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This is about the electric field of a ring with radius r, at a distance z from center, along the axis of the ring. The ring carries a uniform line charge [tex]\lambda[/tex]. We always say that the radial component of the field cancels out due to symmetry. Can somebody tell how to prove it mathematically (using cylindrical coordinate system only)?
[tex]dE_{rad}=-\frac{1}{4\pi\epsilon_0}\frac{r\lambda d\theta}{(r^2+z^2)}\hat{\textbf{r}}[/tex]
[tex]E_{rad}=-\frac{1}{4\pi\epsilon_0}\frac{r\lambda}{(r^2+z^2)}\int_0^{2\pi} \hat{\textbf{r}} d\theta[/tex]
???
[tex]dE_{rad}=-\frac{1}{4\pi\epsilon_0}\frac{r\lambda d\theta}{(r^2+z^2)}\hat{\textbf{r}}[/tex]
[tex]E_{rad}=-\frac{1}{4\pi\epsilon_0}\frac{r\lambda}{(r^2+z^2)}\int_0^{2\pi} \hat{\textbf{r}} d\theta[/tex]
???