Magnetic Field & π: Uncovering the Mystery

[Amir H.Saba]

Hi
as you know according to ampere's law ∫B.dl=μ₀I
but why μ₀ that appears in Maxwell's equations is exactly 4π *10^-7 ?
for example in electric field ε₀ is 8.85 *10^-12
and μ₀ like ε₀ is a constant that is related to material properties and why this constant is a special number as π that B is special ?

 

[tiny-tim]

Welcome to PF!

Hi Amir! Welcome to PF!

From the PF Library on magnetic field …​

What is µ₀?

µ_o is the conversion factor between tesla ($T\ =\ N/A.m$) and amp-turns per metre ($A/m$): so it has units of $N/A^2$.

Why isn't µ_o = 1 N/A² (so that it needn't be mentioned)?

well, it would be , buuuut …

i] in SI units, a factor of 4π keeps cropping up! … so we multiply by 4π

ii] that would make the amp that current which in a pair of wires a metre apart would produce a force between them of 2 N/m …

which would make most electrical appliances run on micro-amps! …

so, for practical convenience only, we make µ_o 10⁷ smaller, and the amp 10⁷ larger!

(so the amp is that current which in a pair of wires a metre apart would produce a force between them of 2 10^-7 N/m, and µ_o is 4π 10^-7 N/A² (= 4π 10^-7 H/m))

(for historical details, see http://en.wikipedia.org/wiki/Magnetic_constant)​

(And the electric constant (permittivity of free space), $\varepsilon_o$, is defined as 1/µ₀c², = 10⁷/4πc² C²/Nm² (or F/m).)

 

[Amir H.Saba]

yes, your sentences is true ,but my query is that π is a number that related to sphere.but µo is a constant that related to environment.and why is a special number that has π ?
µo is exactly 4π 10^-7 but if it is a constant about environment , why it is exactly a special number ? I think it should related to microscopic vacuum properties ,if isn't it ,why for other environments µ is not 4π *10^-7 ? and is a other number? but for free space is a special number like π !
sorry for my english is not good.

 

[tiny-tim]

yes, your sentences is true ,but my query is that π is a number that related to sphere.but µo is a constant that related to environment.and why is a special number that has π ?

No, µ_o is not related to the environment.

µ_o is simply the conversion factor between tesla and amp-turns per metre.

(of course, µ (for a material) is related to the material)

The 4π is a standard factor in SI units, since it naturally relates the strength of a source to its flux (because the field from the source spread out over an area 4πr² instead of r²).