Question

An infinitely long string uniformly charged with a linear charge density ${\lambda }_1$ and a segment of length uniformly charged with linear charge density ${\lambda }_2$ lie in a plane at right angles to each other and are separated by a distance $r_0$ as shown in figure. The force with which these two interact is $\frac{{\lambda }_1{\lambda }_2}{4\pi {ϵ}_0}\ln \left(x\right)$, then find the value of $x$ , if $l=r_0$.

Answer 1

Electric field due to infinitely long charged wire at distance $r$ from it is $\frac{2k\lambda }{r}$

Now, we take small element $dr$ on wire (2) which is having distance $r$ from wire (1)


Now, force on this element $=\left({\lambda }_{2}dr\right)\times \frac{2k{\lambda }_1}{r}$

or $dF=\frac{2k{\lambda }_1{\lambda }_{2}dr}{r}$

Total force $=\int dF={\int }_{r_0}^{r_0+l}\frac{2k{\lambda }_1{\lambda }_2}{r}dr=2k{\lambda }_1{\lambda }_2\ln \left(\frac{r_0+l}{r_0}\right)=\frac{{\lambda }_1{\lambda }_2}{4\pi {ϵ}_0}\ln \left(4\right)$