Fourier transform of potential from point charge

**TriKri** · February 28th 2011, 05:29 PM

Hi! How can I calculate the Fourier transform of the potential generated by a point charge in the origin with the charge q?

**chisigma** · February 28th 2011, 05:57 PM

How many dimensions?... one, two, three... or more?...

Kind regards

$\chi$ $\sigma$

**TriKri** · February 28th 2011, 07:26 PM

Three dimensions!

**chisigma** · March 1st 2011, 05:07 AM

Originally Posted by TriKri

Three dimensions!

All right!... but for semplicity sake may be it is better to start with the two-dimensional case

...

The potential function generated by a charge q in the origin is...

$\displaystyle V(x,y)= \frac{q}{4 \pi \varepsilon_{0}\ \sqrt{x^{2}+y^{2}}}$ (1)

The two-dimension Fourier tranform of V(x,y) by definition is...

$\displaystyle \mathcal{F} \{V(x,y)\}= \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} V(x,y)\ e^{-j 2 \pi (u\ x + v\ y)}\ dx dy$ (2)

Now if we pass to polar coordinates the (2) becomes...

$\displaystyle \mathcal{F} \{V(r, \theta)\}= \frac{q}{4 \pi \varepsilon_{0}}\ \int_{0}^{\infty} \int_{- \pi}^{\pi} e^{-j 2 \pi (u\ r \cos \theta + v\ r\ \sin \theta )}\ dr d \theta$ (3)

Now it is easy to verify that the integral...

$\displaystyle \int_{- \pi}^{\pi} e^{-j 2 \pi (u\ r \cos \theta + v\ r\ \sin \theta )}\ d \theta$ (4)

... vanishes everiwhere with the only exception of the case $u=v=0$ where is $2 \pi$ so that the final result is...

$\displaystyle \mathcal{F} \{V(x, y)\}= \frac{q}{2 \varepsilon_{0}}\ \delta(u,v)$ (5)

Kind regards

$\chi$ $\sigma$

**TriKri** · March 1st 2011, 11:47 AM

Thank you, but does the integral really vanish? If $r\sqrt{u^2+v^2} < 1/4$ the exponent over e will be in the interval $(-j\pi/2, +j\pi/2)$ , so the real part of the integrand will be positive for all $\theta$ and then the integral can't vanish, can it?

**TriKri** · March 5th 2011, 07:20 AM

I'm still wondering what the Fourier transform is; the delta function can't be the answer since it's the Fourier transform of 1 (or at least of a real constant). The Fourier transform of 1/r in three dimensions has to be some kind of standard transform, hasn't it?

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