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Motivation: I'm doing a homework problem involving a rod sliding freely down a pair of parallel conducting rails. I've got a quantity of unit $\mathrm{A \cdot m}$ and want to know what I should name and call it.

Is there a conventional letter for that quantity (like $q$ for charge) or a conventional name for that unit. (I'm thinking "charge velocity" - as in "the charge velocity is $1.0\,\mathrm{A \cdot m}$" - would be a good name, but am hesitant to "coin a unit" as part of my homework problem.)

RE comment: The problem is to calculate the terminal velocity of said rod when it is placed on a 15-degree inclined plane, given the resistance of the loop is 15 ohms. I found the emf using black magic Faraday's Law, obtained the current, and multiplied by the length vector of the rod to get a vector quantity in Ampere-meters, and took the cross product of this with the B field to get the drag on the rod.

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    $\begingroup$ Could you please explain what quantity you were asked to calculate? $\endgroup$ – G. Smith Oct 23 at 5:04
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    $\begingroup$ I know it's not ammeter $\endgroup$ – user47014 Oct 23 at 5:21
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    $\begingroup$ When calculating $\vec{F}=\vec{I}L\times\vec{B}$, there is no particular need to break out $\vec{I}L$ as an intermediate quantity and give it a name. Just specify $\vec{I}$, $L$, and $\vec{B}$ and use them to calculate $\vec{F}$. $\endgroup$ – G. Smith Oct 23 at 6:05
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Such a quantity does not have a commonly-used name. See https://en.wikipedia.org/wiki/SI_derived_unit

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The units of H-field (magnetic field strength, or magnetic field intensity) are A/m.

So Ampere metres could represent a flux of H-field.

Unlike the B-field, there can be "sources" and "sinks" of H (we call them magnetic poles), and so the total H-field flux into or out of a closed volume can be positive or negative, but would be zero in vacuum.

I'm not really sure how this applies to the problem at hand.

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As others have stated, there is no common word for your quantity. As Rob Jeffries pointed out, it does have units of flux, but what you're describing is not a flux. I can, however, give you some intuition about the quantity. This is just one way of thinking about it:

You mentioned that you're getting this term from the Lorentz force law for a test wire in a magnetic field, $\vec{F}_\text{wire} = I\vec{L}\times\vec{B}$. You can read this equation as $I\left| L \right| \left|B\right|$ for the components of $I$ that are orthogonal to $B$. Using an analogy with gravity, the force on a test mass in a gravitational field is $m\vec{G}$, where $m$ is the gravitational charge (the mass of the test body).

Using this analogy, you can see that your term is something akin to a magnetic charge. Notice that the magnetic charge has two weird properties: It has a direction, and it's direction effects the force it feels in a magnetic field.

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