Electric Charges, Fields, and Gauss's Law
Six topics on the force between charges and the field that carries it. Coulomb's law as a magnitude plus a direction read from the geometry, conservation of charge through friction, contact and induction, the electric field and what its lines do and do not mean, fields of continuous distributions by integration, electric flux through an oriented surface, and Gauss's law, which turns enough symmetry into a field in one line.
AP exam 15-25%1 topics
Topics
Key forms For every problem in this unit
Coulomb's law (magnitude)
F = k|q₁||q₂| / r², direction from the picture: unlike attract, like repel
Coulomb constant
k = 1 / (4πε₀) = 8.99 × 10⁹ N·m²/C². Use k OR ε₀, never both in one expression
Electric field
E = F / q₀ (force per unit POSITIVE test charge); F = qE
Point charge
E = kq / r², radially out from +q, in toward −q
Superposition
add the contributions as VECTORS, components first. A charge never contributes to the field acting on itself
Continuous distribution
dE = k dq / r², integrate WITH direction. r and the projection angle usually vary element to element
Charge densities
dq = λ dx (line), σ dA (surface), ρ dV (volume)
Elementary charge
e = 1.6 × 10⁻¹⁹ C. Free charge is an integer multiple of it
Electric flux
Φ = EA cosθ, with θ measured to the surface NORMAL. Surface parallel to E carries zero flux
Gauss's law
net flux through a CLOSED surface = q(enclosed) / ε₀
What E means there
the TOTAL field, outside sources included. Outside charge contributes zero NET flux, not zero field
When it hands you E
only when symmetry makes E constant in magnitude and normal to the surface: sphere, infinite cylinder, infinite plane
Uniform shell
inside: E = 0. outside: E = kQ / r², as if all the charge sat at the center
Uniform solid sphere (insulating)
inside: E = kQr / R³, rising linearly. outside: E = kQ / r²
Infinite sheet
E = σ / (2ε₀), the SAME at every distance
Just outside a conductor
E = σ / ε₀, twice the lone-sheet value, because the interior field is zero
Infinite line
E = 2kλ / r = λ / (2πε₀r)