The problem reads is from a Schaum's Outlines of EMFT and reads:
A total charge of \$1\mathrm{nC}\$ is equally distributed among \$2n\$ points which are placed equidistantly on a circle with \$1\mathrm{m}\$ radius centered at the origin in the \$xy\$ plane. Find the electric field intensity on the axis of the circle at \$z = \pm 1\mathrm{m}\$.
The answer given is \$\overline{E} = 3.18 \hat{a}_z \ \mathrm{Vm^{-1}}\$. But I keep getting \$\overline{E} = 4.5 \hat{a}_z \ \mathrm{Vm^{-1}}\$.
My method: By symmetry of a \$2n\$-gon at the origin, the field intensities parallel to the \$xy\$ plane will cancel each other out when you add them all together, so that the intensity experienced by something at \$z = \pm 1\mathrm{m}\$ is all due to intensity along \$\hat{a}_z\$.
So far we have:
$$\overline{E} = 2n \times \dfrac{\dfrac{10^{-9}}{2n}}{4\pi\dfrac{10^{-9}}{36 \pi} (\sqrt{2})^2} \hat{a}_z = \pm 4.5 \hat{a}_z \ \mathrm{Vm^{-1}}$$
where \$\epsilon \approx 10^{-9}{36 \pi}\$ is the permittivity.
I think it may be because I'm using \$\sqrt{2}\$ for distance when I should be using \$1\$? But that would give \$9 \ \mathrm{Vm^{-1}}\$, so I'm not sure what to do. Thanks.
