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7.3: The Permeability of Free Space

Last updated

Mar 5, 2022
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- 7.2: Force Between Two Current-carrying Wires
- 7.4: Magnetic Moment

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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 $12.5664 \times 10^{−7} \text{T m A}^{−1}$ .

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