# What was defined first and how? The ampere or the vacuum permeability?

I've been looking up the history and evolution of the seven base units and am currently checking out the ampere. What I've found is that 1A is defined as the current in a wire which would experience a force equivalent to this formula:

$$F=\frac{\mu_0 I_1 I_2l}{2\pi r}$$ which is $$2\times 10^{-7}$$N when all other values are 1.

But since this definition is dependent on vacuum permeability, I tried looking for how the value of this was defined and either came across the same formula or Ampere's Circuital Law... which also has current in it.

I understand the force can be measured with a balance or a weight, but currents are measured with an ammeter, which if I'm not wrong, is a modified/evolved galvanometer, which is also calibrated based on weight balances(?) upon the force exerted by a current carrying wire, so it's all starting to feel like an ouroboros to me.

Any help? My knowledgebase is basically at the level of a Cambridge A Levels graduate, with a bit more due to compulsory courses in university.

• Maybe HSMSE is better suited for history questions: hsm.stackexchange.com Commented Jan 12 at 11:51
• @Mauricio Oh, I'll give it a try then. Is there a way to crosspost or do I need to copy it all? Commented Jan 12 at 12:15
• You can just copy it all and add a note to say that you also posted it here. Commented Jan 12 at 13:24
• Link to the crosspost in HSM: hsm.stackexchange.com/q/16113 Commented Jan 12 at 14:22

Although the historical part of this question does belong on the history of science and mathematics site, there is also a portion of this question that is a question of physics. Specifically, there is a frequent confusion amongst students regarding the distinction between a quantity and a unit. This is particularly prevalent with the SI units. For some reason, students may view the SI units as more than merely a very popular set of unit conventions.

For your specific question, the ampere is a unit of current. It is not current. Currents can be measured by a galvanometer. In experiments with wires and galvanometers you can establish that $$F \propto \frac{I_1 I_2 l}{r}$$ without ever using SI units. You can simply note the deflection of the galvanometer without ever labeling any specific amount of deflection as "1 ampere".

The physics is contained in this proportionality (which contains neither any specific unit of current nor the vacuum permeability). Eventually it may be useful to measure currents in other types of experiments and to relate currents measured in different laboratories. For that purpose, it is convenient to have a standard current that could be replicated in different contexts and locations. For that we can choose to use a system of units, the SI, such that the unit current, the ampere, is defined as a specific number of elementary charges passing in a fixed amount of time.

In SI units the proportionality holds, and we can write the constant of proportionality as $$\mu_0/2\pi$$. It is then a matter of experiment to determine the value of $$\mu_0$$ when the currents, distances, and force are all measured in SI units.

• My question might've been misunderstood. Hopefully my comment on Alfred's answer clarifies things. Commented Jan 17 at 21:54
• @SpectraXCD yes, I answered that. It is the last paragraph of my answer.
– Dale
Commented Jan 17 at 22:19

Not so long ago, the permeability of vacuum was defined as exactly $$4 \pi 10^{-7}$$. And then your formula was an experimental definition of the Ampere.
But recently, that changed. The charge of the electron was measured with a precision better than the inevitable error in measuring the force and the distance between two wires. So the definition of the Coulomb was changed to a fixed number of electron charges, exactly 6.241 509 629 152 65 × $$10^{18}$$. And one Ampère is one Coulomb per second. From that point on, the formula is now an experimental definition of the permeability of vacuum which is not anymore exactly $$4 \pi 10^{-7}$$. Of course the values that can be measured are extremely close to that value, but in principle it is now an experimental definition, not an a priori one as it was before.
So during my entire career as a physicist, the permeability of vacuum was exactly $$4 \pi 10^{-7}$$ since I retired in 2016...
• This answer does not explain the origin of $\mu_0$. Commented Jan 15 at 8:55
• @Mauricio I don't understand your remark. You mean, the value of $\mu_0$ ? Certainly if the unit of charge is defined in terms of the charge of the electron, units of distance and force are given, the formula above defines experimentally this value. And it will not be exactly $4 \pi 10^{-7}$ But if you mean "the origin" in a metaphysical sense, then I am not the one who created the world... Commented Jan 15 at 14:39
• By origin, they mean how $\mu_0$ was first defined. While we all know there is nothing special about our chosen set of units, there is undoubtedly some form of convenience to them that led to their creation. For example, 'the meter' is currently described as the distance travelled by light in 1/299792458th of a second. Before that, it was the length of the prototype meter. Before that it was 1/10Mth of the distance from equator to north pole. And before that, it was the length of the pendulum taking a second to swing from one extreme to the other. And there lies the origin, the 'convenience'. Commented Jan 17 at 21:35