A very long uniformly charged, thin rod is oriented along x-axis with its one end at origin and it extends along +x-axis. The charge per unit length of the rod is $\lambda$. Calculate the electric field vector at x = -a  on x-axis.

Field at x = -a due to any selection of the rod will be directed along -x - axis.

Consider a very small section of length dx at a distance x from point P. the charge on this section is dq such that dq = $\lambda$dx. The electric field at P due to this section is:

$$\begin{gathered} d\overrightarrow{E }= \frac{-1}{4{\mathrm{\pi \epsilon }}_0}\frac{dq}{x^2}\hat{i} (along -x-axis) \\ Net field at P is \overrightarrow{E } = -{\int }_a^{\infty }\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{dq}{x^2}\hat{i} = \frac{-\lambda \hat{i}}{4{\mathrm{\pi \epsilon }}_0}{\int }_a^{\infty }\frac{dx}{x^2} = \frac{-\lambda }{4\pi {\epsilon }_0}\left[\frac{1}{a} - \frac{1}{\infty }\right] = \frac{-\lambda \hat{i}}{4{\mathrm{\pi \epsilon }}_{\mathrm{o}}\mathrm{a}} \end{gathered}$$