Abstract
The rise of dimensional analysis in the latter part of the nineteenth century occurred largely in the context of electromagnetism. It soon appeared that the subject, albeit seemingly straightforward, was in fact wrought with difficulties. These revealed deep conceptual issues regarding the character of physical quantities. Usually, whether or not these problems actually constituted inconsistencies was itself a matter of debate. In one instance, however, regarding the electrostatic dimensions of the magnetic pole, all protagonists agreed that the matter required attention. A controversy ensued in 1882. Its resolution partly relied on the realization that it arose from differences between the scientific cultures prevalent on the Continent versus in Great Britain. These cultural differences concerned the possible relevance of the medium in which interactions involving magnetic poles take place, as well as the understanding of permeability in Ampère’s model of magnetism. The controversy around the electrostatic dimensions of the magnetic pole entailed crucial issues that were soon to play just as central a part in a wider debate about the dimensions of electromagnetic quantities in different systems of units. Why the latter topic was never raised during the 1882 controversy provides insights into the early understanding of dimensional analysis.
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Notes
SI stands for “Système International d’Unités.” The others are the units of: time (the second), mass (the kilogram), temperature (the kelvin), electric current (the ampere), light intensity (the candela) and amount of substance (the mole).
As if to highlight the arbitrary character of its definition, the meter went through several: originally, in 1793, it was defined as one ten-millionth of the distance from the Earth’s equator to its North Pole; then in 1889 it became the length of a prototype bar, and in 1960 the 11th “Conférence Générale des Poids et Mesures” (CGPM) redefined this as 1,650,763.73 times the wavelength of the emission line of krypton-86. The current definition has been in use since 1983, since it was decided at the 17th CGPM.
For example, Wilhelm Weber defined the unit of e.m.f. as follows: “As an absolute unit of measure of electromotive force, may be understood that electromotive force which the unit of measure of the earth’s magnetism exerts upon a closed conductor, if the latter is so turned that the area of its projection on a plane normal to the direction of the earth’s magnetism increases or decreases during the unit of time by the unit of surface” (Weber and Wilhelm Eduard 1861, p. 227).
That is, a numerical value and the unit, such as “4 kg”.
That is, taking a single derivative with respect to a quantity yields the same dimensions as dividing by it, and taking the double derivative, the same dimensions as dividing by its square.
Note that this is trivially the case in the examples given above: there are only two terms, one on each side of the equal sign, and both have the same dimensions.
It is by no means the only one however. Notably, in quantum field theory dimensional analysis is used to determine what terms can rightfully appear in a theory. See footnote 8 below for more details.
These relations are:
- \(L = ct\), where L is the distance traveled by a photon, t its time of travel and c the speed of light in vacuum.
- \(E = h f\), where E is the energy of a photon of frequency f, and h is Planck’s constant.
- E = mc \(^{2}\), where E is the energy of a particle of mass m (and c still the speed of light).
Both are also given the numerical value of 1. Since \(L = ct, [L] = \hbox {L}\) implies \([t] = \hbox {L}\). Since \(E = h f\) and f is as always given by inverse of a time, this in turn implies \([\hbox {E}] = \hbox {L}^{-1}\). \(E = mc^{2}\) then implies \([\hbox {M}] = [\hbox {E}] = \hbox {L}^{-1}\). Because particle physics deals more with energies than lengths, it has become customary to use M rather than L as the base quantity: \([L] = [t] = \hbox {M}^{-1}\).
An important application of this is the following. The action, which is the time integral of an energy, is dimensionless: \(\hbox {M}^{-1} \hbox {M} = \hbox {M}^{0}\). The Lagrangian density \({\mathcal {L}}\) is related to the action S by: \(S=\int {\hbox {d}^{4}} x{\mathcal {L}}\), where the integral is over spacetime coordinates. Since both time and space have dimension \(\hbox {M}^{-1}\), each term in \({\mathcal {L}}\) must have dimension \(\hbox {M}^{4}\). Taking a scalar theory as an example, that is, a theory where the field, \(\phi \), is a scalar quantity, one of these terms is \(\frac{1}{2}( {\partial _t \varphi })^{2}\), which implies that \([\varphi ] = \hbox {M}\). Knowing this puts constraints on what form the other terms in \({\mathcal {L}}\) can take, i.e., what functions of the field they can be, since they all are required to have dimension \(\hbox {M}^{4}\).
See Roche (1998, pp. 197–202). The period was characterized by a preoccupation for standards: the metric system endured after the Napoleonic wars and motivated several German states to organize reforms of weights and measures. To this end, they enlisted the help of scientists, notably Gauss himself who in 1836 was put in charge of such a program in Hanover. Defining new standards of measurement called for precision instruments, and also the respect of well-defined procedures, both when duplicating material standards and when actually making measurements. See notably Olesko (1995) and Olesko (1996).
