Electro

Permittivity of free space (ε0)

8.854×10⁻¹² C²N⁻¹m⁻² — represents the resistance encountered forming an E-field in vacuum

Electric field (E) definition

Force per unit charge; units N/C or V/m

Equipotential lines

Lines/surfaces of constant potential V. The E-field is always perpendicular to them. Work moving a charge along one is ZERO

Electric potential φ (or V)

Work done per unit charge moving a test charge from infinity to a point. E = −∇φ, with V(∞)=0 by convention

Total charge from volume charge density

Q = ∭ρ dτ, where ρ is volume charge density (C/m³)

Total charge from surface charge density

Q = ∬σ da, where σ is surface charge density (C/m²). For a homogeneous disk: Q = σπR²

Electric field, superposition of point charges

E = (1/4πε0)·Σ(qi/ri²)·r̂i

Electric flux (ΦE)

A measure of the number of field lines passing through a surface

Electric flux formula

ΦE = ∬E·da. For a uniform field: ΦE = EA·cosθ

Gauss's Law (integral form)

∮E·da = Q(enclosed)/ε0 — depends ONLY on the net enclosed charge, never on the surface's shape or size

Gauss's Law (differential form)

∇·E = ρ/ε0

Electric dipole moment (p)

p = qd, the product of charge magnitude q and separation distance d. Units: C·m

Electrostatic energy — discrete charges

U = ½ Σ(i≠j) (1/4πε0)(qiqj/rij)

Electrostatic energy — continuous charge distribution

U = ½∫ρφ dV

Magnetic field (B) / magnetic flux density

Units: Tesla (T)

Current density (j)

Current per unit area. Units: A/m²

Vacuum permeability (μ0)

4π×10⁻⁷ N/A² — constant for vacuum's ability to support magnetic fields

Ampère's Law (integral form)

∮B·dl = μ0·I(enclosed)

Ampère's Law (differential form)

∇×B = μ0·j

Vector potential (A)

B = ∇×A, using the Coulomb gauge ∇·A = 0

Magnetic flux (ΦB)

Total magnetic field through an area. Units: Weber (Wb). ΦB = ∬B·dA

EMF (electromotive force)

NOT a force — the work done per unit charge to maintain current. Units: Volts. EMF = ∮E·dl

Faraday's Law

EMF = −dΦB/dt

Lenz's Law

Induced current flows so as to oppose the change in flux that created it — the minus sign in Faraday's law

Displacement current

Maxwell's correction to Ampère's Law: a changing E-field also generates a B-field. I_d = ε0·(dΦE/dt)

Ampère-Maxwell Law

∇×B = μ0j + μ0ε0(∂E/∂t)