearer) would have the largest parallax.

Bessel correctly reasoned that stars with large proper motions were more likely to

be nearby ones and selected such a star, 61 Cygni, for his historic measurement. His

measured parallax for that star diﬀers by less than 8% from the currently accepted

value.

Given such an impressive record in astronomy, it seems only ﬁtting that the

famous functions that bear Bessel’s name grew out of his investigations of pertur-

bations in planetary systems. He showed that such perturbations could be divided

into two eﬀects and treated separately: the obvious direct attraction due to the

perturbing planet and an indirect eﬀect caused by the Sun’s response to the per-

turber’s force. The so-called Bessel functions then appear as coeﬃcients in the series

treatment of the indirect perturbation. Although special cases of Bessel functions

were discovered by Bernoulli, Euler,andLagrange the systematic treatment by Bessel

clearly established his preeminence, a ﬁtting tribute to the creator of the most fa-

mous functions in mathematical physics.

27.2 Solutions of the Bessel DE

The Frobenius method is an eﬀective way of ﬁnding solutions for ODEs. We

rewrite (27.6) by multiplying it by v

to turn all its coeﬃcients into polyno-

mials as suggested by Equation (26.7). This yields

+ v

− m

R =0. (27.7)

Since v

vanishes at v = 0, we must assume a solution of the form

R(v)=v

∞



k=0

∞



k=0

k+s

from which we obtain

∞



k=0

(k + s)v

k+s

∞



k=0

(k + s)(k + s − 1)v

k+s

27.2 Solutions of the Bessel DE 643

Substituting these as well as (v

− m

)



∞

k=0

k+s

in the DE yields

∞



k=0

[k + s +(k + s)(k + s − 1)



 

=(k+s)

−m

k+s

∞



k=0

k+s+2

=0.

To ﬁnd the recursion relation, we need to have the same power of v in the

sum. We do this by rewriting the ﬁrst sum as

− m

+ c

[(s +1)

− m

s+1

∞



k=2

[(k + s)

− m

k+s

= c

− m

+ c

[(s +1)

− m

s+1

∞



n=0

n+2

[(n +2+s)

− m

n+2+s

where in the second line, we introduced n = k −2. Since n is a dummy index,

we can change it back to k. It then follows that

− m

+ c

[(s +1)

− m

s+1

∞



k=0

k+2

[(k +2+s)

− m

]+c

k+2+s

=0.

Assuming that c

= 0 and setting the coeﬃcients of all powers of v equal to

zero, we get

= m

[(s +1)

− m

]=0,

k+2

[(k +2+s)

− m

]+c

=0.

The ﬁrst equation gives m = ±s. Inserting this in the second equation gives

(2s +1)=0 ⇒ c

=0 or s = −

The choice s = −

gives m = ∓

which is not acceptable,

as we decided

that m is to be a positive integer. We therefore conclude that s = ±m and

= 0. It follows from the recursion relation that all odd c’s are zero. The

Frobenius series will therefore look like

R(v)=v

±m

∞



k=0

2k+2

= −

(2k +2+s)

− m

. (27.8)

The ratio test for the convergence of series yields

lim

k→∞



2k+2



= lim

k→∞



(2k +2+s)

− m



=0,

Actually, problems arising from other areas of physics beyond electrostatics and steady-

state heat transfer allow noninteger values of m. However, we shall not deal with such

problems here.

644 Laplace’s Equation: Cylindrical Coordinates

which indicates that

Box 27.2.1. The series of Equation (27.8) is convergent for all values

of v.

We now use the recursion relation to obtain the coeﬃcients of expansion.

Rewrite the recursion relation as

recursion relation

for Bessel

equation

k+2

= −

(k +2+s)

− s

= −

(k + 2)(2s + k +2)

where we substituted s

for m

.Thisgives

= −

2(2s +2)

= −

4(2s +4)

=(−1)

4(2s +4)

2(2s +2)

= −

6(2s +6)

=(−1)

6(2s +6)

4(2s +4)

2(2s +2)

and, in general,

=(−1)

2k · (2k −2) ...2



 

(2s +2k)[2s +(2k − 2)] ...(2s +2)



 

(s+k)(s+k−1)...(s+1)

Multiplying the numerator and denominator by s!, we obtain

=(−1)

k!(s + k)!

. (27.9)

Substituting (27.9) in (27.8) yields

R(v)=c

s!v

∞



k=0

(−1)

k!(s + k)!

= c

s!2





∞



k=0

(−1)

k!(s + k)!





where we substituted s for ±m in the exponent of v outside the summation.

We also absorbed the powers of 2 in the denominator of the sum into the

powers of v, and outside the sum, we multiplied and divided by 2

.Itis

customary to choose the arbitrary constant c

to be equal to 1/(s!2

). This

leads to

Box 27.2.2. The Bessel function of order s is denoted by J

and is

given by the series

(x)=





∞



k=0

(−1)

k!(s + k)!





(27.10)

which is convergent for all values of x.

27.3 Second Solution of the Bessel DE 645

Although Equation (27.10) was derived assuming that m—and therefore Equation (27.10)

is valid not only

for integer s, but

also for real and

even complex s.

s—was an integer, lifting this restriction will still yield a series which is con-

vergent everywhere, and one can deﬁne Bessel functions whose orders are

real or even complex numbers. The only diﬃculty is to correctly interpret

(s + n)! for non-integer s. But this is precisely what the gamma function was

invented for (see Deﬁnition 11.1.1). Thus, we let Equation (27.10) stand for

Bessel functions of all orders.

