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Symbol Meaning Type of Object ~E Electic Field Vector field ~B Magnetic Field Vector field e0 permitivity constant constant µ0 Permeability constant constant c speed of light constant k Coulomb’s constant: σ Closed orientable surface C Curve simple closed piecewise smooth curve ~n Outward unit normal vector, normal to the surface σ vector-valued function ~T vector-valued function qenc Unit tangent vector, tangent to the curve C Charge enclosed by the surface ρ charge density charge per unit volume ~J current density vector vector-valued function: magnitude = current per unit area, direction = same or opposite the velocity of the moving charges, depending on their sign icondenc or ienc conduction current enclosed by the closed curve C electric flux ΦE ΦB RR constant magnetic flux dS σ R 1 4πe0 ds C Surface integral; integrate with respect to the surface area of σ Line integral; integrate with respect to arc length of C; Note that integrals H of the form ~F • ~Tds can also be exHC ~ where r(t) is the pressed as ~F • dr, C H C ds S ~ dA S ~ dA C ~ ds H vector equation of C Line integral over a closed curve ~ dA Physicist’s symbol for a surface integral over a closed surface Physicist’s symbol for a surface integral over a surface that is not closed. Physicist’s symbol for a line integral over a closed curve. “area vector” ~ ds “arc length vector” R H vector-valued function; this is the outward unit normal vector scaled so that its magnitude accounts for the area of the surface; it encompasses the ~ndS from the 3c formulas involving surface integrals vector-valued function; this is the unit tangent vector scaled so that its magnitude accounts for the arc length; it encompasses the ~Tds or the ~ from the 3c formulas involving dr line integrals Maxwell’s Equations 1. Gauss’ Law for Electricity Integral Form RR ~E • ~ndS = ΦE = |{z} σ 3c Differential Form qenc e0 5 • ~E = div(~E) = ρ e0 f lux ΦE = |{z} 4b H S ~ = ~E • dA qenc e0 f lux 2. Gauss’ law for magnetism 3c 4b Integral Form Differential Form RR ~B • ~ndS = 0 ΦB = 5 • ~B = div(~B) = 0 σ H ~ =0 ΦB = S ~B • dA 3. Faraday’s Law of Induction Ingegral Form H 3c ~E • ~Tds = C I H C ~ = − dΦB ~E • dr dt ~ = − dΦB = − d ~E • ds dt dt | {z } 4b Differential Form C R S ~ 5 × ~E = curl (~E) = − ∂∂tB ~ ~B • dA em f 4. Ampere/Maxwell’s Law Ingegral Form 3c 4b H C H ~B • ~Tds = H C ~ = µ0 e0 dΦE + µ0 ienc ~B • dr dt dΦE ~ ~ C B • ds = µ0 e0 dt + µ0 icondenc H R d ~ ~ ~ ~ C B • ds = µ0 e0 dt S E • dA + µ0 icondenc Differential Form 5 × ~B = curl (~B) = µ0 ienc + e0 µ0 = = ∂~E ∂t 4πk~ 1 ∂~E J+ 2 c2 c ∂t ~J 1 ∂~E + 2 e0 c 2 c ∂t Some formulas were taken from the following website: hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html Thanks to Kaz Tarui and Katherine Meyer-Canales for their help in developing this handout.