Thursday, 25 July 2019

electromagnetism - Why does the vacuum permeability have the value of pi in it?


The vacuum permeability, or the capability of the vacuum to permit magnetic field lines, contains the value of π. Why? What does this have to do with the ratio of a circle's circumference to its diameter?



Answer



This is nothing but a choice of units.


Let me make that (hopefully) more clear by explaining more about how choosing units in electromagnetism works:





  • Coulomb's Law is F=kq1q2r2ˆr for the force between two charges q1 and q2. k is different in the various systems of units - essentially, it depends on how the unit of charge is defined.




  • Ampere's Law is F=kI1I2C1C2dr1×(dr2×r)|r|3 for the force between two currents along C1 and C2. k is different in the various systems of units - essentially, it depends on how the unit of current is defined.




In electrodynamics, we find out that k/k=c2

otherwise, we are free to choose. In CGS, we define k=1, and in SI k=107VsAm.


Now we introduce constants μ0=4πk,ε0=14πk

That is nothing more than a definition - the factors of 4π simplify some formulas later on, e.g. when fields are integrated over the surface of a sphere.


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