27.3 Second Solution of the Bessel DE

As in the case of Legendre polynomials, we can obtain a second solution of

the Bessel DE using Equation (24.6). For the Bessel DE, we have p(x)=1/x.

Using J

(x) as our input, we can generate another solution. With C =0in

Equation (24.6), we obtain

(x)=KJ

(x)

(u)

exp



−



du = A

(x)

(u)

where A

≡ Kc and α are arbitrary constants determined by convention.

Note that, contrary to J

(x), Z

(x)isnotwellbehavedatx = 0 due to the

presence of u in the denominator of the integrand.

Although the above procedure manufactures a second solution for the

Bessel DE, it is not the customary procedure. It turns out that for non-

integer s, the Bessel function J

−s

(x) is independent of J

(x)andcanbeused

as a second solution.

However, a more common second solution is the linear

combination

(x)=

(x)cossπ − J

−s

(x)

sin sπ

(27.11)

called the Bessel function of the second kind, or the Neumann function.

For integer s the function is indeterminate because of Equation (11.32) and

the identity cos nπ =(−1)

. Therefore, we use l’Hˆopital’s rule and deﬁne Bessel function of

the second kind or

Neumann function

(x) ≡ lim

s→n

(x) = lim

s→n

∂

∂s

(x)cossπ − J

−s

(x)]

π cos nπ

lim

s→n



∂J

∂s

− (−1)

∂J

−s

∂s



From (27.10) we obtain

∂J

∂s

= J

(x)ln





−





∞



k=0

(−1)

Ψ(s + k +1)

k!Γ(s + k +1)





where

Ψ(x) ≡

ln[(x −1)!] =

ln Γ(x)=

dΓ(x)/dx

Γ(x)

See Hassani, S. Mathematical Physics: A Modern Introduction to Its Foundations,

Springer-Verlag, 1999, Chapter 14 for details.

646 Laplace’s Equation: Cylindrical Coordinates

Similarly,

∂J

−s

∂s

= −J

−s

(x)ln









−s

∞



k=0

(−1)

Ψ(−s + k +1)

k!Γ(−s + k +1)





Substituting these expressions in the deﬁnition of Y

(x) and using J

−n

(x)=

(−1)

(x) [Equation (11.32)], we obtain

(x)=

(x)ln





−





∞



k=0

(−1)

Ψ(n + k +1)

k!Γ(n + k +1)





−

(−1)





−n

∞



k=0

(−1)

Ψ(k −n +1)

k!Γ(k − n +1)





. (27.12)

It should be clear from (27.12) that the Neumann function Y

(x) is ill deﬁned

at x = 0, as expected of the second solution of the Bessel DE such as Z

(x)

discussed above.

Since Y

(x) is linearly independent of J

(x) for any s, integer or noninteger,

it is convenient to consider {J

(x),Y

(x)} as a basis of solutions for the Bessel

DE. In particular, the solution of the radial equation in cylindrical coordinates,

i.e., the ﬁrst equation in (27.1), becomes

R(ρ)=AJ

(v)+BY

(v)=AJ

(lρ)+BY

(lρ). (27.13)

27.4 Properties of the Bessel Functions

We have already considered some properties of the Bessel functions in Chap-

ter 11. In this subsection, we quote those results and obtain other useful

properties of the Bessel functions.

27.4.1 Negative Integer Order

Equation (11.32) gives a relation between a Bessel function of integer order

and the Bessel function whose order is negative of the ﬁrst one

−m

(x)=(−1)

(x). (27.14)

27.4.2 Recurrence Relations

A number of recurrence relations involving Bessel functions of integer orders

and their derivatives were derived in Chapter 11 which we reproduce here.

The ﬁrst one, involving no derivatives is

m−1

(x)+J

m+1

(x)=

(x). (27.15)

27.4 Properties of the Bessel Functions 647

The second one, which includes derivatives of Bessel functions, is

m−1

(x) −J

m+1

(x)=2J



(x). (27.16)

Combining these two equations, one obtains

recurrence

relations involving

derivatives

m−1

(x)=

(x)+J



(x),

m+1

(x)=

(x) −J



(x). (27.17)

We can use these equations to obtain new—and more useful—relations.

For example, by diﬀerentiating x

(x), we get

(x)]



= mx

m−1

(x)+x



(x)

= x

(x)+J



(x)



 

m−1

(x) by (27.17)

= x

m−1

(x).

Integrating (really, antidiﬀerentiating) this equation yields

m−1

(x) dx = x

(x). (27.18)

Similarly, the reader may check that

−m

m+1

(x) dx = −x

−m

(x). (27.19)

27.4.3 Orthogonality

Bessel functions satisfy an orthogonality relation similar to that of the Leg-

endre polynomials. However, unlike Legendre polynomials, the quantity that

determines the orthogonality of diﬀerent Bessel functions is not the order but

a parameter in their argument (also see Example 24.5.3).

Consider two solutions of the Bessel DE corresponding to the same az-

imuthal parameter, but with diﬀerent radial parameter. More speciﬁcally, let

f(ρ)=J

(kρ)andg(ρ)=J

(lρ). Then

dρ



−



f =0,

dρ



−



g =0.

The reader may check that by multiplying the ﬁrst equation by ρg and the

second equation by ρf and subtracting, one gets

dρ

[ρ(fg



− gf



)] = (k

− l

)ρfg,