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7.3: The Permeability of Free Space

If each of the currents in the arrangement of Section 7.2 is one amp, and if the distance \(r\) between to two wires is one metre, and if the experiment is performed in a vacuum, so that  \(\mu\ = \mu_0\), then the force per unit length between the two wires is \(\mu_0/(2 \pi)\) newtons per metre. But we have already (in Chapter 6) defined the amp in such a manner that this force is 2 × 10−7 N m−1. Therefore it follows from our definition of the amp that the permeability of free space, by definition, has a value of exactly

\[\mu_0=4\pi \times 10^{-7}\text{ T m A}^{-1},\label{7.3.1}\]

or, as we shall learn to express it in a later chapter, \(4 \pi \times 10^{-7}\) henrys per metre, \(\text{H m}^{-1}\) .

It was mentioned briefly in Chapters 1 and 6 that there is a proposal , likely to become official in 2018, to re-define the coulomb (and hence the amp) in such a manner that the magnitude of the charge on a single electron is exactly \(1.60217 \times 10^{−19} \text{C}\). If this proposal is passed (as is likely), \(\mu_0\) will no longer have a defined value, but will have a measured value of approximately \(12.5664 \times 10^{−7} \text{T m A}^{−1} \).