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.
Best Answer
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.