Something we did in electrostatics that's a source of confusion for me:(adsbygoogle = window.adsbygoogle || []).push({});

We learned to use caution when taking the divergence of the (all important) radial vector field:

[tex] \vec{v} = \frac{1}{r^2} \hat{r} [/tex]

Applying the formula in spherical coords gave zero...a perplexing result. The problem, it was revealed, was that the function increases without bound as the origin is approached (I don't know, I tried to phrase it properly. My physics prof just said...it "blows up").

So we were reminded of the Dirac Delta function, which saved the day:

[tex] \nabla \cdot \vec{v} = 4\pi\delta^3(r) [/tex]

Fine. I tried to understand why there was a singularity at the centre. In other words, 1. The divergence tended to infinity at the origin, and 2. it was zero everywhere else. My reasoning was:

1. The divergence in this particular example is the rate at which the radial component of the vector field changes as r changes. So...from the centre, not only is the magnitude of the field "infinitely large" but, it immediately changes to having some finite value at some finite distance r from the origin. So the rate of change is also infinitely large...there is "infinite divergence" from the centre.

2. The divergence has a magnitude of zero everywhere else but the origin, because at any of these other points, the vector field is not "diverging away" from that point. It is confined to pass through it in only one direction (the radial direction).

There are two problems that I subsequently discovered with #1 and #2. My reasoning as it has been stated should apply to any vector field in the radial direction whose magnitude was inversely proportional to radial distance from the centre. This was clearly not true!

[tex] \nabla \cdot \left(\frac{1}{r}\hat{r}\right) = \frac{1}{r^2} [/tex]

and in general:

[tex] \nabla \cdot \left(r^n \hat{r}\right) = n+2(r^{n-1}) [/tex]

(except for n = -2)

So my reasoning is shot to pieces! I don't understand this...there is no singularity at the centre for other radial vector fields of this form with n < 0

(even though they still "blow up" at the origin!). Not only that, but suddenly we have a non-zero divergence everywhere else. The field is still radial only, so from what point in space other than the origin could the field possibly be "diverging"? It still passes through these points in only one direction! Admittedly, its magnitude is diminishing as it does so, but that is true for 1/r^2 as well!

The math says what it says...so what is wrong with #1 and #2?

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# Divergence of a Radial Vector Field

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