f(r) = f(r) f(r) where |f(r) = − ▼ | Απ and V fir) d³r |r-r'| f₁(r) = Vx [ — ƒ d³r' - Απ V' X f(r') ] r-r Consider f(k), the Fourier transform of f(r): 1 +00 f(r) d³k eik r f(k) (2x)-00 Using the results above, find expressions for the Fourier transforms of the curlfree (longitudinal) and divergenceless (transverse) parts of f(r). Your results should tell you why these parts have the names that they do! It may help to know that the Fourier transform of 1/|r-r'| is (47/k²)e-ik·r', and that the Fourier transform of the Dirac delta function ô(r-r') is e-ik·r'
f(r) = f(r) f(r) where |f(r) = − ▼ | Απ and V fir) d³r |r-r'| f₁(r) = Vx [ — ƒ d³r' - Απ V' X f(r') ] r-r Consider f(k), the Fourier transform of f(r): 1 +00 f(r) d³k eik r f(k) (2x)-00 Using the results above, find expressions for the Fourier transforms of the curlfree (longitudinal) and divergenceless (transverse) parts of f(r). Your results should tell you why these parts have the names that they do! It may help to know that the Fourier transform of 1/|r-r'| is (47/k²)e-ik·r', and that the Fourier transform of the Dirac delta function ô(r-r') is e-ik·r'
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Katz, Debora M.
Chapter39: Relativity
Section: Chapter Questions
Problem 59PQ
Related questions
Question
![f(r) = f(r) f(r)
where
|f(r) = − ▼ |
Απ
and
V fir)
d³r
|r-r'|
f₁(r) = Vx [ — ƒ d³r' -
Απ
V' X f(r')
]
r-r
Consider f(k), the Fourier transform of f(r):
1
+00
f(r)
d³k eik r f(k)
(2x)-00
Using the results above, find expressions for the Fourier transforms of the curlfree (longitudinal) and divergenceless (transverse) parts of f(r). Your results should tell you why these
parts have the names that they do!
It may help to know that the Fourier transform of 1/|r-r'| is (47/k²)e-ik·r', and that the Fourier transform of the Dirac delta function ô(r-r') is e-ik·r'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97224513-2bed-410c-8663-4b491c74af5a%2F62b72269-ee30-4da3-8604-168915cfa658%2Fnwh7pfz_processed.png&w=3840&q=75)
Transcribed Image Text:f(r) = f(r) f(r)
where
|f(r) = − ▼ |
Απ
and
V fir)
d³r
|r-r'|
f₁(r) = Vx [ — ƒ d³r' -
Απ
V' X f(r')
]
r-r
Consider f(k), the Fourier transform of f(r):
1
+00
f(r)
d³k eik r f(k)
(2x)-00
Using the results above, find expressions for the Fourier transforms of the curlfree (longitudinal) and divergenceless (transverse) parts of f(r). Your results should tell you why these
parts have the names that they do!
It may help to know that the Fourier transform of 1/|r-r'| is (47/k²)e-ik·r', and that the Fourier transform of the Dirac delta function ô(r-r') is e-ik·r'
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