Field of Point Charges
Imagine dropping a stone into still water. Ripples spread outward in every direction, and if you placed a cork anywhere on the surface, it would bob — the water carries a 'disturbance' through space. A charged particle does something conceptually similar, but instead of disturbing water it disturbs the electric field that fills all of space around it. The electric field due to a point charge is one of the most fundamental ideas in electrostatics: a single charged particle, sitting alone in space, creates an invisible influence that reaches out in all directions and can exert forces on any other charge that enters that region. A point charge is an idealization — we treat the charged object as if all its charge Q is concentrated at a single geometric point with no physical size. This is a very good approximation whenever you are far from a charged object compared with its actual dimensions (like treating the Earth as a point mass when calculating orbital mechanics). The electric field E at any location around the point charge tells you the force per unit charge that a tiny positive 'test charge' would experience if placed there. Crucially, the field exists in space independently of whether a test charge is actually present. The field of a point charge has two beautiful properties: its magnitude follows an inverse-square law (doubling the distance reduces the field to one-quarter its value), and its direction is always radially outward from a positive charge or radially inward toward a negative charge. These two features make it possible to draw clean field-line diagrams, solve real problems analytically, and build up the fields of complex charge distributions by superposition — adding the individual fields of each charge together as vectors.
- Start with Coulomb's Law
- Define the Electric Field
- Assign Direction
- Understand the Inverse-Square Law
- Superposition for Multiple Charges
Start with Coulomb's Law
Coulomb's Law gives the force between two point charges q₁ and q₂ separated by distance r: F = k|q₁||q₂|/r². The constant k = 8.99 × 10⁹ N·m²/C² is Coulomb's constant (sometimes written 1/4πε₀ where ε₀ = 8.85 × 10⁻¹² C²/(N·m²) is the permittivity of free space). This force is a push or pull along the line connecting the two charges.
Define the Electric Field
To isolate the influence of a single source charge Q, imagine placing a very small positive test charge q₀ at a point P. The force on q₀ is F = kQq₀/r². The electric field at P is defined as E = F/q₀, the force per unit charge. This definition strips away the test charge itself, leaving only the property of the space created by Q: E = kQ/r². Because q₀ is taken to be infinitesimally small, it does not disturb the field it is measuring.
Assign Direction
The electric field is a vector. If Q is positive, the force on a positive test charge points away from Q, so E points radially outward. If Q is negative, the force on a positive test charge points toward Q, so E points radially inward. In vector form, writing r̂ as the unit vector pointing from Q to the field point P: E⃗ = kQ/r² · r̂. A negative Q automatically reverses the direction because k and r² are always positive, so the sign of Q controls the sign of the field vector.
Understand the Inverse-Square Law
The 1/r² dependence is not arbitrary — it is a consequence of geometry. The field lines spreading from a point charge distribute themselves uniformly over a sphere of surface area 4πr². As r doubles, that area quadruples, so the field strength (lines per unit area) falls to one-quarter. This is exactly the same reason light from a bulb dims as 1/r² and gravity weakens as 1/r². Nature uses the inverse-square law whenever something spreads outward in three dimensions from a point source.
Superposition for Multiple Charges
If several point charges Q₁, Q₂, Q₃… are present, the total electric field at any point P is simply the vector sum of the individual fields: E⃗_total = E⃗₁ + E⃗₂ + E⃗₃ + …. Each charge contributes its own kQᵢ/rᵢ² in the appropriate direction, and you add them as vectors (breaking into x- and y-components if needed). This superposition principle holds exactly in classical electrostatics and makes it possible to compute the field of any charge distribution, no matter how complex.
The Key Equation
- E — Electric field magnitude (N/C or V/m): The strength of the electric field at the observation point; equals force per unit positive test charge.
- k — Coulomb's constant (N·m²/C²): k = 8.99 × 10⁹ N·m²/C²; also written as 1/(4πε₀). Sets the scale of electromagnetic forces in SI units.
- Q — Source charge (Coulombs (C)): The charge creating the field. Positive Q gives an outward field; negative Q gives an inward field.
- r — Distance from source charge (metres (m)): The straight-line distance from the point charge Q to the point where the field is being evaluated.
- \varepsilon_0 — Permittivity of free space (C²/(N·m²)): ε₀ = 8.85 × 10⁻¹² C²/(N·m²); describes how easily electric fields form in a vacuum.
Worked Example
Problem. A small sphere carries a charge of Q = +3.0 × 10⁻⁶ C (3.0 μC). (a) Calculate the magnitude of the electric field at a point P that is r = 0.20 m from the sphere. (b) If a proton (charge e = +1.6 × 10⁻¹⁹ C) is placed at P, what force does it experience and in which direction?
Answer. The electric field at P has magnitude approximately 6.7 × 10⁵ N/C directed away from the sphere. The proton experiences a repulsive force of approximately 1.1 × 10⁻¹³ N directed away from Q.
Origins · 1 minCharles-Augustin de Coulomb and how this was discovered
Coulomb used a delicate torsion balance of his own invention to measure the forces between charged spheres with unprecedented precision. His 1785 experiments established quantitatively that the electrostatic force is proportional to the product of the charges and inversely proportional to the square of the distance — the law that now bears his name and forms the foundation for the point-charge electric field formula E = kQ/r². Before turning to electricity, Coulomb spent years as a military engineer and wrote an influential treatise on structural mechanics. His interest in measuring tiny mechanical forces — from friction to magnetic repulsion — led him to invent the torsion balance, which was sensitive enough to detect forces millions of times smaller than a newton.
Related Concepts
- Coulomb's Law — The electric field formula E = kQ/r² is derived directly from Coulomb's Law by dividing the force between two charges by the test charge. Understanding Coulomb's Law is essential before studying point-charge fields.
- Electric Field Lines — Field lines are the visual language for representing the field of point charges. Their density encodes field strength and their direction encodes the vector direction of E⃗ around charge distributions.
- Superposition of Electric Fields — Real situations involve multiple point charges. The superposition principle allows you to vector-add individual point-charge fields to find the total field of any charge distribution.
- Electric Potential of a Point Charge — The electric potential V = kQ/r is the scalar companion to the vector field E = kQ/r². Learning both together deepens understanding of how energy and force are encoded in the field of a point charge.