Hi as you know according to ampere's law ∫B.dl=μ_{0}I but why μ_{0} that appears in Maxwell's equations is exactly 4π *10^{-7} ? for example in electric field ε_{0} is 8.85 *10^{-12} and μ_{0} like ε_{0} is a constant that is related to material properties and why this constant is a special number as π that B is special ?
Welcome to PF! Hi Amir! Welcome to PF! From the PF Library on magnetic field … What is µ_{0}? µ_{o} is the conversion factor between tesla ([itex]T\ =\ N/A.m[/itex]) and amp-turns per metre ([itex]A/m[/itex]): so it has units of [itex]N/A^2[/itex]. Why isn't µ_{o} = 1 N/A^{2} (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! :yuck: … so, for practical convenience only, we make µ_{o} 10^{7} smaller, and the amp 10^{7} 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/A^{2} (= 4π 10^{-7} H/m)) (for historical details, see http://en.wikipedia.org/wiki/Magnetic_constant) (And the electric constant (permittivity of free space), [itex]\varepsilon_o[/itex], is defined as 1/µ_{0}c², = 10^{7}/4πc² C²/Nm² (or F/m).)
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.
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 4πr^{2} instead of r^{2}).