Sign of the charge from field lines

Problem

An electric field is represented by field lines that start from a central region and go to infinity, denser at the top than at the bottom. What can you say about the sign of the charge contained in the region and about the magnitude of the field at points at equal distance but above and below? Can the lines intersect? Justify your answer.

Solution

Sign of the charge. The field lines exit from the central region and head towards infinity. By convention lines exit from positive charges and enter negative ones, so the charge contained in the region is positive. $\ev{Q > 0}$

Comparing the magnitudes. The density of lines (number of lines per unit area perpendicular to them) is proportional to the magnitude of the field: where the lines are denser the field is stronger. Since at the top the lines are denser than at the bottom, $\ev{|\vec{E}_{\text{top}}| > |\vec{E}_{\text{bottom}}|}$ at equal distance from the central region. A non-uniform density also signals that the charge distribution is not spherically symmetric.

Intersection of the lines. Field lines cannot intersect. If two lines crossed, at the crossing point the field $\vec{E}$ would have two distinct directions (the two tangents), but the electric field is a vector with a single well-defined value at every point in space. The intersection is therefore impossible.

Connections

Topics: Electric field and potential Concepts: Electric field · Electric charge