See notably Everett (1879), chapters X and XI. An analogous description in terms of units rather than dimensions ran as follows: “In the electrostatic system the most important unit, which serves as the basis of all the others, is the unit of electricity. This is determined by the following definition: The unit of electricity is that amount of electricity which exerts the unit of force upon an equal amount of electricity at the unit of distance” (Clausius 1882, p. 383).
He naturally attributed to \(r^{2}\) the dimensions L\(^{2}\).
Maxwell (1873, Vol. 2, chapter 1, § 373). Magnetism was attributed to microscopic electric currents. In 1882 (the time when the controversy of interest took place) Lodge believed “universally accepted” that “the properties of magnetic substances of all kinds [were] explained by molecular electric currents, and no magnets or magnetic substances other than those consisting of current-conveying molecules exist.” (Lodge 1882, p. 363)
Again, this can be expressed in terms of units in the following way:
“The unit of magnetism is that amount of magnetism which exerts upon an equal amount of magnetism at the unit of distance the unit of force.” (Clausius 1882, p. 384)
Clausius actually favored referring to this system as the “electrodynamic system,” because he agreed with Ampère that magnetism arose from microscopic electric currents, hence was electrodynamic in nature, and consequently uses the subscript “d” instead of “m”; however, he made clear that it was standardly referred to as the electromagnetic system. (Clausius 1882, p. 383)
What we would now call the auxiliary magnetic field.
As for the electrodynamic system, it was based on Ampere’s law between current elements (Weber 1851).
British Association for the Advancement of Science (1863).
Energy and work have the same dimensions, as required by the work–energy theorem. Since work is given by force \(\times \) displacement, it has dimensions: \([\hbox {work}] = [\hbox {force}] \times \hbox {L} = \hbox {L} \hbox {T}^{-2} \hbox {M} \times \hbox {L} = \hbox {L}^{2} \hbox {T}^{-2} \hbox {M}\) . Energy density then has dimensions: \([\hbox {energy density}] = [\hbox {energy}] \div [\hbox {volume}] = \hbox {L}^{2} \hbox {T}^{-2} \hbox {M} \div \hbox {L}^{3} = \hbox {L}^{-1} \hbox {T}^{-2} \hbox {M}\).
Following the standard usage I shall use italics for quantities (for instance “e” for quantity of electricity), square brackets to denote the dimensions of a derived quantity (such as [e]) and no brackets, nor italics for the fundamental quantities in terms of which the dimensions are expressed (for instance “L” for length). (Maxwell 1873, Vol. 2, chapter 10, § 621–622.)
Neither can be the second equality in Eq. 17, \([ {\mathcal {F}}]=\frac{\hbox {m}}{\hbox {LT}}\). See footnote 26 below.
Indeed Maxwell wrote:
(3) and (5) \([p] = [m] = {\left[ {\frac{\hbox {L}^{2}\hbox {M}}{\hbox {e}\hbox {T}}} \right] }_{}=[ m]\),
where the “(3)” and “(5)” refer to the numbering of m and p, respectively, when he listed earlier the electromagnetic quantities themselves. Furthermore, he never listed separately (i.e., on different lines) quantities that have the same dimensions. The situation occurs for only m and p, and C and \(\varOmega \) , and in both cases he presented the results the same way, i.e., again with:
(4) and (6) \([C] = [O]= \left[ {\frac{\hbox {e}}{\hbox {T}}} \right] =\left[ {\frac{\hbox {L}^{2}\hbox {M}}{\hbox {m}\hbox {T}^{2}}} \right] \).
If \([C] = [O]\) were to be also interpreted as a relation additional to the fifteen he gave explicitly, we would now have an abundance of riches, for either \([p] = [m]\) or \([C] = [O]\) is necessary to find all the dimensions, but not both (they do yield the same results).
Neither Maxwell nor Clausius comments on why [p] should be equal to [m], either.
And as does \([C] = [\varOmega ]\).
Again there are many other possible combinations, including another four that involve the same number of equations, with \({\left[ {\frac{\mathcal {U}}{\mathcal {B}}} \right] }_{}=\hbox {L}\). The reason why using \(\left[ {\frac{\mathcal {U}}{\mathcal {B}}} \right] =\hbox {L}\) as opposed to \([ p]=[m]\) demands the use of more equations for the derivation of the relation between e and m may be related to the fact that three of Maxwell’s fifteen dimensional relations involve p, whereas only two contain \(\mathcal {U}\) (both m and \(\mathcal {B}\) appear twice).
