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- Electrodynamics - ED
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Coming up... 1 Maxwell’s equations in vacuum 2 Maxwell’s equations inside matter

In the language of differential vector calculus Gauss’s law: ~ = ρ ∇·E 0 (1) Gauss’s law for magnetism: ~ =0 ∇·B (2) ~ ~ = − ∂B ∇×E ∂t (3) Maxwell-Faraday equation Ampere’s law, with Maxwell’s correction ~ = µ0 ∇×B ~ ~J + 0 ∂ E ∂t ! We shall now look at interpretations of these expressions by using their integral forms. (4)

Gauss’s law: enclosed charges ~ = ρ/0 : ∇·E Integrate over a closed volume: Z Z ~ (∇ · E)dV = V V ρ dV 0 Use a mathematical identity (Gauss’s theorem) I ~ = Qenclosed ~ · dS E 0 (5) (6) Relationship between electric field on a closed surface and the charge enclosed inside it The part in red: source of the electric field ~ Leads to Coulomb’s law if Q is a point charge at the centre of S, a sphere of radius r : Er · 4πr 2 = Q/0

Gauss’s law: no magnetic monopoles ~ =0: ∇·B Integrate over a closed volume: Z ~ (∇ · B)dV =0 (7) V Use a mathematical identity (Gauss’s theorem) I ~ =0 ~ · dS B (8) Relationship between magnetic field on a closed surface and the magnetic charge enclosed inside it The part in red: source of the magnetic field. Vanishing of the source ⇒ no magnetic monopoles

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