I hope this question is appropriate for this site, if not, just leave a comment and I will delete.
I am interested in knowing how to derive the electric field due to a spherical shell by Coulomb's law without using double integrals or Gauss Law.
Relevant equations are — Coulomb's law for electric field and the volume of a sphere:
$\vec{E} = \frac{1}{4\pi \epsilon_0}\frac{Q}{r^2}\hat{r}$, where $Q =$
charge, $r=$ distance.$V = \frac{4}{3}\pi r^3.$
From my book, I know that the spherical shell can be considered as a collection of rings piled one above the other but with each pile of rings the radius gets smaller and smaller.
I am not interested in the final formula, just the derivation of it. Thank you!
Best Answer
Well you can first calculate the field of a ring centered at $z=z_0$ on the $z$ axis with radius $r$ (using CGS, multiply by ugly factors later). By symmetry, on the $z$ axis the field is only in the $z$ direction and can be shown to be: $$E_z(z)=\frac{q(z-z_0)}{((z-z_0)^2+r^2)^{3/2}}$$ Now each ring has charge $q=Q\cos \theta d\theta$, and $z_0 = R\cos \theta$. This means you can integrate the expression $E_z(z)$ over $\theta$ to get the field at any point on the $z$ axis. By symmetry, you can choose the ring direction as you wish, so that this expression is true for points not on the $z$ axis as well, with $r$ replacing $z$.
As I mentioned in the comments, since the field of each ring contains an integral, this is really a double integral, even if you decide to call this "two single integrals".