Electric field
The electric field describes how source charges alter the space around them. Once the field is known, the force on any test charge follows immediately. This separates the properties of the sources from the charge used to probe them.
Field means force per unit positive charge
E⃗ = F⃗/q so F⃗ = qE⃗
The SI unit is N C⁻¹, equivalent to V m⁻¹. The direction of E⃗ is defined as the direction a positive test charge would be pushed. A negative charge feels force opposite to E⃗.
Field of a point charge
E = k|Q|/r², k = 8.9875517923 × 10⁹ N m² C⁻²
The inverse-square dependence means doubling distance reduces field magnitude to one quarter. The sign of Q determines inward or outward direction.
Point-charge field
Magnitude for a point charge: E = k|q|/r².
Superposition: fields add as vectors
For several source charges, find the field produced by each source at the observation point as if the other sources were absent, then add all field vectors. Symmetry can make this much easier.
Do not add magnitudes unless all field vectors happen to point along the same direction.
How to read field lines
Field lines are a visualization, not physical threads. They leave positive charges and end on negative charges or extend to infinity. The electric field is tangent to a field line at each point, and a denser drawing of lines represents a stronger field.
Field lines never cross: one point in space cannot have two different electric-field directions at the same instant.
The number of drawn lines is a convention, so do not count individual lines as though they were physical objects. What matters is the pattern: tangent gives direction, and relative line density conveys relative field strength.
Fields, forces and motion
Once E⃗ is known, a charged particle's electric force is qE⃗. If electric force dominates, its acceleration is a⃗ = qE⃗/m. Because q can be positive or negative, two particles placed in the same field can accelerate in opposite directions.
The field is a property of the source configuration; changing the test charge changes the force on that test charge, not the pre-existing field of the sources.
In electrostatic equilibrium, the electric field inside the conducting material is zero; otherwise free charges would keep moving. Excess charge resides on the conductor's surface, and the field just outside is perpendicular to that surface. This makes conductors a useful boundary case for checking field reasoning.
Worked examples
1. Field of a point charge
Find the electric-field magnitude 0.20 m from a +3.0 nC point charge.
Solution
E = k|q|/r² = (8.99 × 10⁹)(3.0 × 10⁻⁹)/(0.20)² ≈ 674 N C⁻¹, directed away from the positive charge.
2. Force on a test charge
A −2.0 μC charge is placed in a uniform field of 300 N C⁻¹ directed east. What electric force acts on it?
Solution
F⃗ = qE⃗. Magnitude = 2.0 × 10⁻⁶ × 300 = 6.0 × 10⁻⁴ N. Because q is negative, the force points west, opposite the field.
3. Midpoint between equal positive charges
What is the net electric field exactly halfway between two equal positive point charges?
Solution
The two field magnitudes are equal at the midpoint and point in opposite directions, so the net field is zero. The individual fields are not zero; they cancel vectorially.