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I'm learning about torque on a conductive coil in a magnetic field. I have been taught that $\vec\tau = \vec\mu \times \vec{B}$, where $\vec\mu$ is the magnetic dipole moment. Also, $\mu = I\vec{A}$, where $\vec A$ is the area vector of the loop.

To find the direction of the area vector, I am told to use the right hand rule with regards to the current in the loop (curl your fingers in the direction of current, and your thumb points in the direction of the area vector).

My question is: Why does this give the correct direction for the area vector? Is the area vector just defined to be this way to avoid nasty usage of minus signs, or is there some other reason for this?

My guess is that whoever formalized this law/equation (not sure what correct term is for this instance) started with the direction of torque, and worked backwards defining the direction of $\vec\mu$ and $\vec{A}$ to reduce or eliminate stray minus signs in the equations. However, this is, of course, just a guess; I want to know what the true reason is.

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The area vector is typically (in the treatments I have encountered) simply defined this way and then all other facts are written in such a way that they are consistent with this convention.

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As far as I know, the area vector is a purely mathematical object whose definition is related to the orientability of the surface (in this case, a disk). This is a property of surfaces embedded in an Euclidean space that allows to choose surface normal vector to the surface at every point. For an oriented surface, this normal is determined so that we can use the right-hand rule to define a clockwise direction of loops on the surface, which by the way, is needed if we want to apply Stokes' theorem.

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  • $\begingroup$ Are you saying Stokes' theorem is dependent on the right-hand rule being imposed in particular? Or that some convention is necessary simply to use it in practice? $\endgroup$
    – Muphrid
    Commented Mar 30, 2013 at 17:42
  • $\begingroup$ I love math, but in my humble opinion, this response confuses the issue. It is true that for an oriented, two-dimensional surface, the right hand rule can be used to define directions of loops on the surface and that this convention affects Stoke's theorem. These statement don't tell us anything about magnetic moments. In particular, they don't tell us why, given a certain planar current loop, the orientation of the surface is chosen such that the mathematical loop orientation points in the same direction as the current. $\endgroup$
    – joshphysics
    Commented Mar 30, 2013 at 17:52
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    $\begingroup$ @Muphrid Yes, I just was saying that you need to impose the same convention as here in Stokes' theorem to perform the line integral. $\endgroup$
    – Dani
    Commented Apr 1, 2013 at 18:37
  • $\begingroup$ @joshphysics: I understood the question as related just to the definition of the area vector. What you say is also interesting, intuitively I would say that they have the same direction because both the magnetic field and the surface normal are pseudovectors and that in the Biot-Savart law $\mathbf{B}$ just happens to present the same convention as $\mathbf{A}$. $\endgroup$
    – Dani
    Commented Apr 1, 2013 at 18:43
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Well now, that depends: are you asking why the area vector is defined to be in that direction rather than the opposite direction, or are you asking why it's perpendicular to the loop at all?

If the latter, then hopefully it makes sense that when you have a special plane, like the plane of a current loop, then the axis perpendicular to the plane is also special in the sense that it is the only direction not in the plane. So we can decide on the convention that we will use a vector perpendicular to the plane to represent the orientation of the plane. Technically, this vector is called a pseudovector; a pseudovector represents the directions perpendicular to it, rather than than the direction it points in.

Now, there are two vectors perpendicular to a plane, and we can choose either one to be the vector we'll use to represent that plane. The right-hand rule is a way of remembering how to make the choice. We could just as well use the left-hand rule, and make the opposite choice, and physics would still work the same, but we had to choose one rule or the other, and it happened to be the right-hand rule.

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