Question

Drive an expression for electric field due to infinite charge straight wire at a distance 'r'.

Answer 1

Let charge density be $P$.

Charge in $dx$ element be $Pdx$.

$E$ due to $dx$

$=\frac{kPdx}{x^2+r^2}=\frac{kPr}{r^2}\frac{dx/r}{1+{\left(\frac{x}{r}\right)}^2}=\frac{kPr}{r^2}\left(\frac{dt}{1+t^2}\right)=\frac{kPr}{r}{\tan }^{-1}t={\left[\frac{kPr}{r}{\tan }^{-1}\left(\frac{x}{r}\right)\right]}_{-\infty }^{\infty }=\frac{\pi kP}{r}$