1. Understand the charge distribution

A long cylindrical shell has:

• upper half surface charge density $+\sigma$
• lower half surface charge density $-\sigma$

Since the cylinder is very long, the problem is effectively two-dimensional in the cross-sectional plane.

So we only need to think about the electric field pattern in a plane perpendicular to the axis.

2. Key physical idea

Electric field lines:

• originate from positive charge
• terminate on negative charge
• are perpendicular to the conducting/equally charged surface locally only in special symmetric cases; here the charge is prescribed on the shell, so we determine the field qualitatively from symmetry.

Because the top half is positive and the bottom half is negative, the field lines must generally go from the upper semicircular surface to the lower semicircular surface.

Thus, near the cylinder:

• field lines emerge from the upper half,
• curve around the sides,
• and enter the lower half.

3. Symmetry analysis

The configuration is:

• symmetric about the vertical diameter of the cross-section,
• antisymmetric about the horizontal diameter.

So the correct field pattern must satisfy:

• Left and right sides should look mirror images.
• Field direction should be from top to bottom.
• On the vertical axis above the cylinder, the field should point upward or downward?

Let us check carefully.

Take a point far above the cylinder. The upper positive half is closer, the lower negative half is farther. The field due to the positive part points away from it (upward component), while the field due to the negative part points toward it (downward component). Since the positive half is nearer, the net field above tends to be upward.

Similarly, far below the cylinder, net field tends to be upward? No: below the cylinder, the negative half is closer, so the field points toward the negative charges, i.e. upward toward the cylinder. Thus below also the field is directed upward near the vertical axis toward the negative lower half.

This means field lines can come out from the top region, bend around, and terminate on the lower half.

4. Inside the cylinder

This is not a uniformly charged full cylinder, so the field inside is not zero.

In fact, by symmetry, inside the cavity the field should be roughly from the positive upper half toward the negative lower half, i.e. downward.

So the correct figure must show field lines passing through the interior from top to bottom.

5. Which schematic is correct?

The correct field-line diagram must have:

• lines starting on the upper semicircle,
• ending on the lower semicircle,
• symmetry about the vertical axis,
• nonzero field inside the cylinder,
• curved lines outside connecting upper and lower halves.

Among the given options, this corresponds to Option C.

6. Final answer

Therefore, the correct option is $\boxed{\text{C}}$

7. Comparison with stored answer

Stored correct answer: C

My derived answer: C

So they agree.