How is the constant of the Biot-Savart Law derived?

In my A-level textbook there is no explanation regarding how the constant in the Biot-Savart law is derived!

So how is the constant, $k =\frac {\mu_0}{4\pi}$ derived, and what's the intuition behind this derivation?

• I don't think this is a duplicate. This one asks about the constant; the proposed dupe is broader and regards the law itself and its place within the broader electromagnetic theory. – Emilio Pisanty Aug 3 '16 at 13:23
• I agree with Emilio Pisanty. The duplicate (67445) does not answer the present question. – sammy gerbil Aug 3 '16 at 13:47
• I don't think my question is dupe. I didn't know Maxwell's equation or how to derive the biot-savart law from it. That's why I asked it! So please unflag it from dupe. – Mockingbird Aug 3 '16 at 13:51
• Consider it history, dupe reference is gone, my sincere apologies. – user108787 Aug 3 '16 at 14:00
• I retracted my close vote; sorry about that! – heather Aug 3 '16 at 14:35

The constant in the Biot-Savart law isn't really derived from anything - it is essentially defined with a fixed value, which then serves as a definition of the SI ampere.

The Biot-Savart law for a current loop $C$, $$\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\int_C \frac{I\mathrm d\mathbf l'\times(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3},$$ is not particularly useful on its own, and you need to couple it with the Lorentz force to give you a measureable consequence. In particular, the force on a second current loop $C'$ is then $$\mathbf F_{C'} = \int_{C'} I'\mathrm d\mathbf l'\times\mathbf B(\mathbf r') = \frac{\mu_0}{4\pi}\int_{C'} I'\mathrm d\mathbf l'\times\int_C \frac{I\mathrm d\mathbf l\times(\mathbf r'-\mathbf r)}{|\mathbf r'-\mathbf r|^3}.$$ This force - the force on a current loop $C'$ carrying a current $I'$ caused by a current $I$ in a loop $C$ - depends on two factors: the geometry, and the values of the currents. Happily, these two factors separate completely: $$\mathbf F_{C'} = \frac{\mu_0}{4\pi} I'I\int_{C'}\int_C \frac{\mathrm d\mathbf l'\times\left(\mathrm d\mathbf l\times(\mathbf r'-\mathbf r)\right)}{|\mathbf r'-\mathbf r|^3} =\frac{\mu_0}{4\pi}I'I \times \boldsymbol{\mathcal G}_{C'C}.$$ Here the geometrical factor $\boldsymbol{\mathcal G}_{C'C}$ is the same regardless of the currents, and it is only a function of the loops. The implication is then that we can fix the constant for a single geometry, a simple case that's easy to analyze and to implement, and then this will work for all geometries. In particular, the reference geometry is two infinite wires a distance $L$ apart, in which case the force on a unit length $\Delta l$ of each wire is $$\frac{\Delta \mathbf F}{\Delta l} = \frac{\mu_0 I'I}{2\pi L}\hat{\mathbf u}.$$

This is where the definition of the ampere comes in: we fix the value of $\mu_0$ at $4\pi\times10^{-7} \:\mathrm{N/A^2}$, and this lets us define the ampere as

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per metre of length.

There are many ways to define electromagnetic units but at some point you are going to need to make some new definition, and fix a standard for measuring electric charge. Generally, this can be done either through a charge standard (essentially by fixing the constant on Coulomb's law) or through a current standard as above. The SI chose the latter because it allows for nicer metrological properties of the resulting system: it makes it easier to produce measurement systems which are consistently accurate everywhere.

That said, the form $\mu_0/4\pi$ definitely looks very suspicious, as does the value of the vacuum magnetic permeability, $\mu_0=4\pi\times10^{-7} \:\mathrm{N/A^2}$; surely those came from somewhere, right? The cleanest answer is that, since they're definitions, we can define whatever we want, and we only need to justify them later (if at all) by showing they're convenient choices.

Of course, the SI system of electromagnetic units has a long and storied history, which is where those values came from, but that history isn't actually relevant at the moment: the SI constants like $\mu_0/4\pi$ have the form they have because it makes the 'real' equations of electromagnetism (and, in particular, Maxwell's equations) have a cleaner form. Similarly, the numeric values of the constants (e.g. why $4\pi\times10^{-7}$ instead of just $4\pi$?) were chosen because they strike a good middle point for most everyday electrical measurements to have reasonable values.

And, as a final word: the above is the situation as of this writing, but the SI standards are being redefined, with the new system probably coming into effect around 2018. In the new system, we will change from fixing $\mu_0$ to fixing the value of the charge of an electron at roughly $$e=1.602\,176\,487\,186\times 10^{-19}\:\mathrm C,$$ or more specifically to whatever specific numerical value is most accurate according to CODATA at the time of the transition, much like what was done with the meter when the speed of light was fixed to its current exact value in 1983.

The redefinition of the ampere and the coulomb will have some pretty far-reaching consequences in terms of how the standards are implemented (which you can read more about in this answer of mine), and it also changes what happens to the constant in the Biot-Savart law. In particular, $\mu_0$ goes from having an exact, fixed value, to being experimentally determined, with the value $$\mu_0=\frac{2h}{ce^2}\alpha.$$ Here $h$ is Planck's constant, $c$ is the speed of light, and $e$ is the electron charge, all of which have fixed values. The uncertainty comes in $\alpha$, the fine structure constant, which is a dimensionless natural constant that must be experimentally measured, and which roughly speaking measures how strong the electromagnetic coupling is (in the form of the Coulomb constant $e^2/4\pi\varepsilon_0$) compared to the quantum-relativistic standard measure of coupling strength, $\hbar c$.

This looks like an even more complicated way to define the constant in the Biot-Savart law, but it's for a reason: as a result of the long history of electrical measurement standards, and to remain backwards compatible with over a century of electrical measurements and technology, while still allowing for the best of cutting-edge technology in precision measurements to go forward and enable better, more precise, more stable, and more consistent scientific measurements going forward.