Assume a test charge with charge $Q$ is placed at the center of a 13-sided regular polygon, the vertices of which are charges with equal charge $q$. What is the electrical force on the center charge?
To start we write this as our electrical force... $$ \vec{F}_{Q} = \sum^{13}_{n} \frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{R^{2}} \left(cos\left(\frac{2n\pi}{13}\right)\hat{i} , sin\left(\frac{2n\pi}{13}\right)\hat{j}\right) $$
and with some simplification we get... $$ \vec{F}_{Q} = \frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{R^{2}} \left\{1-2cos\left(\frac{\pi}{13}\right) + 2cos\left(\frac{2\pi}{13}\right)-2cos\left(\frac{3\pi}{13}\right) + 2sin\left(\frac{\pi}{26}\right)-2sin\left(\frac{3\pi}{26}\right)+2sin\left(\frac{5\pi}{26}\right), 0\right\}$$ ...more simplification... $$ \vec{F}_{Q} = \frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{R^{2}} \left\{ 1.11022\times10^{-16}, 0 \right\} $$ Now we have an electrical force on the test charge, however, in a ring of equidistant charges such as the one mentioned above, shouldn't the $\hat{i}$ component be equal to zero? Does anyone have any ideas as to why I'm getting a non-zero number for my $\hat{i}$ component?
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