Animated Solution for Physics - Electrostatics: A long cylindrical shell carries positive surface charge $\sigma$ in the upper half and negative surface charge $-\sigma$ in the lower half. The electric field lines around the cylinder will look like figure given in (figures are schematic and not drawn to scale)

Imagine a long, hollow pipe. Now, imagine painting the entire upper half of this pipe with a perfectly uniform positive charge, and the lower half with an identical but negative charge. What you have just created is a beautiful, continuous, two-dimensional electric dipole.

This problem isn't about crunching numbers or solving complex integrals; it is a test of your physical intuition and your deep understanding of the fundamental properties of electric field lines. Let's break down the logic step-by-step to see why only one of these patterns can exist in reality.

The Journey of a Field Line

The most absolute, unbreakable rule of electric field lines is their origin and destination. Electric field lines must always originate from positive charges and terminate at negative charges. They are the visual representation of the path a tiny positive test charge would take if released in the field.

If we look at some of the incorrect options, we see field lines shooting straight out into the void of infinity, or arriving from nowhere. But why would a field line travel all the way to infinity when there is a perfectly good negative charge sitting right next door? The positive charges on the upper half are strongly "pulling" the field lines towards the negative charges on the lower half. Therefore, the lines must connect the two halves directly.

The Rule of Smoothness

Another critical property of electric field lines is that in any charge-free region of space, they must form continuous, smooth curves.

Why is this so important? The tangent to an electric field line at any given point tells us the exact direction of the net electric field vector $\vec{E}$ at that point. If a field line were to have a sharp corner or a kink, it would mean that at that specific vertex, you could draw multiple tangents. This would imply that the electric field has two or more different directions at the exact same point in space!

Since the electric field is a single, unique vector sum of all forces at any point, having multiple directions is physically and mathematically impossible. This immediately disqualifies any option that shows field lines taking sharp, ninety-degree turns.

By applying these two fundamental rules—originating at positive/terminating at negative, and maintaining smooth curves—we are left with only one logical conclusion.

The electric field lines must gracefully emerge from the positive upper half, curve smoothly through the surrounding space, and terminate on the negative lower half. This creates a beautiful, continuous pattern that perfectly mirrors the dipole-like nature of our cylindrical shell.

As a fascinating bonus thought: what happens inside the cylinder? Because of this specific surface charge distribution, the electric field inside the hollow region doesn't curve at all. It is perfectly uniform, pointing straight down from the positive ceiling to the negative floor. Physics is full of these elegant surprises!