A student draws a field-line diagram with 8 lines densely packed near a positive charge and 4 more spread out further away, then concludes: 'There are exactly twice as many field lines on the left, so the field is twice as strong there.' What is wrong with this reasoning?
AField line density tells you nothing about field strength — only arrows can encode magnitude
BThe number of lines drawn in a diagram is arbitrary; only the relative spacing (density) between lines reflects relative field strength — absolute line counts cannot be compared
CThe field must be calculated from Gauss's law; field lines are only qualitative
DField lines must be drawn in equal numbers throughout a diagram for it to be valid
The number of field lines in any diagram is a choice made by the person drawing it — there is no physical quantity that determines 'how many lines to draw.' What encodes field strength is the density of lines: how tightly packed they are relative to each other. The same pattern could be drawn with 4 lines or 40 lines; only the relative spacing conveys information. Near a point charge, lines naturally diverge as 1/r², so their density decreases, correctly encoding the 1/r² fall-off of the field — regardless of the absolute number chosen.
Question 2 Multiple Choice
Why is it impossible for two electric field lines to cross each other in a valid field diagram?
ACrossing field lines would imply equal and opposite fields that cancel to zero
BIf two field lines crossed at a point, the electric field at that point would simultaneously point in two different directions, which is impossible since the field is a single-valued vector
CCrossed field lines would violate Gauss's law by implying net charge at the crossing point
DField lines must remain parallel in regions of uniform field strength
The electric field is a single-valued vector at every point in space — it has one magnitude and one direction at each location. A field line traces the direction of the field along its path, so its tangent points in the field direction. If two lines crossed, the field at the crossing point would need to simultaneously point in two different directions (one tangent per line), which is a contradiction. The no-crossing rule is not a drawing convention but a direct consequence of the field being well-defined.
Question 3 True / False
A positive charge released from rest in a static electric field will move along an electric field line.
TTrue
FFalse
Answer: True
If a positive charge starts from rest, the only force on it is F = qE, which is directed along the local electric field — that is, tangent to the field line. As the charge accelerates, the force at each subsequent point is again tangent to the local field line passing through that point. This means the trajectory of a charge starting from rest exactly traces a field line. (A charge with an initial velocity that is not along the field will follow a curved path that is NOT a field line, since its velocity carries it off the line.)
Question 4 True / False
In electrostatics, electric field lines can form closed loops under certain charge distributions.
TTrue
FFalse
Answer: False
Electrostatic field lines never form closed loops. A closed loop would mean a nonzero circulation of the electric field — the field would do net work on a charge moved around the loop. But the electrostatic field is conservative (its curl is zero everywhere in charge-free regions), so any closed-loop integral of E·dl = 0. This is a fundamental property of static electric fields, not just a diagramming convention. Closed field loops can occur for induced electric fields in changing magnetic fields (Faraday's law), but never in electrostatics.
Question 5 Short Answer
State the three strict rules governing valid electric field-line diagrams and identify the physical principle that underlies each rule.
Think about your answer, then reveal below.
Model answer: (1) Field lines never cross — if they did, the electric field would have two directions at a single point, contradicting the fact that the field is a single-valued vector. (2) Field lines never form closed loops in electrostatics — a closed-loop field would allow net work to be extracted by moving a charge around the loop, violating energy conservation; the electrostatic field is conservative (zero curl). (3) The number of lines originating on or terminating at a charge is proportional to the charge's magnitude — this encodes Gauss's law: the total electric flux through any closed surface is proportional to the enclosed charge, and the count of field lines crossing the surface represents that flux.