Question

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:

• A. zero
• B. $\frac{2\lambda }{{\pi \epsilon }_{0}R}$
• C. $\frac{\lambda }{{\pi \epsilon }_{0}R}$
• D. $\frac{\lambda }{{2\pi \epsilon }_{0}R}$

Answer 1

The correct option is B $\frac{2\lambda }{{\pi \epsilon }_{0}R}$

Electric field at point $P$ due to line charge distribution $+\lambda$
$E_{+\lambda }=\frac{\lambda }{2\pi {\epsilon }_0\left(\frac{R}{2}\right)}$
Electric field at point $P$ due to line charge distribution $-\lambda$
$E_{-\lambda }=\frac{\lambda }{2\pi {\epsilon }_0\left(\frac{R}{2}\right)}$
$E_{+\lambda }$ and $E_{-\lambda }$ have same direction
$\therefore E_{net}=E_{+\lambda }+E_{-\lambda }=2\left(\frac{\lambda }{2\pi {\epsilon }_0\left(\frac{R}{2}\right)}\right)=\frac{2\lambda }{\pi {\epsilon }_{0}R}$