The second equality in Eq. 17, \([ {\mathcal {F}}]=\frac{\hbox {m}}{\hbox {LT}}\), can be derived equally simply with \(\left[ {\frac{\mathcal {U}}{\mathcal {B}}} \right] =\hbox {L}\) as with \([ p]=[m]\):
\(\left[ {\frac{E}{{\mathcal {F}}}} \right] =\hbox {L}\left[ {\frac{p}{E}} \right] =\hbox {T}, \hbox { with } [ p]=[m],\)
or: \(\left[ {\frac{m}{\mathcal {B}}} \right] =\hbox {L}^{2}, \left[ {\frac{\mathcal {U}}{{\mathcal {F}}}} \right] =\hbox {T}, \hbox {with} \left[ {\frac{\mathcal {U}}{\mathcal {B}}} \right] =\hbox {L}\).
There are many other possibilities, most more complicated. To give but one by way of example:
\(\left[ {\mathcal {DF}} \right] =\frac{\hbox {M}}{\hbox {LT}^{2}}, \quad \left[ {\frac{p}{E}} \right] =\hbox {T}, \left[ {\frac{\hbox {E}}{{\mathcal {F}}}} \right] =\hbox {L}, \left[ {\frac{e}{\mathcal {D}}} \right] =\hbox {L}^{2} \hbox { with } [ p]=[m]\).
Analogously, the electromagnetic system was based on Coulomb’s law for magnetism: \(\mathcal {H}=\frac{m}{L^{2}}\), where \(\mathcal {H}\) was the “magnetic force” on a magnetic pole of unit magnitude.
\(\left[ {\mathcal {F}} \right] =\frac{\hbox {e}}{\hbox {L}^{2}}=\frac{\hbox {L}^{2}\hbox {M}}{\hbox {m}\hbox {T}}\times \frac{1}{\hbox {L}^{2}}=\frac{\hbox {M}}{\hbox {m}\hbox {T}}\) and \([ {\mathcal {F}}]=\frac{\hbox {m}}{\hbox {L}\hbox {T}}\) imply \(\frac{\hbox {M}}{\hbox {m}\hbox {T}}=\frac{\hbox {m}}{\hbox {L}\hbox {T}}\), so that \(\left[ \hbox {m} \right] =\hbox {L}^{1/2}\hbox {M}^{1/2}\).
Similarly, having obtained above in Eq. 17 \(\left[ {\mathcal {F}} \right] =\frac{\hbox {L}\hbox {M}}{\hbox {e}\hbox {T}^{2}}\), and using \({\mathcal {F}}=\frac{\hbox {e}}{\hbox {L}^{2}}\) yielded for the electrostatic dimensions of the electric charge: \(\left[ \hbox {e} \right] =\hbox {L}^{3/2}\hbox {M}^{1/2}\hbox {T}^{-1}\).
Nowadays this would be expressed as the requirement that the magnet and the circuit must have the same magnetic moments.
Joseph David Everett (1831–1904) had been secretary of the Committee for the selection and nomenclature of dynamical and electrical units, which had been appointed by the British Association (British Association for the Advancement of Science 1875), and had published two treatises entirely devoted to units and dimensions (Everett 1875, 1879). He was professor of natural philosophy at Queen’s College, Belfast for most of his career, from 1867 until 1897.
From Coulomb’s law between electrostatic charges, \(f=\frac{e_1 e_2 }{r^{2}}\), we have in terms of dimensions:
\(\begin{array}{l} [ {e^{2}}]=[f] [{r^{2}}_{}]=\hbox {M L T}^{-2}\times \hbox {L}^{2}=\hbox {M L}^{3}\hbox {T}^{-2}, \\ \hbox {i.e.}[e]=\hbox {M}^{1/2}\hbox {L}^{3/2}\hbox {T}^{-1}. \\ \end{array}\)
Current has dimensions of charge divided by time, therefore in the electrostatic system of units: [current] = \(\hbox {M}^{1/2} \hbox {L}^{3/2} \hbox {T}^{-2}\).
Using the relation Pole = Current \(\times \) Length then implies \([pole] = \hbox {M}^{1/2} \hbox {L}^{5/2} \hbox {T}^{-2}\).
These were the electromagnetic dimensions of current given by Maxwell (Maxwell 1873, Vol. 2, chapter 10, § 626). In order to obtain them he doubtless divided by T the electromagnetic dimensions for the electrostatic charge, \(\left[ {e_\mathrm{m} } \right] =\hbox {M}^{1/2}\hbox {L}^{1/2}\). In turn, the latter were derived by taking the dimensional form of the expression for the magnetic form of Coulomb’s law, i.e., \(\mathcal {H}=\frac{m}{\hbox {L}^{2}}\), and substituting in it the dimensional expressions for [m] and \(\left[ \mathcal {H} \right] \) in terms of e valid in all systems of units:
\([m]=\frac{\hbox {L}^{2}\hbox {M}}{e\hbox {T}_{}}\)( Eq. 15 above);\(\quad [\mathcal {H} ]=\frac{\hbox {e}}{\hbox {LT}}\).
