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Old Feb25-05, 01:27 AM       Last edited by Reshma; Feb25-05 at 01:31 AM..            #1
Reshma

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Dirac Delta function

Can someone explain me the Dirac Delta function for the function:

LaTeX Code: \\vec A = \\frac{\\hat r}{r^2}
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Old Feb25-05, 02:37 AM                  #2
James R

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Your question doesn't seem to make sense. The Dirac Delta Function is always the same. It doesn't rely on any other function.
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Old Feb25-05, 03:00 AM                  #3
Reshma

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I'm sorry, the given function is the Dirac delta function. Can someone explain it to me?
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Old Feb25-05, 05:00 AM                  #4
maverick280857

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The best thing an electrical engineer could do for physics: http://mathworld.wolfram.com/DeltaFunction.html (might help you)
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Old Feb26-05, 03:27 AM                  #5
Reshma

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Thanks for the link, Vivek. But it did not completely solve my problem. The proofs given in most texts are too mathematical. I need a more physical interpretation of the problem.
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Old Feb26-05, 03:42 AM                  #6
vincentchan

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what is your question?
your question doesn't make sense at all?
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Old Feb26-05, 03:53 AM       Last edited by himanshu121; Feb26-05 at 03:57 AM..            #7
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Originally Posted by Reshma
Can someone explain me the Dirac Delta function for the function:

LaTeX Code: \\vec A = \\frac{\\hat r}{r^2}
I believe You want to interpret its curl or div in terms of Dirac Delta Function

LaTeX Code: \\vec{\\nabla} X \\vec A
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Old Feb26-05, 04:12 AM                  #8
maverick280857

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Originally Posted by Reshma
Thanks for the link, Vivek. But it did not completely solve my problem. The proofs given in most texts are too mathematical. I need a more physical interpretation of the problem.
Yes they are mathematical because of the very definition of DDF. Strictly, it is not a function but it is considered a function. If you want good physical interpretations of its applications, get a copy of Classical Electrodynamics by Griffiths and read the first chapter (I think its called mathematical preliminaries but I'm not very sure).

Hope that helps...

cheers
vivek
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Old Feb26-05, 04:13 AM                  #9
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Perhaps you (Reshma) are asking for a proof that the charge (density) distribution that produces this field is a dirac-delta function (about the origin) ? The given field itself is not a dirac-delta.
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Old Feb26-05, 06:09 AM                  #10
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It can be proven really easily that the Green's function for an oO domain (R^{3}) for the Poisson equation:
LaTeX Code:  \\Delta V(\\vec{r})=f(\\vec{r})  (1)

is:LaTeX Code:  G(\\vec{r},\\vec{rsingle-quote})=\\frac{1}{4\\pi|\\vec{r}-\\vec{r}single-quote|}  (2)

And incidentally,the field,being the -gradient of the solution of (1),can be put in connection to (2)...

Daniel.
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Old Feb26-05, 07:28 AM                  #11
Reshma

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Originally Posted by himanshu121
I believe You want to interpret its curl or div in terms of Dirac Delta Function

LaTeX Code: \\vec{\\nabla} X \\vec A
Yes, you are right. I want an interpretation of the divergence of the given function.
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Old Feb26-05, 07:30 AM                  #12
Reshma

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Originally Posted by maverick280857
Yes they are mathematical because of the very definition of DDF. Strictly, it is not a function but it is considered a function. If you want good physical interpretations of its applications, get a copy of Classical Electrodynamics by Griffiths and read the first chapter (I think its called mathematical preliminaries but I'm not very sure).

Hope that helps...

cheers
vivek
Yes I do have Griffith's book which has described the above function over a sphere using Green's theorem.
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Old Feb26-05, 07:35 AM                  #13
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You want the proof that the LaTeX Code:  \\nabla \\cdot \\frac{\\vec{r}}{r^{3}}  is proportional (it's a "-1" the coefficient of proportionaliry,IIRC) to delta-Dirac...?

That's a pretty delicate matter.It's not really for physicists...Any book on PDE-s should have it,when discussing Laplace & Poisson equations.

Daniel.
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Old Feb26-05, 08:01 AM                  #14
Reshma

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Originally Posted by dextercioby
You want the proof that the LaTeX Code:  \\nabla \\cdot \\frac{\\vec{r}}{r^{3}}  is proportional (it's a "-1" the coefficient of proportionaliry,IIRC) to delta-Dirac...?

That's a pretty delicate matter.It's not really for physicists...Any book on PDE-s should have it,when discussing Laplace & Poisson equations.

Daniel.
I am extremely sorry for stretching this thread this far
I only want to know the proof for:

LaTeX Code:  \\nabla \\cdot\\frac{\\vec{r}}{r^{2}}

With a little physical interpretation...
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Old Feb26-05, 11:50 AM       Last edited by dextercioby; Feb26-05 at 02:59 PM..            #15
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That is something else...As u can yourself check...

Turning to the original question,i can add:except for the origin,where the fraction to whom you apply the diff.operator is not defined,the result is zero.However,as LaTeX Code:  \\frac{\\vec{r}}{r^{3}}  is the Green function for the Poisson equation for R^{3},it can be shown that,in fact:

LaTeX Code:  \\nabla\\cdot\\frac{-\\vec{r}}{r^{3}}=-4\\pi\\delta(\\vec{r})

As for physical significance,please,check (as you probably already have) Griffiths' book.Or Jackson's...

Daniel.
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Old Feb26-05, 01:57 PM                  #16
kanato

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LaTeX Code:  \\nabla \\cdot \\frac{\\hat{r}}{r^2} = <BR>\\frac{\\partial}{\\partial r} \\left( r^2 \\cdot \\frac{1}{r^2} \\right) = 0

everywhere, except the origin, where we have a point of non-differentiability.

By integrating over the volume of a sphere, and applying the divergence theorem, we see

LaTeX Code:  \\int_V \\nabla \\cdot \\frac{\\hat{r}}{r^2} r^2 \\sin \\theta dr d\\theta d\\phi<BR>= \\oint_S \\frac{\\hat{r}}{r^2} \\cdot \\hat{r} r^2 \\sin \\theta d\\theta d\\phi<BR>= \\oint_S \\sin \\theta d\\theta d\\phi = 4\\pi<BR>

independent of the radius of the sphere. Thus integration over any volume including the origin gives 4*pi, and any other volume gives zero. A function which satisfies this would be 4*pi times a delta function located at the origin. Thus, LaTeX Code:   \\nabla \\cdot \\frac{\\hat{r}}{r^2} = 4 \\pi \\delta({\\vec{r}})
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