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One of the quantities appearing in the integral form of Maxwell's Equations is the line integral of the magnetic field around a closed loop. (The relevant equation states that this is equal to the current through any surface bounded by that loop, plus the displacement current in the case of changing electric fields, with some constant coefficients possibly thrown in, depending on how you manage your units. This is usually called Ampère's Law, with Maxwell's correction for the displacement current.)

Is there a name for this quantity? More generally, is there a name for the line integral of the magnetic field along an arbitrary non-closed curve? (Then this is not equal to any other named quantity or sum of quantities.) Ideally, I'd want a name for the integral of the H-field (rather than the B-field, when one distinguishes them) in amperes (or the equivalent in non-SI units), but I am not really picky about those details.

It seems that every other quantity in the integral form of Maxwell's Equations has a name (magnetic flux, electric flux, charge, etc), so I'd hope that this one does too. Of course, the term for the line integral of the electric field (‘electromotive force’) is somewhat of a historical oddity, so maybe this quantity is too obscure to have a name. Still, you'd think that somebody would have given it one sometime!

ETA: A theoretical way to measure this quantity is to place a conductive tube (as conductive as possible, with as small diameter as possible) around the curve in question. The magnetic flux through the tube will induce a current running around (not along) the inside of the tube, which will in turn serve to cancel (or shield) the source of the magnetizing field, in accordance with Lenz's Law (in the broad sense). The current so induced, serving to completely cancel the magnetic field near the curve in question, should equal this quantity, if I'm thinking about this correctly. (Well, I'm having a little trouble getting the sign straight, but one way or the other it should work!)

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  • $\begingroup$ "Of course, the term for the line integral of the electric field (electromotive force) is somewhat of a historical oddity" and misleading - electromotive force isn't always given by line integral of the electric field. "Contour integral of electric/magnetic field" is accurate and comprehensible. I do not think there is commonly used shorter expression. $\endgroup$
    – Ján Lalinský
    Commented Mar 1, 2014 at 9:44
  • $\begingroup$ The term ‘electromotive force’ is used on Wikipeda for this line integral, citing ‘David M. Cook (2003). The Theory of the Electromagnetic Field. Courier Dover. p. 157. ISBN 978-0-486-42567-2’ (which I have not looked at). In light of Tobias's answer below, perhaps ‘voltage drop’ would be a better term. $\endgroup$
    – Toby Bartels
    Commented Mar 3, 2014 at 22:29
  • $\begingroup$ Would it be appropriate to create the tag magnetic-circuits and add it to this question? That seems to be the name of the relevant topic. (Not that I could create it, since I don't have enough reputation on this StackExchange.) $\endgroup$
    – Toby Bartels
    Commented Mar 4, 2014 at 3:37

2 Answers 2

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Line integrals of the magnetic field strength are magnetic voltage drops. Just google for "magnetic voltage drop" (including the double-quotes).

In the quasi-static case ($\dot{\vec{D}}=\vec{0}$) the $\vec{H}$-field within a simply path-connected domain with zero current density has a magnetic potential. In this case you can calculate the magnetic voltage drops as potential differences.

The magnetic voltage caused by a winding is called current-linkage or ampere-tuns. From the mathematical point of view the closed path integral of the magnetic field strength is a circulation.

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  • $\begingroup$ Thanks, this is a nice clear term! One link that I found through searching on this term is the Wikipedia article on ‘reluctance’ (= magnetic resistance), where they use the term ‘magnetomotive force’ instead; but that is pretty ugly, so I like your term better. $\endgroup$
    – Toby Bartels
    Commented Mar 3, 2014 at 22:32
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The English Wikipedia article for reluctance uses the term ‘magnetomotive force’. I like Tobias's answer better, so I'm accepting it, but I'm also recording this one. If somebody else adds another better answer, then maybe I'll switch!

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