How Maxwell obtained the latter from his fifteen dimensional relations (Eqs. 9–13) is again unclear.
\([ \mathcal {H}]=\left[ {\frac{m}{{L}^{2}}} \right] \) then led to \(\frac{\hbox {e}}{\hbox {LT}}=\frac{\hbox {L}^{2}\hbox {M}}{\hbox {e}\hbox {T}}\times \frac{1}{\hbox {L}^{2}}\), and solving this for e indeed results in \(\left[ {e_\mathrm{m} } \right] =\hbox {M}^{1/2}\hbox {L}^{1/2}\).
This force had been discovered by Oersted and studied by Ampère.
From Coulomb’s electrostatic law the dimensions of charge in the electrostatic system of units are \(\left[ \hbox {e} \right] =\hbox {L}^{3/2}\hbox {M}^{1/2}\hbox {T}^{-1}\) (see footnote 30 above), and those of current are simply these divided by T.
More precisely, the relation that this force law implies for the magnetic intensity, i.e., [B] = [pole] \(\times \hbox {L}^{-2}\).
Recall from Eq. 17 that Maxwell had somehow obtained for the electric field \({\mathcal {F}}\) the dimensional relation \([ {\mathcal {F}}]=\frac{\hbox {L}\hbox {M}}{\hbox {e}\hbox {T}^{2}}\), valid in all systems of units. In the electrostatic system charge has dimensions \(\left[ \hbox {e} \right] =\hbox {L}^{3/2}\hbox {M}^{1/2}\hbox {T}^{-1}\) (see footnote 30 notably), so in that system the electric field has dimensions:
\([ {\mathcal {F}}]=\frac{1}{\hbox {L}^{3/2}\hbox {M}^{1/2}\hbox {T}^{-1}}\times \frac{\hbox {L}\hbox {M}}{\hbox {T}^{2}}=\hbox {L}^{{-1}/2}\hbox {M}^{1/2}\hbox {T}^{-1}\),
and therefore the electrostatic dimensions of its line integral are \([ {\hbox {e.m.f.}}]=\hbox {L}^{{-1}/2}\hbox {M}^{1/2}\hbox {T}^{-1}\times \hbox {L}=\hbox {L}^{1/2}\hbox {M}^{1/2}\hbox {T}^{-1}\).
\([\hbox {Pole}] \times \) [Length] = \([k_{3}] \times [\hbox {Current}] \times [\hbox {Area}]\) yields in the electrostatic system of units, with \([\hbox {Current}] = \hbox {M}^{1/2} \hbox {L}^{3/2^{}} \hbox {T}^{-2}\) (see notably footnote 37):
\(\left[ {\hbox {Pole}} \right] \times \hbox {L}=\left[ {k_3 } \right] \times \hbox {M}^{1/2}\hbox {L}^{3/2}\hbox {T}^{-2}\times \hbox {L}^{2}\Rightarrow \left[ {\hbox {Pole}} \right] =\left[ {k_3 } \right] \times \hbox {M}^{1/2}\hbox {L}^{5/2}\hbox {T}^{-2},\)
so with \(\left[ {k_3 } \right] =\hbox {L}^{-2}\hbox {T}^{2}:\quad [ {\hbox {Pole}}]=\hbox {L}^{-2}\hbox {T}^{2}\times \hbox {M}^{1/2}\hbox {L}^{5/2}\hbox {T}^{-2}=\hbox {M}^{1/2}\hbox {L}^{1/2},\)
but when \(\left[ {k_1 } \right] =\left[ {k_2 } \right] =\hbox {L}^{-2}\hbox {T}^{2},\hbox {i.e.}\left[ {k_3 } \right] =1:\quad [ {\hbox {Pole}} ]=\hbox {M}^{1/2}\hbox {L}^{5/2}\hbox {T}^{-2}.\)
Everett (1882b, p. 434). Italics in the original.
See concluding remarks.
This article was published in the same issue as Everett’s second paper, i.e., Everett (1882b).
More specifically, Thomson discussed the magnetic force between a current and a pole, and the force between two poles. In order to conclude from these considerations their implications regarding Clausius’s relation, what is at stake is the effect of, respectively, the circuit and of one of the two poles; one could describe the other pole involved in these two forces as a “test pole,” and what Thomson calls “forces” actually corresponds to our field H—and indeed, the H generated by a current-carrying wire is independent of the permeability \(\mu \).
Indeed, although Thomson did not make this explicit, his results implied:
\(F_\mathrm{current~loop}~ \alpha ~\hbox {current}\times \hbox {area} \quad F_\mathrm{pole}~\alpha \quad \frac{1}{\mu } \times \hbox {magnetic moment}\)
so \(F_\mathrm{current~loop} =F_\mathrm{pole}\) requires:
\(\hbox {current}\times \hbox {area} \quad \alpha \quad \frac{1}{\mu }\times \hbox {magnetic moment,}\)
which motivated Thomson’s modification of the relation magnetic \(\hbox {moment} = \hbox {current} \times \hbox {area}\) used by Clausius.
