Electric Charge, Force, and Field
Everything in this course traces back to one idea introduced here: charges create a field, and the field pushes on other charges. Coulomb's law and superposition handle point charges directly; continuous distributions (a charged rod, ring, or disk) need the same idea applied through an integral, ∫k dq/r², which is the calculus this course's title promises and the algebra-based version simply cannot do. Gauss's Law in the next unit is a faster route for symmetric distributions, but it's this unit's integral that explains why Gauss's Law works at all.
What you'll learn
- Apply Coulomb's law (vector form) to find the net electrostatic force on a charge from one or more point charges.
- Calculate the electric field due to a point charge or a collection of point charges using superposition.
- Set up and evaluate a definite integral to find the electric field of a continuous charge distribution (a rod, ring, or disk).
- Analyze the motion of a charged particle in a uniform electric field using kinematics.
- Describe how charge redistributes on a conductor in electrostatic equilibrium, and contrast this with an insulator.
- Calculate the torque and potential energy of an electric dipole in a uniform external field.
1. Charge, Conductors, and Coulomb's Law
Electric charge comes in two signs and is quantized — every charge in nature is an integer multiple of the elementary charge e=1.6×10⁻¹⁹ C — and it's conserved: the total charge of an isolated system never changes, even as charge freely redistributes within it.
Materials fall into two broad categories by how easily charge moves through them. In a conductor, some electrons are free to move throughout the material; in an insulator, they're bound to their atoms and stay put. This is why a charged rod brought near a neutral conductor induces a redistribution of charge on it (the near side polarizes opposite to the rod), while a neutral insulator only polarizes at the molecular level.
Coulomb's law gives the force between two point charges: F=kq₁q₂/r², directed along the line joining them — repulsive for like signs, attractive for opposite signs. When more than two charges are present, the superposition principle applies: find the force from each charge independently, then add the results as vectors.
2. The Electric Field and Superposition
The electric field at a point is defined as the force per unit charge a small positive "test charge" would feel there: E=F/q₀. Crucially, the field exists at that point regardless of whether a test charge is actually there — it's a property of space set up by the source charges, the same way a river's current exists whether or not a leaf is floating in it to reveal it.
For a single point charge, E=kq/r², pointing radially away from a positive source (toward a negative one). When multiple source charges are present, superposition applies to fields exactly as it does to forces: compute each charge's field independently and add the vectors.
3. Continuous Charge Distributions
A real charged object isn't a point — it's a continuous distribution of charge spread over a line, surface, or volume. The strategy is always the same: slice the object into infinitesimal pieces of charge dq, each acting like its own point charge at some distance r from the field point, then integrate: E=∫k dq/r².
The specific setup changes with geometry. For a rod with uniform linear charge density λ=Q/L, dq=λ dx, and you integrate over the length. For a ring or disk, symmetry often cancels some components before you even integrate, which is exactly what happened for the ring's field at its own center.
4. Electric Dipoles
An electric dipole — two equal and opposite charges separated by a small distance — is described by its dipole moment p=qd, a vector pointing from the negative to the positive charge. Placed in a uniform external field, a dipole feels zero net force (the forces on the two charges cancel) but a nonzero torque, τ=pE sinθ, that tries to rotate it into alignment with the field.
The potential energy of a dipole in a field is U=−pE cosθ, which is minimized (most stable) when the dipole is aligned with the field and maximized (least stable, and unstable to any perturbation) when anti-aligned — the same behavior a compass needle shows in a magnetic field, which is no coincidence.
Key equations
- F = kq₁q₂/r² — The force between two point charges — repulsive if q₁q₂>0, attractive if q₁q₂<0. Acts along the line joining them.
- E = F/q₀ — The definition of electric field — force per unit charge. Exists at a point in space whether or not a test charge is actually there.
- E = kq/r² — Field of a single point charge, directed radially away from a positive source (or toward a negative one).
- E = ∫ k dq/r² — Field of a continuous distribution — break it into point-charge pieces dq, write r in terms of one variable, and integrate. The specific setup (what dq is, how r depends on position) changes with the geometry.
- p = qd, τ = pE sinθ, U = -pE cosθ — Electric dipole in a uniform field: p is the dipole moment, τ is the torque trying to align p with E, and U is minimized (most stable) when p is parallel to E.