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I am doing a problem in which I am given a uniformly charged disk, charge density $\sigma$, with radius $b$ and a concentric hole of radius $a$ in the $xy$ plane. A part of the problem I would have to show what happens to the electric field in the $z$ component as $s = a \to b$.

So far I have:

$$ E_{z} = \frac{1}{2\epsilon_0} \int^b_a \frac{\sigma z}{(s^2 + z^2)^{3/2}}s ds$$

I am expecting the disk charge turn into a ring charge, but following the fundamental theorem of calculus, $E_z \to 0$ as $a \to b$. Am I missing something?

Thanks,

turo

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Your issue is keeping the charge density constant. If you start with a disk charge with a hole in it (an annulus) and keep shaving off more material to make the hole bigger, without changing the charge distribution... well, you're also shaving the charge off the annulus, and you will eventually end up with an infinitesimal ring with just as infinitesimal charge on it. No charge, no field, which is exactly what you predict with this calculation.

If you want to end up with the formula for a ring charge, you must send $a\to b$ while keeping the total charge $Q=\sigma\pi(b^2-a^2)$ constant. I.e. you must not shave off material from inside the annulus until it is a thin ring, but compress the annulus so that the charge stays on it and gets (infinitely) concentrated onto the resulting ring.

To redo your mathematical setup, relabel the original surface charge density to $\sigma_0,$ so the charge on the annulus is $Q=\sigma\pi(b^2-a^2).$ The surface charge density of the compressed annulus is a function of the hole size $s$: $\sigma(s)=\frac Q{\pi(b^2-s^2)}=\sigma_0(b^2-a^2)/(b^2-s^2)$ (the shaved annulus would have $\sigma(s)=\sigma_0$). The electric field of the new annulus (with hole size $s$) is $$E_z(s)=\frac1{2\epsilon_0}\int_s^b\frac{\sigma(s)z}{(r^2+z^2)^{3/2}}\,dr.$$ The limit $\lim_{s\to b}E_z(s)$ should recover the formula for a ring charge of linear charge density $Q/(2\pi b).$

Note that both these results are correct answers to the question "what happens if I start with an annulus with uniform charge density and then turn it into a thin ring?". The behavior of the resulting ring depends on how you turned the annulus into a ring: did you keep $\sigma$ constant (shaving), or did you keep $Q$ constant (compressing)? With the information you've presented, it's not entirely clear which of these cases you're in. You should carefully inspect the situation in your problem to figure out which is correct in your context. In general, whenever you take a limit, be careful of what variables are being controlled (here $a$/$s$), which are being held constant ($\sigma$ or $Q$), and which are being allowed to vary (the other one of $\sigma$ or $Q$, as well as $E_z$). This is especially important in physics problems, where there are many variables and the functional relationships (which variables determine which) are often left implicit.

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