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    $\begingroup$ What makes you think it is not the distance between the two dipoles? In the derivation of the gradient expression, r is the distance from the point where the current density is being evaluated. $\endgroup$ Commented Mar 19 at 23:24
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    $\begingroup$ @Ruffolo but what does it mean for a dipole to be located somewhere? Isn't it a current distribution in the space? I don't quite get what you mean, a distance need two endpoints to measure the distance between them, if in this case one is R (the point where the 1st current distribution is distributed around), what is the other? We have already integrated along the 2nd current distribution, there is no 2nd point explicitly written in the notes. $\endgroup$ Commented Mar 20 at 0:07
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    $\begingroup$ Well then what is the force on? In your original expression for a force on a dipole in a field, where is the force applied, if the dipole is not located somewhere? $\endgroup$ Commented Mar 20 at 0:13
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    $\begingroup$ In the treatment by Tong you are supposed to consider the current distributions to be “localized” even in the original expression. Then in the dipole-dipole case you are supposed to consider r to be much larger than the size of the localized current distributions that make up the two dipoles. It’s not spelled out, but that’s the way to think of it. $\endgroup$ Commented Mar 20 at 0:33
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    $\begingroup$ For the general case you’d have varying forces at every current element. It wouldn’t look clean like in Tong. $\endgroup$ Commented Mar 20 at 0:34