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## Charge distribution of a uniformly charged disk

1. The problem statement, all variables and given/known data
The problem can be found in Jackson's book, I think in chapter 1 problem 3 or something like this.
I must determine the charge distribution of a uniformly charged disk of radius R in spherical coordinates (I've done it in cylindrical coordinates and had no problem). The total charge is Q.
I've found a solution on the internet but the answer is different from mine.
I forgot to mention that I have to use Dirac's delta.

2. Relevant equations
$\int _{\mathbb{R}^3} \rho (\vec x )=Q$.

3. The attempt at a solution
Since the charges are over a 2d surface, there will be 1 Dirac's delta in the expression for rho, the charge density. I will use Heaviside's step function because the surface is limited.
Let $(r, \theta , \phi )$ be the coordinates. I make the ansatz/educated guess that $\rho$ is of the form $C \delta \left ( \theta - \frac{\pi }{2} \right ) \Theta (r \sin \theta -R)$.
Integrating this distribution in all the space, I reach that C is worth $\frac{3Q}{2\pi R^3}$.
Therefore $\rho (r, \theta )=\frac{3Q\delta \left ( \theta - \frac{\pi }{2} \right ) \Theta (r \sin \theta -R)}{2\pi R^3}$.
However the solution provided on the internet is $\rho (\vec x )=\frac{q }{\pi R^2r} \delta \left ( \theta - \frac{\pi }{2} \right ) \Theta (r -R)$.
Are they both equivalent (I doubt it), if not, did I do something wrong? If so, what did I do wrong? Thanks a lot!
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