Electric Fields and Field Line Patterns
An electric field is the region around a charge in which another charge experiences a force; the electric field strength is E = F ÷ q (in N/C or V/m), for a point charge Q it is E = kQ ÷ r2, and field lines point away from positive charges and toward negative charges, with the density of lines showing the strength. In brief: the direction of the field is the direction of the force on a small positive test charge. Between two parallel charged plates, the field is uniform, E = V ÷ d. Fields from several charges add as vectors. Field lines never cross, and are perpendicular to the surface of a conductor.
The field concept lets us describe electric forces without referring to a second charge. This guide defines the electric field and its strength, describes the patterns of field lines for point charges, dipoles and parallel plates, compares the patterns in a table, works through field strength calculations and corrects common mistakes.
What is an electric field?
A charge Q alters the space around it, so that any other charge placed there feels a force. The electric field strength at a point is defined as the force per unit positive charge: E = F ÷ q, where q is a small positive test charge. It is a vector, and the unit is the newton per coulomb (N/C), which is the same as volt per meter (V/m). A charge q in a field E experiences a force F = qE: in the direction of E for positive charges, opposite for negative charges.
What is the field of a point charge?
From Coulomb’s law, the force on the test charge q at distance r from a point charge Q is kQq ÷ r2, so
E = kQ ÷ r2.
The field points radially outward for Q > 0 and inward for Q < 0, and decreases as the square of the distance. For several charges, the net field at a point is the vector sum of the fields due to each charge (superposition). Outside a uniformly charged sphere, the field is the same as that of a point charge at its center. Inside a conductor in equilibrium, the field is zero, and any excess charge sits on the surface. This is the basis of the shielding effect of a Faraday cage: the interior of a hollow conductor is protected from external electric fields.
What do field line patterns look like?
- Isolated positive point charge: straight lines radiating outward, evenly spaced; a negative charge has lines pointing radially inward.
- Two opposite charges (dipole): lines begin on the positive charge and end on the negative one, curving between them. The field between them is strong, and the lines are dense.
- Two like charges: lines curve away from each other; there is a point midway where the field is zero (a neutral point).
- Parallel plates (opposite charges): uniform field between the plates, with parallel, equally spaced lines from the positive to the negative plate, E = V ÷ d. At the edges, the lines bulge outward (fringing).
- Charged conductor: lines leave the surface at right angles; the field is strongest at sharp points, which is why lightning rods have points and why sharp points produce corona discharge.
Rules for drawing: lines start on positive and end on negative charges (or at infinity); the tangent to a line gives the direction of E; the number of lines is proportional to the charge; lines never cross, because the field has one direction at each point.
Field patterns compared
| Configuration | Field strength | Field lines |
|---|---|---|
| Point charge | E = kQ ÷ r2 | Radial |
| Parallel plates | E = V ÷ d (uniform) | Parallel, evenly spaced |
| Dipole | Vector sum | Curved from + to − |
| Inside a conductor | Zero | None |
Worked example: field strength calculations
- Point charge. Q = +5.0 nC. At r = 0.10 m: E = 8.99 × 109 × 5.0 × 10−9 ÷ 0.102 = 4,495 N/C, directed away from the charge. At 0.20 m, E = 1,124 N/C.
- Force from the field. A charge of +2.0 µC placed 0.10 m from the charge in item 1 experiences F = qE = 2.0 × 10−6 × 4,495 = 9.0 × 10−3 N.
- Parallel plates. A potential difference of 100 V is applied across plates 2.0 cm apart. E = V ÷ d = 100 ÷ 0.020 = 5,000 V/m = 5,000 N/C. An electron in this field feels F = eE = 1.602 × 10−19 × 5,000 = 8.0 × 10−16 N.
- Dipole. Charges +2.0 µC and −2.0 µC are 0.20 m apart. At the midpoint (0.10 m from each), the field due to each is 8.99 × 109 × 2.0 × 10−6 ÷ 0.102 = 1.8 × 106 N/C, and both point from the positive toward the negative charge, so the total is 3.6 × 106 N/C.
- Like charges. For two +2.0 µC charges 0.20 m apart, the fields at the midpoint cancel: 1.8 × 106 − 1.8 × 106 = 0.
How did the field concept develop?
Michael Faraday introduced “lines of force” in the 1830s to 1840s to represent the electric and magnetic influences, and treated them as real. James Clerk Maxwell expressed the field theory mathematically (1861 to 1865) and predicted electromagnetic waves. Today the field is regarded as a basic physical entity, and quantum field theory describes forces as exchanges of particles between charges.
Which mistakes are common?
Mistake 1: Drawing field lines that cross
The field has a unique direction at each point, so lines cannot cross.
Mistake 2: Using the direction of force on a negative charge as the direction of the field
The field direction is defined by the force on a positive test charge.
Mistake 3: Adding field magnitudes without direction
Fields are vectors. Add components.
Mistake 4: Saying that the field is zero where there are no field lines drawn
The field exists everywhere. Only a finite number of lines are drawn.