A long conducting cylinder is divided into two equal halves along its axis. One half of the cylinder is maintained at a constant potential \( V_0 \), while the other half is at a potential of 0, as shown in Fig.
The system as a whole carries no net charge. The electric potential distribution in the region surrounding the cylinder will be
1
\( V = \frac{V_0}{\pi} \left(\arctan\frac{2 a r \sin \theta}{|r^2 - a^2|} \right) \)
2
\( V = V_0 \left(\frac {\pi}{2}+\arctan\frac{2 a r \sin \theta}{|r^2 - a^2|} \right) \)
3
\( V = \frac{V_0}{\pi} \left(\frac {\pi}{2}+\arctan\frac{2 a r \sin \theta}{|r^2 - a^2|} \right) \)
4
\( V = \frac{V_0}{\pi} \left(\pi+\arctan\frac{2 a r \sin \theta}{|r^2 - a^2|} \right) \)