On Wikipedia it says:

This force is used in the formal definition of the ampere, which states that it is "the constant current that will produce an attractive force of $2 × 10^{-7}$ newton per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum."

In reference to the definition of an Ampere, why was $2 × 10^{–7}$ chosen?


The definition of Ampere is obtained by the below equation of force between two infinitely long parallel current carrying conductors.
enter image description here

Where $F$ is force, $\triangle{L}$ is small length element, $\mu_0$ is absolute permeability of vaccum or free space, $I_1, I_2$ are current flowing through two conductors.

By calculation we can obtain that $\frac{\mu_0}{4\pi}=10^{-7} T A^{-1} m$

When $\triangle{L}=1m, I_1=I_2=1A, r=1m$. By substituting the values in the above equation given in the figure, we obtain $\frac{F}{\triangle{L}}$ to be equal to $2X10 ^{-7}N m{-1}$.

Thus we have the definition of one ampere as: One ampere is that current, which when flowing through each of the two parallel conductors of infinite length and placed in free space of one metre from each other, produces between them a force of $2X10^{-7}$ newton per metre of their lengths.

  • $\begingroup$ I think the permeability constant was based on the definition of Ampere... $\endgroup$ – user24082 Mar 5 '14 at 7:40
  • $\begingroup$ Yes, ampere defines vaccum permeability. See here. I hope this cleared your doubts. $\endgroup$ – Immortal Player Mar 5 '14 at 12:57
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    $\begingroup$ I don't see how this answers the question i.e. "why was a factor of $2 \times 10^{-7}$ used?". $\endgroup$ – John Rennie Mar 5 '14 at 18:15
  • $\begingroup$ @JohnRennie: Thank you for the comment. I answered the question considering definition of ampere. Considering comments to answer "why was a factor $2$ X $10^{-7}$ used?" I think it is not used, it is obtained from measurement. If one coulomb of charge per unit time is made to flow through both the parallel conductors separated by distance 1m, force between them is "measured" to be $2$ X $10{-7}$ N. If any thing wrong please let me know. $\endgroup$ – Immortal Player Mar 5 '14 at 19:50

The ampere is used, among other things, to derive the volt, which is the electric potential difference that a current of one ampere has to flow across in order to deliver a power of one watt. (The watt itself is defined mechanically, from second, meter and kilogram).

One volt is around the same order of magnitude as the potential differences commonly encountered in electrochemistry -- one cell of a chemical battery typically delivers around one volt. This is not coincidental; the size of a volt was explicitly selected to make this true around 1880 when chemical batteries were the main source of electricity for laboratory work and telecommunication.

Back when the volt was first defined, it was based on centimeter-gram-second units rather than the meter-kilogram-second system that would become SI. But even then, the definition involved an "arbitrary" power of $10$ selected to make the unit come out conveniently for laboratory voltages.

The factor of $2$ is there because then the scale factor in the Biot-Savart law, the basic law of magnetostatics, becomes an exact power of $10$. (This factor is written as $\frac{\mu_0}{4\pi}$ in modern contexts where electrodynamics is the fundamental theory, which is why the exact numeric magnitude of $\mu_0$ in SI units is $4\pi\times 10^{-7}$).


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