Thomson’s demonstration of this point was somewhat circular, in so far that the way he obtained the electrostatic dimensions of the permeability \(\mu \) required using \(\hbox {M}^{1/2}\hbox {L}^{1/2}\) for the electrostatic dimensions of the magnetic pole (Thomson 1882a, p. 428). Indeed, \(\mu \) is given by the ratio of the “magnetic induction” (i.e., our modern B) and the “magnetic force” (our field H, which he represented by F); now Thomson took the “magnetic induction” to be given by \(m{/}r^{2}\), which clearly involves the pole m:
\([ \mu ]=\frac{[B]^{}}{[H]}=\frac{[m]}{[H]}\hbox {L}^{-2}\)
Thomson had already derived the electrostatic dimensions of the magnetic pole to be \(\hbox {M}^{1/2}\hbox {L}^{1/2}\), without referring to the relation of interest (i.e., \(\hbox {magnetic moment} = \mu \times \hbox {current} \times \) area): from the expression for the force between a current and a pole, he deduced that the product current \(\times \) pole must have dimensions of energy. Indeed, he took for what we would now call H:
\(H=2\frac{\hbox {current}}{\hbox {distance from wire}_{}}\),
so that the work required to move a magnetic pole along a circle around the wire must be:
\(\hbox {work}=H\times \hbox {pole}\times \hbox {distance moved}=2\frac{\hbox {current}}{\hbox {distance from wire}}\times \hbox {pole}\times 2\pi \times \hbox {distance from wire}.\)
Hence the dimensions of the product current \(\times \) pole must be those of work, i.e., energy: ML\(^{2}\)T\(^{-2}\). Since the electrostatic dimensions of current (from the relation between current and charge, and Coulomb’s force law between electrostatic charges) were \(\hbox {M}^{1/2}\hbox {L}^{3/2}\hbox {T}^{-2}\), it follows that the pole had electrostatic dimensions:
\([m]=\frac{[\hbox {energy}]}{[\hbox {current}]}=\frac{\hbox {ML}^{2^{}}\hbox {T}^{-2}}{\hbox {M}^{1/2}\hbox {L}^{3/2}\hbox {T}^{-2}}=\hbox {M}^{1/2} \hbox {L}^{1/2}.\)
This indeed implied that the dimensions of \(\mu \) were \(\hbox {L}^{-2} \hbox {T}^{-2}\) as noted in the main text:
\(\left[ \mu \right] =\frac{[m]}{\left[ H \right] }\hbox {L}^{-2}=\frac{[m]}{\left[ {\frac{\hbox {current}}{\hbox {distance from wire}}} \right] }\hbox {L}^{-2}=\frac{\hbox {M}^{1/2} \hbox {L}^{1/2}}{\left[ {\frac{\hbox {M}^{1/2}\hbox {L}^{3/2}\hbox {T}^{-2}}{\hbox {L}}} \right] }\hbox {L}^{-2}=\hbox {L}^{-2}\hbox {T}^{-2}\)
It is worth noting, however, that the dimensions of \(\mu \) can also be obtained in a way that does not rely upon the dimensions of m: in the electrostatic system the dimensions of the force between two charges are \(\left[ {\frac{q q'}{d^{2}}} \right] \), and the dimensions of the force between two current-carrying wires are \(\left[ {\mu i i'} \right] \), hence \(\left[ {\mu \frac{q q'}{t t'}} \right] \). Therefore, requiring that these forces have the same dimensions implies \(\left[ {\mu \frac{q q'}{t t'}} \right] =\left[ {\frac{q q'}{d^{2}}} \right] \), and \(\left[ \mu \right] =\hbox {L}^{-2}\hbox {T}^{2}\).
