Question

Two mutually perpendicular long straight conductors carrying uniformly distributed charges of linear charge densitites ${\lambda }_1$ and ${\lambda }_2$ are positioned at a distance a from each other. How does the interaction between the rods depend on a?

Answer 1

Let us make two-dimensional view of the situation. Because of symmetry we can say the force on wire 2 (${\lambda }_2$) will be along the negative y-direction.

$dF=E_{x}dqcos\theta$

$F=\int dF=\int \frac{{\lambda }_1}{2\pi {\epsilon }_0\sqrt{a^2+x^2}}\left({\lambda }_{2}dx\right)cos\theta$

where $cos\theta =\frac{a}{\sqrt{a^2+x^2}}$

$F=\frac{{\lambda }_1{\lambda }_2}{2\pi {\epsilon }_0}a{\int }_{-\infty }^{\infty }\frac{dx}{a^2+x^2}=\frac{{\lambda }_1{\lambda }_{2}a}{2\pi {\epsilon }_0}\times \frac{1}{a}\left[ta{n}^{-1}\frac{x}{a}\right]$

$F=\frac{{\lambda }_1{\lambda }_2}{2\pi {\epsilon }_0}$.

It is independent of a.