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Converting E&M Formulae from MKS To Converting E&M formulae from MKS to cgs Dana Longcope 1/18/17 Maxwell’s equations assume famously different forms in the two systems of units. MKS cgs ∂ ∂ Faraday Bmks = Emks Bcgs = c Ecgs ∂t −∇× ∂t − ∇× 4π Amp`ere Bmks = µ0Jmks Bcgs = Jcgs ∇× ∇× c 1 Gauss Emks = ρmks Ecgs = 4πρcgs ∇ · ǫ0 ∇ · Each formula on the left (MKS) can be converted to the corresponding formula on the right (cgs) using the substitutions in the left column of the following table convert cgs/MKS factor note 4 4 Bmks Bcgs 10 Tesla = 10 Gauss → 7 −7 µ0 4π 10 µ0 = 4π 10 → × 1 −11 2 ǫ0 10 ǫ0µ0 = 1/c → 4πc2 6 Emks c Ecgs 10 from Faraday → 1 −5 Jmks Jcgs 10 from Amp`ere → c 1 −7 ∂ρ ρmks ρcgs 10 from Gauss and/or = J → c ∂t −∇ · 1 −1 qmks qcgs 10 (qE)mks (qE)cgs → c → 1 −1 Imks Icgs 10 I = J da → c · Z 2 11 ηmks c ηcgs 10 Ohm’s law: E = η J → The first three are fundamental relations between the unit systems. The next three are obtained using the equations listed, and the last three are related variables. These same substitutions can be used to convert any E&M formula or expression from MKS to cgs. Some common examples are v 1 (q v B) q B , (J B) J B mks c mks c × → × cgs × → × cgs q1q2 B B (q1q2)cgs , 4πǫ → µ ρ → √4πρ 0 mks √ 0 !mks cgs 1 1 ǫ 1 B2 B2 , 0 E2 E2 2µ0 mks → 8π cgs 2 mks → 8π cgs 1 2 2 2 q n 4πq n qB qB ω = = , Ωc = = p ǫ m m m mc 0 !mks !cgs mks cgs In the last row the substitutions yield equalities since the unit of time is the same in both systems. The constants µ0 and and ǫ0 go over to combinations of c and 4π, making the inverse conversion, from cgs to MKS, more problematic: when does 4π become µ0, and when is it just a number that does not change? The “factor” in the table is the ratio of the cgs expression to the corresponding MKS expression. It is a power of 10 accounting for the changes from meters to centimeters and kilograms to grams. The scaling of Faraday’s law, for instance, in each system can be written B E B c E mks mks , cgs cgs ts ∝ ℓm ts ∝ ℓcm where ℓm is the value of some length scale expressed in meters and ℓcm is the same length scale, but expressed in centimeters. The dimensionless constant of proportionality depends on the structure of the functions and is the same for both versions. That constant is eliminated by taking the ratio of the two expressions to obtain the factor listed in the table c E ℓ B cgs = cm cgs = 102 104 = 106 . (1) Emks ℓm · Bmks · Similarly, the scaling of Amp`ere’s law yields Bmks Bcgs 4π Jcgs µ0 Jmks , . ℓm ∝ ℓcm ∝ c Taking the ratio of these two expressions we find the corresponding factor J /c ℓ µ B − − − cgs = m 0 cgs = 10 2 10 7 104 = 10 5 . (2) Jmks ℓcm · 4π · Bmks · · These factors are used for unit-conversions : cgs unit : MKS unit = conversion 4 Bcgs : Gauss Bmks : Tesla = 10 Gauss 6 −5 Ecgs : statvolt/cm Emks : V/m = 10 /c = 3.33 10 statvolt/cm × 2 2 −5 5 2 Jcgs : statamp/cm Jmks : A/m = c 10 = 3 10 statamp/cm × 8 Φcgs = B da : Maxwell Φmks : Weber = 10 Mx · Z 8 −3 Vcgs = E dl : statvolt Vmks : Volt = 10 /c = 3.33 10 statvolt · × Z 9 qcgs : statcoulomb qmks : Coulomb = c/10 = 3 10 statcoulomb × 9 Icgs : statamp Imks :Amp = c/10 = 3 10 statamp × 2.
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