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).

 

[sardar]

Hello, thank you for bringing up this interesting question about the value of μ0 in Maxwell's equations. The value of μ0, also known as the permeability of free space, is indeed a special number just like π, and it has a significant role in understanding electromagnetism.

To understand why μ0 has a value of exactly 4π *10-7, we need to look at the history of how it was discovered. In the 19th century, scientists were trying to understand the relationship between electric and magnetic fields. One of the key experiments was conducted by French physicist André-Marie Ampère, who discovered that the magnetic field around a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire. This relationship is now known as Ampère's law, which you mentioned in your question.

However, to fully understand the relationship between electric and magnetic fields, it was necessary to introduce a constant that would make the units of electric current and magnetic field consistent. This constant was first proposed by British physicist James Clerk Maxwell, who derived a set of equations that unified electricity and magnetism into what we now know as Maxwell's equations. Through his calculations, Maxwell found that the value of μ0 should be exactly 4π *10-7.

So why is this value special? One explanation is that it is related to the fundamental properties of space itself. Just like how the speed of light, c, is a fundamental constant in the universe, μ0 is also a fundamental constant that describes the properties of space. It represents the ability of space to support magnetic fields, just like how ε0 represents the ability of space to support electric fields.

In conclusion, the value of μ0 is a result of the interplay between theoretical insights and experimental observations. It is a fundamental constant that helps us understand the relationship between electric and magnetic fields, and its value of 4π *10-7 is a reflection of the underlying properties of space. I hope this explanation helps to uncover the mystery of μ0 for you.