Q. Two parallel infinite line charges $+\lambda$. and $-\lambda$, are placed with a separation distance $R$ in free space. The net electric field exactly mid-way between the two line charges is
ManipalManipal 1980Electric Charges and Fields
Solution:
Electric field at point $P$ due to line charge distribution $+ \lambda$,
$E_{+\lambda}=\frac{\lambda}{2\pi\varepsilon_{0}\left(\frac{R}{2}\right)}$ away from $+\lambda$
Electric field at point $P$ due to line charge distribution $-\lambda$,
$E_{-\lambda}=\frac{\lambda}{2\pi\varepsilon_{0}\left(\frac{R}{2}\right)}$ towards $-\lambda$
$E_{+\lambda}$ and $E_{-\lambda}$ have same direction,
$\therefore E_{net}=E_{+\lambda}+E_{-\lambda}=2\left(\frac{\lambda}{2\pi\varepsilon_{0}\left(\frac{R}{2}\right)}\right)$
$=\frac{2\lambda}{\pi\varepsilon_{0}\,R}$
