Question

An infinitely long string uniformly charged with a linear charge density ${\lambda }_1=1\mu C/m$ and a segment of length $l$ uniformly charged with linear charge density ${\lambda }_2=\frac{1}{\ln 4}\mu C/m$ lie in a plane at right angle to each other and separated by a distance $l$, as shown in the figure. Determine the force (in $mN$ ) with which these two interact.

Answer 1

Answer: 9

Electric field due to an infinitely long charged wire at distance $r$ from it $=\frac{2K\lambda }{r}$
Now, we take a small element $dr$ on wire (2) which is at distance $r$ from wire (1).

Now, force on this element, $dF=\left({\lambda }_{2}dr\right)\cdot \frac{2K{\lambda }_1}{r}$
or, $dF=\frac{2K{\lambda }_1{\lambda }_{2}dr}{r}$
Total force $=\int dF=\underset{1}{\overset{2l}{\int }}\frac{2K{\lambda }_1{\lambda }_2}{r}dr$
$=2K{\lambda }_1{\lambda }_2\ln \left(\frac{2l}{l}\right)$
Total force $=2\times 9\times 10^9\times 1\times 10^{-6}\times \frac{1}{\ln 4}\times 10^{-6}\times \ln 2$
$=9\times 10^{-3}N$