Need help with EM/Jackson problem

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In summary, the problem is that in Classical Electrodynamics - 2nd Ed., J.D. Jackson on page 50 there is a problem I need help with. Its problem 1.2 which states:The Dirac delta function in three dimensions can be taken as the improper limit as \alpha \rightarrow 0 of the Gaussian functionD(\alpha;x,y,z) = (2\pi)^{-3/2} \alpha^{-3} exp[-\frac{1}{2\alpha^2}(x^2 + y^2 + z^2)]Consider a general orthogonal coordinate system specified by the surfaces, u = constant, v = constant, w = constant
  • #1
pmb_phy
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In Classical Electrodynamics - 2nd Ed., J.D. Jackson on page 50 there is a problem I need help with. Its problem 1.2 which states

The Dirac delta function in three dimensions can be taken as the improper limit as [itex]\alpha \rightarrow 0[/itex] of the Gaussian function

[tex]D(\alpha;x,y,z) = (2\pi)^{-3/2} \alpha^{-3} exp[-\frac{1}{2\alpha^2}(x^2 + y^2 + z^2)][/tex]

Consider a general orthogonal coordinate system specified by the surfaces, u = constant, v = constant, w = constant, with length elements du/U, dv/V, dw/W in the three perpendicular directions. Show that

[itex]\delta[/itex](x - x') = [itex]\delta[/itex](u - u')[itex]\delta[/itex](v - v')[itex]\delta[/itex](w - w')UVW

by considering the limit of the above Gaussian. Note that as [itex]\alpha \rightarrow 0[/itex] only the infinitesimal length element need be used for the distance between the points in the exponent.

I can't seem to get started on this one. Note that [tex]D(\alpha;x,y,z)[/tex] is the product of three Gaussian functions i.e.

[tex]D(\alpha;x,y,z)[/tex] = G(x)G(y)G(z)

The product UVW reminds me of a Jacobian but I'm not quite sure how. :tongue:

Any thoughts/solutions/answers? I want to know the answer more than I want to be walked through it with hints (I have hundreds of more problems to work through besides this one which I gave up on). Thanks.

Pete
 
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  • #2
HINT:( :tongue2: )
[tex] \iiint_{R^{3}} \lim_{\alpha\rightarrow 0} [D(\alpha;x,y,z)] \ dx \ dy \ dz [/tex]

[tex]=\iiint_{R^{3}} \lim_{\alpha\rightarrow 0} [D(\alpha;x-x',y-y',z-z')] \ d(x-x') \ d(y-y') \ d(z-z') =1 [/tex]

Now make the change of coordinates...

Daniel.
 
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  • #3
Pete -- Think of a delta function as the inverse of a volume element, so if the volume at a point changes by dx dy dz -> dx dy dz/UVW, the delta function must transform as advertised.

More detail: as alpha-> zero, the Gaussian gets very narrow, and the transformation from (x,y,z) to (u,v,w) reduces to a constant one. So, the factor x*x + y*y + z*z in the original Gaussian, becomes, from dx=du/U, (u/U)**2 + (v/V)**2 + (w/W) **2. The rest is a bit of algebra.

All a delta function cares about is its own point -- the rest be damned.

Regards,
R
 

1. What is an EM/Jackson problem?

The EM/Jackson problem is a theoretical problem in electromagnetics that involves determining the electric and magnetic fields produced by a charged particle in motion. It is named after its discoverer, John David Jackson.

2. Why is the EM/Jackson problem important?

The EM/Jackson problem is important because it is a fundamental problem in electromagnetics that has practical applications in fields such as telecommunications, electronics, and particle physics. It also helps us understand the behavior of electromagnetic waves and their interactions with matter.

3. What are some common approaches to solving the EM/Jackson problem?

Some common approaches to solving the EM/Jackson problem include using vector calculus, Maxwell's equations, and Green's functions. Numerical methods such as finite element analysis and finite difference time domain methods are also used.

4. What are some challenges in solving the EM/Jackson problem?

Some challenges in solving the EM/Jackson problem include dealing with complex geometries and boundary conditions, as well as finding analytical solutions for more complex systems. Additionally, the problem can become computationally intensive for large systems.

5. How is the EM/Jackson problem related to other problems in electromagnetics?

The EM/Jackson problem is closely related to other problems in electromagnetics, such as determining the electric and magnetic fields of a steady current in a wire. It is also related to other theoretical problems such as the radiation of electromagnetic waves from an antenna or the scattering of electromagnetic waves by an object.

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