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How to write a volume (charge for example) density using delta Dirac function in spherical coordinates in this case:

There are thin charged disk (radius $a$) with surface charge density $\sigma$. The centre of a disk is on $z$-axis. The distance between centre of a disk and origin of a spherical coordinates is $h$. Assume that $\phi$ - is an angle between $z$-axis and edge of a disk. Thus $\tan\phi$=$a/h$.

How to write volume density through delta Dirac function?

I think that $r$ (distance in spherical coordinates) should be from $h$ to $\sqrt{a^2+h^2})$ and angle $\theta$ should be from zero to $\phi$. Note that for each angle $\theta$ from 0 to $\phi$ corresponds $r$. Hence we should consider r as function of $\theta$ $r(\theta)$.

In this way the result is $\rho$ =$\sigma\frac{\delta(\theta-\phi)}{r}\delta(r-\frac{h}{cos(\theta)})$. Is there mistake?

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  • $\begingroup$ Start by figuring this out for a disk centered on the origin in spherical coordinates. Then the only delta function will be for the poloidal angle since the charge distribution is not delta-like in the radial or azimuthal directions. Then perform a translation. Remember, Dirac delta functions are meant to be integrated. $\endgroup$ – honeste_vivere Sep 13 '17 at 14:10

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