PX157 - B2b - electric field from an infinite line of charge

charge uniformly spread along the line
linear charge density, $λ$ is constant
cylindrical coordinates $(R, ϕ, z)$
at point, $P : \vec{r} = R \hat{R} + z \hat{z}$
section at ${\vec{r}}^{'} = z^{'} \hat{z}$
$d Q = λ d z^{'}$
$\vec{r} - {\vec{r}}^{'} = R \hat{R} + (z - z^{'}) \hat{z}$

d i s t a n c e = | \vec{r} - {\vec{r}}^{'} | = \sqrt{R^{2} + (z - z^{'})^{2}}

d \vec{E} = \frac{λ d z^{'}}{4 π ϵ_{0}} \frac{R \hat{R} + (z - z^{'}) \hat{z}}{(R^{2} + (z - z^{'})^{2})^{\frac{3}{2}}}

\vec{E} = \int d \vec{E} = \int_{- \infty}^{+ \infty} \frac{λ}{4 π ϵ_{0}} \frac{R \hat{R} + (z - z^{'}) \hat{z}}{(R^{2} + (z - z^{'})^{2})^{\frac{3}{2}}} d z^{'}

\begin{aligned} \int_{- \infty}^{+ \infty} \frac{a}{(a^{2} + x^{2})^{\frac{3}{2}}} . d x & = \frac{2}{a} \\ \int_{- \infty}^{+ \infty} \frac{x}{(a^{2} + x^{2})^{\frac{3}{2}}} . d x & = 0 \end{aligned}

∴ \vec{E} = \frac{λ}{4 π ϵ_{0}} \frac{2}{R} \hat{R} = \frac{λ}{2 π ϵ_{0} R} \hat{R}