10—Partial Differential Equations 311I picked the axis forθ= 0pointing toward the split between the cylinders. No particular reason, but you have to
make a choice. I make the approximation that the cylinder is infinitely long so thatzdependence doesn’t enter.
Also, the two halves of the cylinder almost touch so I’m neglecting the distance between them.
(b) What is the electric field,−∇V on the central axis? Is this answer more or less what you would estimate
before solving the problem? Ans: (b)E= 4V 0 /πR.
10.17 Solve the preceding problemoutsidethe cylinder. The integerncan be either positive or negative, and this
time you’ll need the negative values. (Andwhymustnbe an integer?) Ans:(4V 0 /π)
∑
nodd(1/n)(R/r)nsinnθ10.18 In the split cylinder of problem 16 , insert a coaxial wire of radiusR 0 < R. It is at zero potential. Now
what is the potential in the domain∑ R 0 < r < R? You will needboth the positive and negativenvalues,
(Anrn+Bnr−n) sinnθ
10.19 Fill in the missing steps in deriving Eq. ( 45 ).
10.20 Analyze how rapidly the solution Eq. ( 45 ) approaches a constant asz increases from zero. Compare
Eq. ( 41 ).
10.21 A broad class of second order linear homogeneous differential equations can, with some manipulation, be
put into the form (Sturm-Liouville)
(p(x)u′)′+q(x)u=λw(x)u
Assume that the functionsp,q, andware real, and use manipulations much like those that led to the identity
Eq. (5.12). Derive the analogous identity for this new differential equation. When you use separation of variables
on equations involving the Laplacian you will typically come to an ordinary differential equation of exactly this
form. The precise details will depend on the coordinate system you are using as well as other aspects of the PDE.
10.22 Carry on from Eq. ( 28 ) and deduce the separated solution that satisfies these boundary condition. Show
that it is equivalent to Eq. ( 29 ).
10.23 The Laplacian in cylindrical coordinates is in problem 15. Separate variables for the equation∇^2 V = 0
and you will see that the equations inzandθare familiar. The equation in thervariable is less so, but you’ve
seen it (almost) in Eqs. (4.16) and (4.17). Make a change of variables in ther-differential equation,r=kr′,
and turn it into exactly the form described there.