Two infinitely long thin straight wires having uniform linear charge densities $\mathrm{\lambda }$ and $2\mathrm{\lambda }$ are arranged parallel to each other at a distance r apart. The intensity of the electric field at a point midway between them is 

Given data: Two infinitely long thin straight wires

Concept used: Electric fields

Detailed solution: 

Distance between two parallel infinite long thin straight wires = r

Electric field due to infinite long thin straight wire, 

$\mathrm{E}=\frac{\mathrm{\lambda }}{2{\mathrm{\pi \epsilon }}_0\mathrm{r}}$

${\mathrm{E}}_1$ Due to first infinite wire $\mathrm{P}=\frac{{\mathrm{\lambda }}_1}{2{\mathrm{\pi \epsilon }}_0{\mathrm{r}}_1}$

${\mathrm{E}}_2$Due to second infinite wire $\mathrm{O}=\frac{{\mathrm{\lambda }}_2}{2{\mathrm{\pi \epsilon }}_0{\mathrm{r}}_2}$

Given ${\mathrm{r}}_1={\mathrm{r}}_2=\frac{\mathrm{r}}{2},{\mathrm{\lambda }}_1=\mathrm{\lambda } \mathrm{and} {\mathrm{\lambda }}_2=2\mathrm{\lambda }$

$\Rightarrow {\mathrm{E}}_1=\frac{\mathrm{\lambda }}{2{\mathrm{\pi \epsilon }}_0\left(\frac{\mathrm{r}}{2}\right)}=\frac{\mathrm{\lambda }}{{\mathrm{\pi \epsilon }}_0\mathrm{r}}$

${\mathrm{E}}_2=\frac{\mathrm{\lambda }}{2{\mathrm{\pi \epsilon }}_0\left(\frac{\mathrm{r}}{2}\right)}=\frac{2\mathrm{\lambda }}{{\mathrm{\pi \epsilon }}_0\mathrm{r}}=2{\mathrm{E}}_1$

$\therefore$ Electric intensity at mid-point, $\mathrm{E}={\mathrm{E}}_2-{\mathrm{E}}_1=2{\mathrm{E}}_1-{\mathrm{E}}_1=\mathrm{E}$

$\therefore \mathrm{E}=\frac{\mathrm{\lambda }}{{\mathrm{\pi \epsilon }}_0\mathrm{r}}$