The problem was to give the charge distribution of a uniformly charged disk of radius R centered at the origin and perpendicular to the z-axis.(adsbygoogle = window.adsbygoogle || []).push({});

The professor explained us how to do so (though many parts were unclear to me and I couldn't copy everything that was on the blackboard because we were too much students and I couldn't see the bottom part of the blackboard).

For a punctual charge, the general form of the charge distribution is [tex]\rho (\vec x) = q \delta (x) \delta (y) \delta (z)[/tex].

For a charged plane: [tex]\rho (\vec x) = q \delta (z)[/tex] and for a charged line: [tex]\rho (x,y,z)=q \delta (x) \delta (y)[/tex].

However for a finite (in extension) plane perpendicular to the z-axis, [tex]\rho (\vec x) = f(x,y)\delta (z)[/tex]. The professor told us that the Dirac's delta "reduces" the dimension (though I don't understand why/how. If you have any comment on his, this would be nice). That's why for a point-like charge we need 3 of these deltas, 1 for a plane and 2 for a line.

The purpose of the function f in the above example is to make the final value coincide with what it should be; that's what I understood.

Using cylindrical coordinates [tex](\rho, z, \phi)[/tex], for the uniformly charged disk of radius R we have that [tex]\rho (\vec x)=\delta (z) \theta (\rho -R) f(\rho)[/tex] where [tex]\theta (\rho - R) = 1[/tex] for [tex]\rho \leq R[/tex] and 0 for [tex]\rho \geq R[/tex]. Thus the theta function is a constraint that delimit the disk of radius R.

Then I don't really know what the professor has done, some integration I believe and he reached a final result I couldn't copy.

In spherical coordinates [tex](r, \theta, \phi )[/tex] and for the same problem, he reached that [tex]\rho (\vec x) =\frac{Q}{\pi R^2 r} \delta (\theta - \frac{\pi}{2}) \hat \theta (r-R)[/tex] which looks different from the result he reached in cylindrical coordinates but it must be the same.

If I understand well the final result, it has no sense to ask for the charge distribution in a particular point in [tex]\mathbb{R}^3[/tex]. However it makes sense to ask for the charge distribution in any surface element; though I don't know how to proceed. I think I should do an integration or so, but I'm not sure and an example would be welcome.

So the final answer to the question is a function that is ill defined because for example in the origin it is not defined nor at the boundaries although there are charges there.

And if I'm not wrong, if I do integrate the final answer over an area (or volume?!) A, then I should get [tex]\sigma _0 A[/tex] ? Namely the amount of charges enclosed into this small area? Is this right?

Why is the function "ill defined" as I call it? Is it because we DO NOT assume charges like points but like a property of a non zero space extension?

If I have the problem "What is the charge distribution of a uniformly charged cube?", I'd rather answer [tex]\rho _0[/tex] for [tex]-a \leq x \leq a[/tex], [tex]-a \leq y \leq a[/tex], [tex]-a \leq y \leq a[/tex]. 0 everywhere else.

My answer wouldn't be ill defined and I could give you the charge distribution in any particular point in space, unlike the method used in more complicated problems.

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# Electrostatics: Charge distribution and Dirac's delta, lots of questions

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