Formula Continuous Charge Distribution in 1d Electric field Linear charge density
$$\boldsymbol{E} ~=~ \frac{1}{4\pi\,\varepsilon_0}\,\int_{L}\frac{\boldsymbol{R}-\boldsymbol{r}}{|\boldsymbol{R}-\boldsymbol{r}|^3}\,\lambda(\boldsymbol{r})\,\text{d}l$$ $$\boldsymbol{E} ~=~ \frac{1}{4\pi\,\varepsilon_0}\,\int_{L}\frac{\boldsymbol{R}-\boldsymbol{r}}{|\boldsymbol{R}-\boldsymbol{r}|^3}\,\lambda(\boldsymbol{r})\,\text{d}l$$
Electric field
$$ \boldsymbol{E}(\boldsymbol{R}) $$ Unit $$ \frac{\mathrm V}{\mathrm m} $$ It indicates the magnitude and direction of the electric force at the location \(\boldsymbol{R}\) that would act on a test charge if it were placed at this location.
Position vector
$$ \boldsymbol{r} $$ Unit $$ \mathrm{m} $$ It goes from the origin to a location within the one-dimensional charge distribution.
Field vector
$$ \boldsymbol{R} $$ Unit $$ \mathrm{m} $$ It goes from the origin to the location (field point) where the electric field is to be calculated.
The connecting vector \(\boldsymbol{R} - \boldsymbol{r}\) is the vector going from a point of charge distribution \(\boldsymbol{r}\) to the considered field point \(\boldsymbol{R}\). Here \(\frac{\boldsymbol{R} ~-~ \boldsymbol{r}}{|\boldsymbol{R} ~-~ \boldsymbol{r}|}\) is the unit vector of the connecting vector.
Linear charge density
$$ \lambda(\boldsymbol{r}) $$ Unit $$ $$ Charge per length at location \(\boldsymbol{r}\) within the considered one-dimensional charge distribution.
Length
$$ L $$ Unit $$ \mathrm{m} $$ The length of the considered one-dimensional charge distribution generating the electric field.
Vacuum Permittivity
$$ \varepsilon_0 $$ Unit $$ \frac{\mathrm{As}}{\mathrm{Vm}} $$ The vacuum permittivity is a physical constant that appears in equations involving electromagnetic fields. It has the following experimentally determined value:$$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$