Thomson also obtained \(\hbox {M}^{1/2}\hbox {L}^{1/2}\) for the electrostatic dimensions of the magnetic pole without referring to the relation of interest (i.e., magnetic moment = \(\mu \times \hbox {current } \times \) area): from the expression for the force between a current and a pole, he deduced that the product current \(\times \) pole must have dimensions of energy. Indeed, he had found for what we would now call the magnetic field H (which he called magnetic force F):
\(H=2\frac{\hbox {current}}{\hbox {distance from wire}_{}}\),
so that the work required to move a magnetic pole a full circle around the wire must be:
\(\hbox {work}=H\times \hbox {pole}\times \hbox {distance moved}=2\frac{\hbox {current}}{\hbox {distance from wire}}\times \hbox {pole}\times 2\pi \times \hbox {distance from wire}.\)
Hence the dimensions of the product current \(\times \) pole must be those of work, i.e., energy: \(\hbox {ML}^{2}\hbox {T}^{-2}\). Since the electrostatic dimensions of current (from the relation between current and charge, and Coulomb’s force law between electrostatic charges) are \(\hbox {M}^{1/2}\hbox {L}^{3/2}\hbox {T}^{-2}\), it follows that the pole has electrostatic dimensions:
\([m]=\frac{[\hbox {energy}]}{[\hbox {current}]}=\frac{\hbox {ML}^{2}\hbox {T}^{-2}}{\hbox {M}^{1/2}\hbox {L}^{3/2}\hbox {T}^{-2}}=\hbox {M}^{1/2} \hbox {L}^{1/2}.\)
“These factors introduce themselves naturally through symbols representing some physical property of the body or medium.” (Thomson 1882b, p. 226)
It is also interesting to note that Larmor not only viewed some relations as more fundamental than others, but he also deemed the electrostatic system of units, and electrostatic phenomena at large, to be more fundamental than their electromagnetic counterparts.
Herwig was then Professor of physics at Darmstadt.
The electrostatic dimensions Wead attributed to I agreed with those for the field Hin Maxwell’s treatise (Maxwell 1873, chapter 10).
The dimensions of torque are given by: \([\hbox {torque}]= [\hbox {force}] \times [\hbox {lever arm}] = \hbox {MLT}^{-2} \times \hbox {L} = \hbox {M}\hbox {L}^{2}\hbox {T}^{-2}\).
i.e., \(\mu _M H=\frac{\pi K}{t^{2}(1+\theta )} \quad \frac{\mu _M }{H}=\frac{1}{2}\frac{\gamma ^{5}\tan \varphi -\gamma '^{5}\tan \varphi '}{\gamma ^{2}-\gamma '^{2}}\)
where \(\mu _M \): magnetic moment of the magnet H: horizontal component of the Earth’s magnetic field
K: moment of inertia t : time \(\gamma \) and \(\gamma \)’: lengths
Moment of inertia has dimensions \(\hbox {M L}^{2}\). Indeed from the angular version of Newton’s second law, torque = moment of inertia \(\times \) angular acceleration:
\([\hbox {moment of inertia}] = [\hbox {torque}] \times [\hbox {angular}~\hbox {acceleration}]^{-1} = \hbox {ML}^{2} \hbox {T}^{-2} \times (\hbox {T}^{-2} )^{-1} = \hbox {ML}^{2}\)
(see previous footnote for the dimensions of torque).
Therefore the two equations yield the dimensional relations:
\(\left[ {\mu _M H} \right] =\left[ {\frac{\pi K}{t^{2}(1+\theta )}} \right] =\frac{ \left[ K \right] }{ \left[ { t^{2}} \right] }=\frac{\hbox {ML}^{2}}{\hbox {T}^{\hbox {2}}} \quad \frac{\mu _M }{H}=\frac{1}{2}\frac{\gamma ^{5}\tan \varphi -\gamma '^{5}\tan \varphi '}{\gamma ^{2}-\gamma '^{2}}=\frac{\left[ { \gamma ^{5}} \right] }{\left[ { \gamma ^{2}} \right] }=\frac{\hbox {L}^{{5}}}{\hbox {L}^{{2}}}=\hbox {L}^{{3}}\)
Since [\(m_\mathrm{s~Clausius}\)] = \(\hbox {L}^{5/2}\hbox {T}^{-2}\hbox {M}^{1/2}\) and [\(e_\mathrm{s~Clausius}\)] = \(\hbox {L}^{3/2}\hbox {T}^{-1}\hbox {M}^{1/2}\) as usual.
Lodge’s paper was followed by an article by E.B. Sargant, but the latter mostly expressed his agreement with Lodge (Sargant 1882, pp. 395–396). Sargant is probably Edmund Beale Sargant (1855–1938), but this is not certain; the latter was educated in Cambridge and later distinguished himself through his contributions to education, notably in South Africa.
Indeed after referring to Clausius’s choice as an “error,” Lodge stated in a footnote:
“In using the term “error” here, I would be understood to mean rather “divergency from opinions commonly held in this country” than absolute incorrectness as to matter of fact. For it would not be becoming to apply the latter term to views held by Prof. Clausius when the experimental foundation of opposing views is confessedly incomplete. The views held by Prof. Clausius are no doubt perfectly consistent and would probably be in accord with fact if only the medium produced no effect such as it is here [i.e., in Great Britain] commonly supposed to produce; and whether the medium does or does not produce such an effect appears to some extent at present a subject of legitimate debate and a matter for experimental investigation.” (Lodge 1882, p. 364)
The situation discussed by Thomson was equivalent to setting \(\mu _\mathrm{inner} =\mu _\mathrm{outer} \) in case 3, so Thomson stated that the magnetic force due to a circuit was the same as in air, hence independent of the medium. He argued that the magnetic force due to a magnet goes as the inverse of the permeability of the medium, but having no way of distinguishing:
Field due to current = \(\mu _\mathrm{inner} \times \) Field due to magnet
from:
Field due to current = \(\mu _\mathrm{outer} \times \) Field due to magnet,
he argued for the latter.
