UY1: Electric Field Lines

Use electric field lines as a visual map of direction and strength, with clear rules tied to superposition and force on test charges.

University Physics Year 1

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Learning objectives

Construct electric-field and potential models for discrete and continuous charge distributions.

This page gives the UY1 working model/result for Electric Field Lines. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

Module path: Electromagnetism (UY1)
Practice: UY1 Electromagnetism Quiz
Full routing: UY1 Assessment Map
Math toolkit: Mathematics for Undergraduate Physics

1) At a glance

Prerequisites: what vector E means (The Electric Field) and superposition (Field of a Point Charge)
Outcomes: read direction/relative strength from a sketch and apply the non-crossing and start/end rules correctly
Key results (rules): tangent gives vector E direction; line density maps | vector E| qualitatively; lines start on + and end on - (or infinity); lines never cross
Common trap: treating field lines as particle trajectories (they are a visualization of the vector field, not paths)
Field lines are a visualization tool, not physical strings.
Tangent to a field line gives local vector E direction.
Line density indicates field strength (qualitatively).
Field lines never cross (one unique field direction at each point).

Motivation / intuition

Field lines are the “vector field picture”: they are drawn so that your eye can quickly see direction (tangent) and relative strength (density) without doing component algebra at every point. Use them for intuition first, then switch to equations for numbers.

2) Setup

Use a small positive test charge to define direction.
For multiple sources, compute

vector Eₙₑₜ = vector E₁ + vector E₂ + …

and sketch lines based on the net field.

Convention: lines start on positive charges and end on negative charges (or infinity).

3) Core derivation/explanation

Field lines come from the field definition:

vector F = q₀ vector E

A positive test charge accelerates along vector E, so field-line arrows point that way.

For many charges:

vector E = ∑ᵢ vector Eᵢ, vector Eᵢ = (1/4πε₀)qᵢ/rᵢ²r hat ᵢ

So shape and direction of field lines follow vector superposition.

Why lines do not cross:

At one point in space, vector E has one direction.
Crossing lines would imply two directions at the same point.

The number of lines you draw is arbitrary; what matters is the pattern and relative density.
Field lines can start/end only on charge (or at infinity). If you see a line ending in empty space, the sketch is inconsistent.
A particle’s actual path depends on its mass, initial velocity, and sign of charge; field lines only encode the instantaneous force direction for a small positive test charge.

Units: [vector E] = N/C, so any “field strength” you compute from a diagram should ultimately match N/C.
Limits/signs: far from a cluster of charges, the pattern approaches that of a single point charge with net Qₙₑₜ (if Qₙₑₜ ≠ 0); if Qₙₑₜ = 0, the far field looks dipolar (lines leave and return).

4) Worked example(s)

Two equal positive charges are fixed on the x-axis, symmetrically about the origin. At the midpoint, fields from each charge have equal magnitude and opposite direction along x, so:

vector E_mid = vector 0

This is a neutral point (unstable for a positive test charge if displaced slightly along certain directions).

Field-line sketch implications:

Lines leave both positive charges.
No line can terminate in empty space; lines curve away from the midpoint region.
The midpoint has no arrow direction because net field is zero there.

5) Practice set (with hints + answers)

Can electric field lines intersect? Hint: think about unique force direction on a test charge. Answer: No.
Around a single positive point charge, are field lines radial inward or outward? Hint: direction of force on a positive test charge. Answer: outward.
If field-line density is larger near one region, what does that indicate? Hint: density is a qualitative map of | vector E|. Answer: stronger electric field magnitude in that region.

6) Summary + next steps

Field lines are a direction/strength map derived from vector E.
Superposition determines the actual line pattern for multiple charges.
Use lines for intuition, then switch to component equations for calculations.

Next: Electric Field Of An Electric Dipole Previous: Electric Dipole Back To Electromagnetism