Electric Charges & Fields

Force between two point charges q₁ and q₂ separated by distance r in vacuum. k = 9×10⁹ N m² C⁻².

Force on charge q₂ due to q₁. r̂₁₂ is the unit vector from q₁ to q₂. Positive F₁₂ means repulsion.

Net force on a charge q due to n other charges is the vector sum of individual Coulomb forces. Each pair interacts independently.

Electric field at a point is the force per unit positive test charge. The limit ensures the test charge does not disturb the source distribution.

Electric field at distance r from a point charge q. Points radially outward for positive q, inward for negative q.

Net field at a point due to a system of charges is the vector sum of fields due to individual charges.

Dipole moment of a pair of charges ±q separated by distance d. Direction: from −q to +q. SI unit: C·m.

Field at distance r from centre along the dipole axis. Direction is along p̂. The approximation holds when r ≫ a (half-length of dipole).

Field at distance r from centre on the equatorial plane. Direction is antiparallel to p̂. Magnitude is half of axial field at same r (far field).

Torque on a dipole with moment p in uniform field E, where θ is the angle between p and E. Torque tends to align p along E.

Potential energy of a dipole in uniform field E. Minimum at θ = 0 (stable equilibrium), maximum at θ = π (unstable equilibrium).

Work done by external agent rotating dipole from angle θ₁ to θ₂ in a uniform electric field E.

Electric flux through a surface. For a uniform field and flat surface: Φ = EA cosθ. SI unit: N m² C⁻¹ (or V·m).

Total electric flux through any closed surface equals the net enclosed charge divided by ε₀. Equivalent to Coulomb's law for static fields.

Field at perpendicular distance r from an infinite line charge with linear charge density λ. Direction: radially outward from the line.

Field due to an infinite plane sheet with surface charge density σ. Uniform and independent of distance. Direction: normal to the sheet.

For two infinite sheets with charge densities +σ and −σ: fields add between the plates and cancel outside. Basis of the parallel plate capacitor.

Field at r > R due to a shell of radius R carrying total charge Q. Behaves as if all charge is concentrated at the centre.

Electric field inside a uniformly charged spherical shell is exactly zero. A direct consequence of Gauss's law: no enclosed charge.

Field outside a uniformly charged solid sphere of radius R and total charge Q. Identical in form to a point charge at the centre.

Field inside a uniformly charged solid sphere grows linearly with r. ρ is the volume charge density. Field is maximum at the surface.

Field at axial distance x from the centre of a ring of radius R and total charge Q. Maximum at x = R/√2. Zero at centre.

Field at axial distance x from centre of a disc of radius R and surface charge density σ. Reduces to infinite plane result (σ/2ε₀) as R→∞.

All observable charges are integer multiples of the elementary charge e. Fractional charges (quarks) are confined and never observed freely.

Charge densities for distributed charges. λ in C/m, σ in C/m², ρ in C/m³. Used to set up field integrals for continuous distributions.