Definition of Electric Field

Imagine you are standing near a bonfire. Even without touching the flames, you feel heat radiating outward — the fire is influencing the space around it. Electric charges do something remarkably similar: a charged object alters the space surrounding it, creating what physicists call an electric field. Any other charge placed in that region will feel a force, even though the two charges never physically touch each other. The electric field is the mathematical tool we use to describe this invisible influence. Formally, the electric field E at a point in space is defined as the electrostatic force F experienced by a small positive test charge q₀ placed at that point, divided by the magnitude of that test charge: E = F / q₀. The test charge must be small enough that it does not disturb the original charge distribution that created the field. This definition transforms a force — which depends on both the source and the test charge — into a property of space itself. Once you know the electric field at every point, you can instantly find the force on any charge placed there. Electric fields are vector quantities: they have both magnitude and direction. The direction of E at any point is defined as the direction of the force that would act on a positive test charge placed there. Fields point away from positive source charges and toward negative source charges. Understanding electric fields is foundational to all of electrostatics, circuit theory, and even electromagnetic waves — Maxwell later showed that changing electric fields can propagate through space at the speed of light.

Key takeaways
  • Why We Need a Field Concept
  • The Test Charge Definition
  • Electric Field of a Point Charge
  • Superposition Principle
  • Direction and Field Lines

Why We Need a Field Concept

Coulomb's law tells us the force between two point charges: F = kq₁q₂/r². But this formula requires you to know both charges. What if you want to describe the electrical environment created by a source charge, independently of whatever other charge might come along later? The electric field answers this: it encodes what the source charge does to surrounding space, so that any incoming charge can immediately calculate its force.

The Test Charge Definition

Place a hypothetical positive test charge q₀ (ideally infinitesimally small) at a location P. Measure the electrostatic force F on it. The electric field at P is E = F / q₀. Because we divide by q₀, the result is independent of the test charge's size — it is a pure property of the point P in space. The SI unit is newtons per coulomb (N/C), which is equivalent to volts per metre (V/m).

Electric Field of a Point Charge

For a single point charge Q sitting at the origin, combining the test-charge definition with Coulomb's law gives E = kQ/r², where r is the distance from Q to the field point. The field points radially outward if Q > 0 and radially inward if Q < 0. At twice the distance, the field is four times weaker — an inverse-square relationship identical to gravity, but electric forces can be either attractive or repulsive depending on the sign of Q.

Superposition Principle

When multiple charges are present, the total electric field at any point is the vector sum of the individual fields from each charge. This is the superposition principle. It means you can handle complex charge distributions by breaking them into simple pieces, computing each piece's contribution, and adding the vectors. This linearity is a deep property of Maxwell's equations and makes electric field calculations tractable.

Direction and Field Lines

Since E is a vector, direction matters. We visualise electric fields using field lines: imaginary curves whose tangent at any point gives the direction of E there. Field lines start on positive charges and end on negative charges. Where lines are densely packed, the field is strong; where they spread apart, the field is weak. Field lines never cross — that would imply two different field directions at the same point, which is impossible.

The Key Equation

E⃗=F⃗q0=kQr2r^\vec{E} = \frac{\vec{F}}{q_0} = \frac{kQ}{r^2}\hat{r}
  • \vec{E} — Electric field (N/C (or V/m)): The electric field vector at the point of interest; encodes the force-per-unit-charge at that location.
  • \vec{F} — Electrostatic force (N): The force experienced by the test charge q₀ due to all other charges in the region.
  • q_0 — Test charge (C): A small positive charge used to probe the field; small enough not to disturb the source distribution.
  • k — Coulomb's constant (N·m²/C²): k = 8.99 × 10⁹ N·m²/C², which equals 1/(4πε₀) where ε₀ is the permittivity of free space.
  • Q — Source charge (C): The charge creating the electric field. Positive Q gives an outward-pointing field.
  • r — Distance (m): The distance from the source charge Q to the field point.
  • \hat{r} — Radial unit vector (dimensionless): Points from the source charge toward the field point; gives the direction of E for a positive charge.

Worked Example

Problem. A small positive charge of Q = +3.0 μC is fixed in space. (a) Calculate the magnitude of the electric field at a point P located 0.20 m from Q. (b) A second charge q = −2.0 nC is placed at P. What force does it experience?

E=6.74×105 N/C;F=1.35×10−3 N (toward Q)E = 6.74 \times 10^5 \text{ N/C};\quad F = 1.35 \times 10^{-3} \text{ N (toward } Q\text{)}

Answer. The electric field at P has magnitude 6.74 × 10⁵ N/C directed away from Q. The force on the −2.0 nC charge is 1.35 × 10⁻³ N directed toward Q (attractive).

Figure
Diagram
Electric Field Lines Around a Positive Point Charge+Qtest charge q₀F = q₀EField lines radiate outward; spacing indicates field strength (closer = stronger)
Origins · 1 minMichael Faraday and how this was discovered
MF
Michael Faraday
1791–1867 · British

Faraday invented the concept of the electric field — or 'lines of force' as he called them — to describe how charges influence surrounding space. Working without a formal mathematical education, he developed an intuitive geometric picture that was later given rigorous mathematical form by James Clerk Maxwell. His field concept replaced the idea of 'action at a distance' with a physical reality pervading space, which became the foundation of classical electromagnetism. Faraday was the son of a blacksmith and had almost no formal schooling. He educated himself by reading books while working as a bookbinder's apprentice, eventually becoming one of the greatest experimental scientists in history and a Fellow of the Royal Society.

  • Coulomb's Law — Coulomb's law provides the force between point charges; combined with the test-charge definition it directly yields the electric field formula E = kQ/r².
  • Electric Potential — Electric potential V is the potential energy per unit charge and is related to the electric field by E = −dV/dr, making it the scalar companion to the vector field E.
  • Electric Field Lines — Field lines are the graphical representation of the electric field vector, showing direction and relative strength throughout space.
  • Gauss's Law — Gauss's law uses the electric field concept to relate the total flux through a closed surface to the enclosed charge, providing a powerful method for calculating E in symmetric situations.