Electric field above a circular loop

1. The problem statement, all variables and given/known data



Find the electric field a distance z above the center of a circular loop of radius r that carries a uniform line charge λ.



2. Relevant equations



$$E=E_r\hat{r}+E_z\hat{z}$$

$$E_r=\frac{\lambda}{4\pi\epsilon_0}\int_0^r\frac{1}{\mathcal{R}^2}\sin {\theta}\,dr$$

$$E_z=\frac{\lambda}{4\pi\epsilon_0}\int_0^r\frac{1}{\mathcal{R}^2}\cos {\theta}\,dr$$

$$\sin{\theta}=\frac{r}{\mathcal{R}}$$

$$\cos{\theta}=\frac{z}{\mathcal{R}}$$

$$\mathcal{R}=\sqrt{r^2+z^2}$$



3. 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:

$$E=-\frac{\lambda}{4\pi\epsilon_0}\frac{1}{\sqrt{r^2+z^2}}\hat{r}+\frac{\la mbda}{4\pi\epsilon_0}\frac{r}{z\sqrt{z^2+r^2}}\hat{z}$$

The solution however is:http://ift.tt/1sLv2c1



I don't really know where I went wrong.