Cylinder surface charge induced by an exterior line ($\sigma =-\frac{\lambda (b^2-a^2)}{2\pi a(b^2+a^2-2ab\cos \theta )}$)

For the real line at radius $b>a$ and polar angle measured toward it, the exterior radial field gives $\sigma (\theta )=-\lambda (b^2-a^2)/[2\pi a(b^2+a^2-2ab\cos \theta )]$. Its integral around the circumference is $-\lambda$ per unit axial length. At $b=2a$, this becomes $-3\lambda /[2\pi a(5-4\cos \theta )]$. The nearer surface has larger induced charge magnitude.