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Radarithm
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Homework Statement
Find the electric field a distance z above the center of a circular loop of radius r that carries a uniform line charge λ.
Homework Equations
$$E=E_r\hat{r}+E_z\hat{z}$$
$$E_r=\frac{\lambda}{4\pi\epsilon_0}\int_0^r\frac{1}{\mathcal{R}^2}\sin{\theta}\,dr$$
$$E_z=\frac{\lambda}{4\pi\epsilon_0}\int_0^r\frac{1}{\mathcal{R}^2}\cos{\theta}\,dr$$
$$\sin{\theta}=\frac{r}{\mathcal{R}}$$
$$\cos{\theta}=\frac{z}{\mathcal{R}}$$
$$\mathcal{R}=\sqrt{r^2+z^2}$$
The Attempt at a Solution
This question is rather simple but I still got it wrong (I checked the solutions manual and it had a different answer which I will post below).
Carrying out the integrations for ##E_r## and ##E_z##:
$$E_r=-\frac{\lambda}{4\pi\epsilon_0}\frac{1}{\sqrt{r^2+z^2}}\hat{r}$$
$$E_z=\frac{\lambda}{4\pi\epsilon_0}\frac{r}{z\sqrt{z^2+r^2}}\hat{z}$$
Therefore the electric field a distance z above a circular loop is:
$$E=-\frac{\lambda}{4\pi\epsilon_0}\frac{1}{\sqrt{r^2+z^2}}\hat{r}+\frac{\lambda}{4\pi\epsilon_0}\frac{r}{z\sqrt{z^2+r^2}}\hat{z}$$
The solution however is:http://i.gyazo.com/e5aba5c7109b6c119b36ad299003b1bc.png
edit: Nevermind, caught my mistake. Can a mentor delete this thread?
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