8/11/2019 Jackson 1 2 Homework Solution

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Jackson 1.2 Homework Problem Solution

Dr. Christopher S. BairdUniversity of Massachusetts Lowell

PROBLEM:

The Dirac delta function in three dimensions can be taken as the improper limit as ! of the"aussian function

D; x , y , z=#$/#$e%p[ # x#y#z#]

Consider a 'eneral ortho'onal coordinate system specified by the surfaces u( constant) v( constant)w( constant) with len'th elements du/U) dv/V) dw/Win the three perpendicular directions. Show that

xx *=uu * vv * ww*UVW

by considerin' the limit of the "aussian above. +ote that as ! only the infinitesimal len'th

element need be used for the distance between the points in the e%ponent.

SOL!"O#:

Start with the 'eneral property of a Dirac delta,

x y zdx dy dz=&

Substitute in our representation,

lim!

D ; x , y , zdx dy dz=&

+ow transform the volume element into the new coordinate system

lim!

D ; x , y , zdu

U

dv

V

dw

W=&

-e do not know e%actly how the one system of coordinates transforms into the other) so we cannottransformDin a direct manner. Let us instead define an intermediate variable function Faccordin' to,

Fu , v ,w =lim!

D; x , y , zu ,v ,w &

U V W

-ith this definition our inte'ral becomes

Fu ,v ,w du dv dw=&

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Because we are inte'ratin' over all space) we are free to make a chan'e of variables which ust shifts

the ori'in.

Fuu *, vv *,ww *dudv dw=&

Comparin' this to the very first e/uation we see that it is identical e%cept with different inte'rationlabels and therefore,

Fuu *, vv *, ww*=uu *vv * ww *

so that) after plu''in' back in) we have

uu * vv * ww *=lim!

D; x , y , zu ,v , w&

U V W

Solve forD,

lim !

D ; x , y , z u ,v ,w =uu * vv *ww *U V W

The entity on the left) no matter what coordinate system it is represented in) is ust what we mean by

the 'eneral three0dimensional Dirac delta,

xx *=uu * vv * ww *UVW

+ow note that we never used the e%plicit form ofD) so we have solved the problem in a way that the

book did not intend. Let us try solvin' the problem usin' a method that uses the e%plicit form ofD.

D; x , y , z=#$/#$e%p[&

##x#y#z#]

Make a chan'e of variablesxx1x*) etc. 23therwise we will not end up with the most 'eneral case.4

D; xx *, yy *, zz*=#$/ #$ e%p[ # xx *#yy *#zz*#]

5s !)Dwill become 6ero unlessx1x* approaches 6ero as well. 7n calculus) we remember that

x1x* approachin' 6ero becomes dx. Therefore we have,

D; xx *, yy *, zz*=#$/ #$ e%p[ # dx

#dy#dz

#]-e reco'ni6e the last part in parentheses as the incremental arc len'th element dss/uared,

D; xx *, yy *, zz*=#$/ #$ e%p[ #ds#]

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8%pand the arc len'th in the new coordinate system,

D=#$ / #

$

e%p[ # du#

U#

dv#

V#

dw#

W# ]

+ote that dxdu /Uand we are not makin' that claim here. 9ather) the entire three0dimensionalincremental arc len'th dsis the same in all ortho'onal coordinate systems. +ow e%pand the increments

back into differences,

D=#$ / #$e%p[ # uu *#

U#

vv *#

V#

ww *#

W# ]

D= e

uu *#/#

#U

#

#

evv *

#/#

#V

#

#

eww *

#/#

#W

#

#

+ow make the substitution &:Uin the first bracket) #:Vin the second bracket) and $:W in the last bracket. -e can do this as lon' as we let &) #) and $'o to 6ero ust like we were

lettin' 'o to 6ero.

D=[ euu *

#/#&

#

#& ][evv *

#/##

#

## ][eww *

#/#$

#

#$ ]U V W-e now let the alpha*s approach 6ero. 8ach term in brackets on the ri'ht side becomes a one0

dimensional linear Dirac delta. The left side becomes the 'eneral e%pression for the three0dimensionalDirac delta,

xx *=uu * vv * ww *UVW

+ow this is a very useful result. Suppose we have a point char'e. 7n spherical coordinates) we can find

the representation of its Dirac delta usin' the above e%pression. ;or spherical coordinates

u=r , v=, w= and the len'th elements are dr ,r d, rsin d so thatU=&) V= &r, W=&

rsin and

xx *=rr* * *&

r#sin

2Spherical Coordinates4

+ote that it is fairly strai'ht0forward to prove usin' Dirac delta properties that

*/ sin=coscos * so that the three0dimensional Dirac delta in spherical coordinates isoften written

xx *=rr* cos cos * *&

r#

as it is on p. ! inJackson.

Similarly in cylindrical coordinates)u=r , v=, w=zand the len'th elements aredr ,r d, dzso that

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U=&) V=&

r, W=& and

xx *=rr* *zz*&

r2Cylindrical Coordinates4

5ssume that instead of a point char'e) we have a line char'e shaped into a rin') centered on theza%is)located at some radius r* and polar an'le *. The char'e is distributed alon' the rin' accordin' to theline char'e density . The total char'e density in this case would be,

=uu * vv *U V w 2Spherical Coordinates4

= rr* *

r2Spherical Coordinates4