Suppose we have an infinite line of charge such that the linear charge distri- bution is given by A(x) = where a is a constant with dimension of length. We can determine the electrostatic potential some perpendicular distance z above the line to be (in SI units) -2² /a² e 2Q V(2) = 4megaz o VI+/dz , VI+ x²/z²" and of course we would like to evaluate the integral. (a) (paper) Make the change of variables x = aq, so that you get e-q? 2Q V (2) = (1) VI+ a²q²/z²° and then with y = z/(/2a), show that this can be written as V(2) = -ev² Ko(y³) , 4t€0a where Ko is a modifed Bessel function. Do this by looking up integral forms of the Bessel function and compare them to Eg. (1).
Suppose we have an infinite line of charge such that the linear charge distri- bution is given by A(x) = where a is a constant with dimension of length. We can determine the electrostatic potential some perpendicular distance z above the line to be (in SI units) -2² /a² e 2Q V(2) = 4megaz o VI+/dz , VI+ x²/z²" and of course we would like to evaluate the integral. (a) (paper) Make the change of variables x = aq, so that you get e-q? 2Q V (2) = (1) VI+ a²q²/z²° and then with y = z/(/2a), show that this can be written as V(2) = -ev² Ko(y³) , 4t€0a where Ko is a modifed Bessel function. Do this by looking up integral forms of the Bessel function and compare them to Eg. (1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Suppose we have an infinite line of charge such that the linear charge distri-
bution is given by
-a² /a²
X(x) :
a
where a is a constant with dimension of length. We can determine the electrostatic potential
some perpendicular distance z above the line to be (in SI units)
20
V(2) =
e-a²/a?
xp:
V1+ x²/z²
4T€oaz
and of course we would like to evaluate the integral.
(a) (paper) Make the change of variables x = aq, so that you get
e-q?
2Q
V (2) =
roo
(1)
V1 +a²q² /zzdq
and then with y = z/(v2a), show that this can be written as
Q
V(z) =
-e²Ko(y²),
4περα
where Ko is a modifed Bessel function. Do this by looking up integral forms of the Bessel
function and compare them to Eq. (1).
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