An uncharged, infinitely long hollow conducting cylindrical shell has inner radius \(a\) and outer radius \(b\). A thin line with uniform linear charge density \(+\lambda\) is placed inside the cavity, parallel to the central cylinder axis but displaced off-center by a distance \(d\), where \(0 < d < a\). The system is allowed to reach electrostatic equilibrium.

Choice	Potential difference \(\Delta V = V_{\text{outer}} - V_{\text{inner}}\)	Total induced charge per unit length \(\int_0^{2\pi} \sigma(\theta) a \, d\theta\)
A	\(0\)	\(-\lambda\)
B	\(0\)	\(-\lambda\left(1 - \dfrac{d}{a}\right)\)
C	\(\dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\)	\(-\lambda\)
D	\(\dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\)	\(-\lambda\left(1 - \dfrac{d}{a}\right)\)

Which of the following correctly pairs the electric potential difference \(\Delta V\) between the outer and inner surfaces of the shell with the total induced surface charge per unit length on the inner cavity wall?