Lodge quoted Maxwell to justify his predictions:
I venture with diffidence to think that Maxwell would have drawn a distinction between the medium inside the region of the solenoid corresponding to the substance of the magnetic shell, and that outside. He over and over again lays stress upon the fact that artificial solenoids can only be compared with magnetic shells for the space outside the shells... (Lodge 1882, pp. 360–361)
The purpose of the proposed experiments was to verify this.
Sargant also stated that an experiment that he had performed at the Cavendish Laboratory in June was in agreement with Lodge’s view (Sargant 1882), but unfortunately he did not give any details. In the same issue of November Lodge had also reported that Sargant had expressed the intent to carry out what I called case 3 above, thereby implying that he believed Sargant had not yet done so.
Whether Sargant actually had this in mind when he spoke of “model of the magnetic system” is unclear. In any case, the fact that Lodge’s experiments involved two types of solenoids (“open” ones whose internal medium changed with the external one and “closed” solenoids with fixed internal medium) made it tempting to assume these could be meant as alternative models for the behavior of Ampèrian currents.
At least, when understood as a relationship between physical quantities, as opposed to dimensions.
The earlier quotation “All that the Ampèrian theory does is to give a physical interpretation to it, and to render one independent of it so soon as one takes account of every current-conveying circuit, whether molecular or other, existing in the field, and does not arbitrarily elect to deal only with those gross solenoids which we can excite and immediately control by batteries” implies something similar.
Lodge stated: “... in media consisting of Ampèrian molecules there is an extra magnetic induction [in addition to that due to macroscopic currents], due to the pointing of these along the lines of force, which is 4\(\pi \) times the magnetization, and which has to be added to the other, thus making the total magnetic induction at any point \(\mu \) times the magnetic force (i.e., \(\mathcal {H}\)) there.” (Lodge 1882, pp. 364–365)
References
Atten, Michel. 1992. Les Théories électriques en France—1870–1900—La contribution des mathématiciens, des physiciens et des ingénieurs à la construction de la théorie de Maxwell. Ph.D. Dissertation, Ecole des hautes études en sciences sociales, Paris.
British Association for the Advancement of Science. 1863. Report of the 33rd meeting of the British Association for the advancement of science, held at Newcastle-upon-Tyne in August and September 1863, London. John Murray, 1864, 111. The early reports were published again together in 1873, with minor modifications (Reports of the Committee on electrical standards appointed by the British Association for the Advancement of Science, London, 1873).
British Association for the Advancement of Science. 1875. Report of the Committee for the selection and nomenclature of dynamical and electrical units. London: British Association for the Advancement of Science.
Charbonneau, Louis. 1996. From Euclid to descartes: Algebra and its relation to geometry. In Approaches to algebra: Perspectives for research and teaching, ed. Nadine Bednarz, Carolyn Kieran, and Lesley Lee, 15–38. Berlin: Springer.
Clausius, Rudolf. 1882. On the different systems of measures for electric and magnetic quantities. Philosophical Magazine and Journal of Science Series 5 13(83): 381–398.
Darrigol, Olivier. 2000. Electrodynamics from Ampère to Einstein, vol. 167. Oxford: Oxford University Press.
Euclid, The Elements, book 5, lemma 3.
Everett, Joseph David. 1875. Illustrations of the Centimetre-gramme-second system of units (London).
Everett, Joseph David. 1879. Units and constants. London: Macmillan and Co.
Everett, Joseph David. 1882a. On the dimensions of a magnetic pole in the electrostatic system of units. Philosophical Magazine Series 5 13(82): 376–377.
Everett, Joseph David. 1882b. On the dimensions of a magnetic pole in the electrostatic system of units. Philosophical Magazine Series 5 13(83): 431–434.
Fourier, Jean-Baptiste. 1822. Théorie analytique de la chaleur. Paris: Jacques Gabay.
Gauss, Carl F. 1833. French translation 1834. Mesure absolue de l’Intensité du Magnétisme terrestre. Annales de chimie et de physique 5: 5–69.
Harman, Peter M. 1987. Mathematics and reality in Maxwell’s dynamical physics: The natural philosophy of James Clerk Maxwell. In Kelvin’s baltimore lectures and modern theoretical physics, ed. Peter Achinstein, and Robert Kagon, 267–297. Cambridge, MA: MIT Press.
