Reference Library

Electric charge is an intrinsic property of matter that determines the electromagnetic force a particle exerts on other matter and experiences in return. Charge exists in two kinds, positive and negative: like charges repel, unlike charges attract. The total charge of an isolated system is conserved. Charge is quantized: every charge is an integer multiple of the elementary charge, q = ±Ne, e = 1.602×10⁻¹⁹ C.

The electrostatic force between two point charges acts along the line joining them, is proportional to the product of the charges, and falls as the inverse square of their separation.

Conductors are materials in which electrons move freely; insulators are materials in which they do not. A charged object held near a conductor redistributes its electrons, charging it by induction without contact. An insulator polarizes instead: its molecules shift without net charge transfer.

The electric field at a point is the electrostatic force per unit positive test charge placed there. It is defined at every point in space, whether or not a charge occupies that point.

A continuous distribution is divided into elements dq, each treated as a point charge. The total field is the vector integral of the element contributions. Charge density is specified per unit length (λ), area (σ) or volume (ρ).

Field lines begin on positive charge and end on negative charge. The tangent to a line gives the direction of E; the line density is proportional to its magnitude. Lines never intersect.

Electric flux is proportional to the number of field lines penetrating a surface. Only the field component normal to the surface contributes.

The net electric flux through any closed surface equals the charge enclosed divided by ε₀. The result is independent of the position of the enclosed charge and of the shape of the surface. Charge outside contributes zero net flux.

Where the distribution is sufficiently symmetric, Gauss's law gives E directly. The dimensionality of the source fixes the distance dependence: point 1/r², line 1/r, plane constant.

For a conductor in electrostatic equilibrium: the field is zero everywhere inside, solid or hollow; excess charge resides on the surface; the field immediately outside is normal to the surface with magnitude σ/ε₀; and on an irregular conductor the surface charge density is greatest where the radius of curvature is smallest.

Potential difference is the change in electric potential energy per unit charge between two points, equal to the negative line integral of E along any path between them. Potential is a scalar. An equipotential surface is a surface of constant V, everywhere perpendicular to the field lines.

In a uniform field the potential decreases linearly with distance along the field direction, and is constant perpendicular to it.

The potential of a point charge varies as 1/r, taking V = 0 at infinite separation. For several charges the potentials add algebraically, with sign.

Each component of the field is the negative derivative of the potential with respect to that coordinate. The field points in the direction of steepest decrease of V.

The potential of a continuous distribution is the scalar integral of dq/r over the distribution. No vector components are involved.

A conductor in electrostatic equilibrium is an equipotential volume: surface and interior are at the same potential, since E = 0 inside.

A capacitor consists of two conductors carrying charges of equal magnitude and opposite sign. Capacitance is the ratio of that charge to the potential difference between the conductors, and depends only on geometry and on the material filling the gap.

Standard procedure: assume a charge Q, obtain E from Gauss's law, integrate E across the gap to obtain ΔV, then form C = Q/ΔV.

Capacitors in parallel share the same potential difference and their charges add. Capacitors in series carry the same charge and their potential differences add.

Charging a capacitor transfers charge across a potential difference that grows as the charge accumulates. The stored energy is located in the electric field between the conductors.

Filling the gap with a dielectric increases the capacitance by the dimensionless factor κ, and raises the voltage the capacitor withstands before breakdown.

An electric dipole is two charges of equal magnitude and opposite sign separated by 2a. Its dipole moment has magnitude p = 2aq and is directed from the negative towards the positive charge.

Current is the rate at which charge passes through a cross-section of a conductor; current density is the current per unit area. The conventional direction of current is that of positive carrier motion, opposite to the electron drift in a metal.

Resistance is the ratio of the potential difference across a conductor to the current it carries. A material is ohmic if its current density is proportional to the applied field, so that resistance is independent of the applied voltage.

In the classical model, free electrons are treated as a gas: the field accelerates them between collisions with the lattice, and each collision randomizes their motion. The mean time between collisions determines both drift speed and resistivity.

Over a limited range, the resistivity of a metal varies approximately linearly with temperature.

The power delivered to a circuit element is the product of the current through it and the potential difference across it. In a resistor this energy appears as heat.

The emf of a source is the voltage across its terminals at zero current — its open-circuit voltage. When current flows, the internal resistance reduces the terminal voltage.

Resistors in series carry the same current and their potential differences add. Resistors in parallel share the same potential difference and their currents add.

For circuits that series and parallel reduction cannot handle. The junction rule expresses conservation of charge; the loop rule expresses conservation of energy.

With a capacitor in series with a resistor, charge and current vary exponentially in time. The product RC is the time constant τ of the circuit.

A magnetic field exerts a force on a charge only while that charge is moving through it, perpendicular to both the field and the velocity.

A charged particle entering a uniform field perpendicular to its velocity moves in a circle, with the magnetic force supplying the centripetal force.

A current-carrying wire in a magnetic field feels a force, since the moving charges composing the current each feel one.

A current loop in a magnetic field experiences a torque that tends to align its magnetic moment with the field.

Every small element of current contributes to the magnetic field at a point, in a pattern that plays the same role for magnetism that Coulomb's law plays for electricity.

Two current-carrying wires exert magnetic forces on each other, since each sits in the field the other creates.

Symmetric current distributions allow the magnetic field to be found without integrating Biot-Savart directly.

A tightly wound coil concentrates and straightens magnetic field lines into a nearly uniform field along its axis.

A changing magnetic flux through a circuit induces an emf around it, whether that change comes from a varying field, a changing area, or a rotating orientation.

A conducting rod moving through a magnetic field has an emf across it, since the field pushes its free charges along its length.

The induced current always flows in the direction that opposes the change in flux that produced it — never reinforcing it. This is a direct consequence of energy conservation: if the induced current reinforced the change instead, the effect would run away without bound, extracting energy from nothing. It is exactly what the minus sign in Faraday's law encodes.

A changing magnetic flux creates an electric field even in empty space, with no wire or circuit required.

A changing current in a coil changes its own magnetic flux, inducing a back-emf that opposes the change.

A resistor and inductor in series delay the rise of current when a source is switched on, on a timescale set by their ratio.

Building up current in an inductor stores energy, exactly as charging a capacitor stores energy — in the field itself rather than in the component.

A capacitor and inductor in series, with no resistance to dissipate energy, oscillate indefinitely.

An alternating source drives a voltage that varies sinusoidally with time; a rotating phasor's projection traces out that variation, letting AC analysis borrow vector-addition techniques.

In an AC circuit, resistors, inductors and capacitors each relate voltage to current differently — resistance alone is not enough to describe the circuit.

Combining R, L and C in series requires adding their voltage drops as phasors, since resistive and reactive drops are out of phase with each other.

A driven RLC circuit responds most strongly when driven at its own natural frequency; a transformer uses induction to step voltage up or down.

Ampère's law as originally stated fails for a charging capacitor; Maxwell's fix restored consistency and led directly to electromagnetic waves.

Four equations completely describe classical electromagnetism: Gauss's law relates electric flux to enclosed charge; Gauss's law for magnetism states that magnetic flux through any closed surface is always zero; Faraday's law relates a changing magnetic flux to an induced electric field; and the Ampère-Maxwell law relates current and changing electric flux to a magnetic field.

Combining Faraday's law with the Ampère-Maxwell law for oscillating fields in empty space yields a wave — and its predicted speed is exactly the speed of light.

An electromagnetic wave carries both energy and momentum, transporting power through space with no medium required.