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

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

Now, force on this element, $d F=\left(\lambda_{2} d r\right) \cdot \frac{2 K \lambda_{1}}{r}$
or, $d F=\frac{2 K \lambda_{1} \lambda_{2} d r}{r}$
Total force $=\int d F=\int\limits_{1}^{2 l} \frac{2 K \lambda_{1} \lambda_{2}}{r} d r$
$=2 K \lambda_{1} \lambda_{2} \ln \left(\frac{2 l}{l}\right)$
Total force $=2 \times 9 \times 10^{9} \times 1 \times 10^{-6} \times \frac{1}{\ln 4} \times 10^{-6} \times \ln 2$
$=9 \times 10^{-3} N$