Question

The linear charge densities on two straight parallel long linear charges are ${\lambda }_1$ and ${\lambda }_2$. Obtain the formula for force acting per unit length between them.

Answer 1

Electric field intensity at distance from a wire of linear charge density ${\lambda }_1$

$E_1=\frac{1}{4\pi {\epsilon }_0}\frac{2{\lambda }_1}{d}=\frac{1}{2\pi {\epsilon }_0}\frac{{\lambda }_1}{d}$

As second conductor is also in this field. Therefore force acting on its unit length (Charge $q={\lambda }_2)$

$F=E_1{\lambda }_2=\frac{1}{2\pi {\epsilon }_0}\frac{{\lambda }_1}{d}\times {\lambda }_2$

or $F=\frac{{\lambda }_1{\lambda }_2}{2\pi {\epsilon }_{0}d}$