Daniel Berwick Evans. Maxwell’s equations and classical electromagentism. Fields: E, B: time dependent vector fields on R^{3}. ÷E = ρ/ϵ_{0}, curl E = −∂B/∂t, ÷B = 0, c^{2} curl B =

j/ϵ_{0} + ∂E/∂t. ρ is “charge density” (coulombs per cubic meter), j is “current” (coulombs per second per square meter).

Gauß’ law is an application of Stokes’ theorem: q/ϵ_{0} = ∫_{U} ρ/ϵ_{0} = ∫_{U} ÷E = ∫_{∂U} E · n. Ampere’s law: − ∫_{D} ∂B/∂t = −dΦ_{B}/dt = ∫_{D} curl E = ∫_{∂D} E · ds. No magenetic charges (monopoles): curl B = 0. If ρ = j = 0 then we get a symmetry between E and B. Charge conservation: ÷j/ϵ_{0} = − ÷ (∂E/∂t) = −(∂/∂t) ÷ E = −(∂/∂t)ρ/ϵ_{0}, hence ÷j = −∂ρ/∂t. Static examples: ∂B/∂t = ∂E/∂t = 0. Point charge: If ρ = δ(x) and j = 0, then E = ±(4πϵ_{0})^{−1}r/|r|^{3} and B = 0. Wave solutions: ρ = j = 0. E = (0, E_{y}, E_{z}), B = (0, B_{y}, B_{z}). E_{y} = f(x − ct) + g(x + ct), E_{z} =

F (x − ct) + G(x + ct), cB_{z} = f(x − ct) − g(x + ct), cB_{y} = −F (x − ct) + G(x + ct). Wave with speed of propogation c.

Solenoid solution: E = 0 outside some cylinder, ρ = 0 everywhere, B is pointing upwards, j goes around the cylinder.

∫ B · ds = |B|h = j/(ϵ_{0}c^{2}) = jnh/(ϵ_{0}c^{2}). |B| = jn/(ϵ_{0}c^{2}). n is the number of turns per length.