Question

(a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
(b) Explain why two field lines never cross each other at any point.

Hint:

The tangent to a field line gives the unique direction of \(\vec E\) at that point. Use single direction per point to argue both no breaks and no crossing.

Answer 1

This question tests two basic properties of electrostatic field lines.

Step 1: Why a field line cannot have a sudden break. An electric field line shows the direction of the electric field \(\vec E\) at every point through which it passes; the tangent to the line at a point gives the direction of \(\vec E\) there. In the region around charges, the electric field exists at every single point in space (except exactly at a point charge). So the field has a definite, continuous direction everywhere.

Step 2: If a field line had a sudden break, it would mean that at the point of the break the electric field suddenly stops, i.e. \(\vec E\) does not exist or has no defined direction there. This is not true in a charge-free region. Hence a field line must be a continuous, unbroken curve. It can only start on a positive charge and end on a negative charge (or run off to infinity), but it never simply breaks in mid-space.

Step 3: Why two field lines never cross. The electric field at any point has one and only one direction. The tangent drawn to a field line at a point gives that single direction of \(\vec E\).

Step 4: Suppose two field lines crossed at a point. Then at that crossing point we could draw two different tangents, giving two different directions for \(\vec E\). A single point cannot have two field directions at the same time. This is impossible, so two field lines can never cross each other.

Answer 2

Expert framing using the field as a single-valued vector function.

Step 1: Treat the electric field as a function \(\vec E(\vec r)\) that assigns one vector to every position \(\vec r\) in space. Field lines are simply the curves whose tangent everywhere is parallel to \(\vec E(\vec r)\). They are the integral curves (streamlines) of this vector field.

Step 2 (no breaks): Because \(\vec E(\vec r)\) is defined and finite at every point of a charge-free region, the streamline can always be continued: from any point you can step a tiny distance along \(\vec E\) to the next point, and so on without interruption. A break would require \(\vec E\) to be undefined at an interior point of empty space, which does not happen. Lines therefore begin and end only on charges (sources and sinks) or at infinity, never in the middle of nowhere.

Step 3 (no crossing): A function is single-valued: \(\vec E(\vec r)\) gives exactly one direction at each \(\vec r\). If two lines crossed at \(\vec r_0\), the streamline through \(\vec r_0\) would have two tangents, so \(\vec E(\vec r_0)\) would point in two directions at once. That contradicts single-valuedness, so crossing is forbidden.

Step 4 (consistency check): The only places this rule relaxes are exactly the points where \(\vec E\) itself is undefined, namely on the point charges themselves, where many lines meet. That exception proves the rule: away from charges, where \(\vec E\) is well defined, lines are continuous and never intersect.