Image line charge for a grounded conducting cylinder ($r_{\mathrm{image}}=a^2/b$)

A real infinite line charge $\lambda$ lies at distance $b>a$ from the axis of a grounded conducting cylinder of radius $a$. An image line $-\lambda$ at radius $a^2/b$ lies inside the excluded conductor. On the circle, real and image distances obey $r_1/r_2=b/a$, so $\varphi =(\lambda /(2\pi {ϵ}_0))\ln [(b/a)r_2/r_1]$ vanishes there. The image field solves the physical exterior problem with bounded far potential. Its additive constant matters; the interior image field is not a physical field inside the conductor.

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.