Hunt, Bruce J. 1994. The Ohm is where the art is: British telegraph engineers and the development of electrical standards. Osiris 2nd Series 9: 48–63.
Hunt, Bruce J. 1996. Scientists, engineers and wildman whitehouse: Measurement and credibility in early cable telegraphy. The British Journal for the History of Science 29(2): 155–169.
Hunt, Bruce J. 1997. Doing science in a global empire: Cable telegraphy and victorian physics. In Victorian science in context, ed. Bernard Lightman, 312–333. Chicago: University of Chicago Press.
Hunt, Bruce J. 2003. Electrical theory and practice in the nineteenth century. In The Cambridge History of Science, Vol. 5 : Modern Physical and Mathematical Sciences, Vol. 5, Mary Jo Nye, 311–328. Cambridge: Cambridge University Press.
Hunt, Bruce J. 2005. The Maxwellians. Cornell: Cornell University Press.
Larmor, Josef. 1882. On the electrostatic dimensions of a magnetic pole. Philosophical Magazine Series 5 13(83): 429–430.
Lodge, Oliver Joseph. 1882. On the dimensions of a magnetic pole in the electrostatic system of units. Philosophical Magazine Series 5 14(89): 357–365.
Macagno, Enzo. 1971. Historico-critical review of dimensional analysis. Journal of the Franklin Institute 292(6): 392.
Maxwell, J. C., 1873. Treatise on electricity and magnetism, 2 Vols.
Mercadier, Ernest, and Vaschy Aimé. 1883a. Remarques sur l’expression des grandeurs électriques dans les systèmes électrostatique et électromagnétique, et sur les relations qu’on en déduit. Comptes rendus de l’académie des sciences 96(2): 118–121.
Mercadier, Ernest, and Vaschy Aimé. 1883b. Sur les unités électriques et magnétiques. Annales télégraphiques 10: 5–16.
Mercadier, Ernest, and Vaschy Aimé. 1883c. Réponse aux observations présentées par M. M. Lévy, dans sa Note du 22 janvier 1883. Comptes rendus de l’académie des sciences 96(5): 334–336.
Mercadier, Ernest, and Vaschy Aimé. 1883d. Sur les dimensions des grandeurs électriques et magnétiques. Journal de Physique 2(2): 245–253.
Mueller, Ian. 1970. Homogeneity in Eudoxus’s theory of proportion. Archive for History of Exact Sciences 7(1): 1–6.
Olesko, Kathryn M. 1995. The meaning of precision : The exact sensibility in early nineteenth-century Germany. In The values of precision, ed. Norton W. Wise, 103–134. Princeton: Princeton University Press.
Olesko, Kathryn M. 1996. Precision, tolerance and consensus: Local cultures in German and British resistance standards. Archimedes 1: 117–156.
Roche, John. 1998. The mathematics of measurement, vol. 189, 70–74. London: Springer.
Sargant, E.B. 1882. On the dimensions of the magnetic pole in electrostatic measure. Philosophical Magazine Series 5 14(89): 395–396.
Schaffer, Simon. 1997. Accurate measurement is an English Science. In The values of precision, ed. Norton W. Wise, 135–172. Princeton: Princeton University Press.
Thomson, Joseph John. 1882a. On the dimensions of a magnetic pole in the electrostatic system of units. Philosophical Magazine Series 5 13(83): 427–429.
Thomson, Joseph John. 1882b. On the dimensions of a magnetic pole in the electrostatic system of units. Philosophical Magazine Series 5 14(87): 225–226.
Wead, Charles Kasson. 1882. On the dimensions of a magnetic pole in the electrostatic system of units. Philosophical Magazine Series 13(84): 530–533.
Weber, Wilhelm Eduard. 1851. Elektrodynamische Maasbestimmungen insbesondere Widerstandsmessungen, Königliche Sächsische Gessellschaft der Wissenschaften zu Leipzig, Mathematisch-physikalische Classe, Abhandlungen, 199–382. English translation by Atkinson, E. 1861. On the measurement of Electric Resistance according to an absolute standard, Philosophical Magazine 4: 226–270.
Weber, Wilhelm Eduard. 1861. On the measurement of electric resistance according to an absolute standard. Philosophical Magazine Series 4 22(146):226–240.
Acknowledgments
The author would like to thank Olivier Darrigol, Nadine de Courtenay, Jan Lacki, John Roche, Scott Walter, and Bruce Hunt for their supportive and helpful comments and suggestions.
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Communicated by: J.Z. Buchwald.
This research was conducted at the University Paris 7, Denis Diderot.
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de Clark, S.G. The dimensions of the magnetic pole: a controversy at the heart of early dimensional analysis. Arch. Hist. Exact Sci. 70, 293–324 (2016). https://doi.org/10.1007/s00407-015-0168-6
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DOI: https://doi.org/10.1007/s00407-015-0168-6