Solution to Laplace's Equation in Cylindrical Coordinates 1 ...
Solution to Laplace's Equation in Cylindrical Coordinates 1 ...
Solution to Laplace's Equation in Cylindrical Coordinates 1 ...
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z<br />
b<br />
c<br />
a<br />
V = f( ρ )<br />
V = 0<br />
y<br />
x<br />
Figure 5: The geometry for the problem of two concentric cyl<strong>in</strong>ders<br />
V = ∑ νn<br />
A νn J ν (k n ρ) s<strong>in</strong>(νφ) s<strong>in</strong>h(k n z)<br />
Here we have discarded solutions <strong>in</strong> N νn (kρ) which are <strong>in</strong>f<strong>in</strong>ite at the orig<strong>in</strong>. To match the<br />
boundry at z = ±L we need <strong>to</strong> have a term s<strong>in</strong>(νφ) which requires ν = 1. Then we require<br />
that the Bessel function, J 1n (k n a) = 0 which determ<strong>in</strong>es the zeros of the Bessel function of<br />
order 1. We write these as α 1n so that k n = α 1n /a. The solution then has the form;<br />
V = ∑ n<br />
A n J 1 (α 1n ρ/a) s<strong>in</strong>h(αz/a) s<strong>in</strong>(φ)<br />
F<strong>in</strong>ally we match the boundry condition at z = ±L where V = V 0 s<strong>in</strong>(φ). Use orthorgonality<br />
<strong>to</strong> obta<strong>in</strong>;<br />
(L/2)[J 1 (α 1n )] 2 A n = 1<br />
s<strong>in</strong>h(αL/a)<br />
∫ a<br />
0<br />
ρdρ J 1 (α 1n ρ/a) V 0<br />
As another example we look at a solution for concentric cyl<strong>in</strong>ders with the boundry<br />
conditions;<br />
r = a, c and z = 0 V = 0<br />
z = b V = f(ρ)<br />
This geometry is shown <strong>in</strong> Figure 5. We choose a solution <strong>to</strong> have the form;<br />
∑<br />
V = ∞ A n s<strong>in</strong>h(k n z) G 0 (k n ρ)<br />
n<br />
Here we have written a superposition of the Bessel and Neumann functions;<br />
6