Difficulty: Introductory | Prerequisites: Basic Coulomb's Law (Week 1), vector notation
Electric fields are the invisible mechanism by which charged objects exert forces on one another across empty space. This week introduces the visual language physicists use to represent those fields: electric field lines. If you can read a field-line diagram, you can estimate where forces are strongest, which direction they point, and how a test charge would move, all without doing a single calculation. This material sits at the foundation of everything that follows in electromagnetism, so it is worth getting comfortable with now.
Electric field lines radiate outward from positive charges and inward toward negative charges. The density of the lines tells you the field strength: closely packed lines mean a stronger field. The electric field at a point depends on the source charge, not the test charge.
Electric field
A vector field produced by electric charges that describes the force per unit charge at every point in space. Formally, E = F/q. In simple terms, it is the "influence zone" a charge creates around itself, telling other charges how hard and in which direction they would be pushed or pulled.
Electric field line
A continuous curve whose tangent at any point gives the direction of the electric field there. Think of these as streamlines that show you where a tiny positive test charge would drift if released.
Test charge
A hypothetical small positive charge (q) used to probe the electric field without disturbing it. In simple terms, it is the measuring instrument you imagine placing at a point to see what force the field delivers.
Field line density
The number of field lines passing through a unit area perpendicular to the lines. Where lines are bunched together, the field is strong; where they spread apart, it is weak.
Uniform electric field
A region where the electric field has the same magnitude and direction at every point. Field lines are parallel and equally spaced. Think of it as a perfectly even push across the whole region.
Non-uniform electric field
A region where the field magnitude or direction (or both) varies from point to point. Field lines converge, diverge, or curve. This is the general case for most real charge arrangements.
A positive source charge Q creates a field that pushes a positive test charge q radially outward.
The magnitude of the force on q at distance r_A from Q is:
|F_A| = kqQ / r_A²
The force is larger at points closer to Q and smaller at points farther away.
For points at different distances: |F_B| > |F_A| > |F_C| > |F_D| when B is closest to Q and D is farthest.
Lines radiate outward from positive charges.
Lines point inward toward negative charges.
The direction of the field at any location is tangent to the field line passing through that point.
Field strength is encoded in the spacing of the lines:
Closely packed lines = strong field.
Widely spaced lines = weak field.
Field lines never cross one another (the field can only point in one direction at a given spot).
The electric field at a point depends only on the source charge and the distance:
E_A = kQ / r_A² (direction: radially outward for positive Q)
The force on a test charge q placed in the field is:
F = qE
Doubling the source charge Q doubles the electric field everywhere.
Doubling the test charge q doubles the force but does not change the field. The field is a property of the source, not of whatever you place in it.
In a uniform field, the field lines are straight, parallel, and evenly spaced.
The field magnitude is the same everywhere inside the region: E_A = E_B for any two points A and B.
A test charge q anywhere in the region feels the same force: F = qE.
In a non-uniform field, line spacing varies across the region.
Where lines are denser, the field (and hence the force on a test charge) is larger.
If two identical test charges are placed at different points, the one in the denser-line region experiences a greater force and accelerates faster.
Example: for field that fans out from left to right, a charge near the dense (left) side is pushed out of the region before a charge placed in the sparse (right) side.
Quantity | Expression | Notes |
|---|---|---|
Force on test charge | |F| = kqQ / r² | Coulomb's law; both charges matter |
Electric field (point charge) | |E| = kQ / r² | Source charge only; test charge irrelevant |
Force from field | F = qE | Connects field back to force on any charge q |
Electric field line maps are used in engineering to design capacitors, cathode ray tubes, and particle accelerators. Anywhere you need to steer or accelerate charged particles, you first map the field lines to predict trajectories, then adjust electrode geometry until the field does what you need.
"Doubling the test charge doubles the electric field." It does not. The field is set by the source charge alone. Doubling q doubles the force, but E stays the same.
"Field lines are paths that charges follow." Not quite. Field lines show the direction of the force (and therefore the acceleration) at each point, but a moving charge has inertia and generally curves rather than tracing a field line exactly.
"If field lines are drawn, there is no field between them." The field exists everywhere in space. The lines are a visual aid; the gaps between them are an artefact of drawing a finite number of lines.
"A uniform field means no field." Parallel, evenly spaced lines mean the field is constant and non-zero everywhere in that region.
⚠️ You will be asked to compare field magnitudes at different points using a field-line diagram, without calculating anything. Practise reading density and direction from pictures.
⚠️ A common exam question: "If the source charge doubles, what happens to E? If the test charge doubles, what happens to E?" Know which variable lives in which formula.
⚠️ Sketching field lines (direction, relative spacing) for simple charge configurations is a standard exam task.
True or false: Electric field lines can cross each other. (False)
If the distance from a point charge doubles, the electric field magnitude becomes ______ of its original value. (one quarter)
True or false: In a uniform electric field, a test charge at position A feels a different force than one at position B. (False, assuming same test charge)
The electric field points ______ from positive charges and ______ toward negative charges. (away; inward/toward)
True or false: Doubling the test charge changes the electric field at that point. (False)
Q: A positive charge Q sits at the origin. Points P1 and P2 are at distances r and 3r, respectively. How does the field at P2 compare to the field at P1?
A: E(P2) = kQ/(3r)² = kQ/9r², which is one-ninth of E(P1). The field falls off as the square of the distance.
Q: A uniform electric field of magnitude E fills a box. A charge q = 5 µC is placed inside. What is the force on the charge?
A: F = qE = (5 × 10⁻⁶ C) × E. The direction of the force is the same as the direction of E (since q is positive).
Q: You are given a field-line diagram around a positive point charge. How can you tell, just from the diagram, that point B (close to Q) has a stronger field than point D (far from Q)?
A: The field lines are more closely spaced near B and more spread out near D. Greater line density corresponds to a stronger field.
Q: If the source charge Q is doubled while the test charge q remains the same, what happens to (a) the electric field and (b) the force on the test charge?
A: (a) The electric field doubles (E is proportional to Q). (b) The force also doubles, since F = qE and q is unchanged.
This material connects directly to Gauss's Law (coming in a few weeks), which provides a powerful shortcut for calculating electric fields by counting field lines through closed surfaces. The concept of field-line density is essentially what Gauss's Law formalises as electric flux.
The distinction between field and force reappears when you study electric potential: potential is energy per unit charge, just as the field is force per unit charge.
electric field lines, Coulomb's law, test charge, source charge, field line density, uniform electric field, non-uniform electric field, point charge field, E = kQ/r², F = qE, P212, PHYS 212, electrostatics, field direction, field magnitude, radial field, superposition of fields