Week 6: Moving Charges and Magnetic Force 213
alwaysbe the veryθthat we want. The right handed normal is the unit vector perpendicular to the
plane of the loop that points in the direction your right hand thumb points when your fingers curl
around the loop in the direction of the current.
This (and the highly suggestive form ofτ) suggests that we define themagnetic dipole moment
of this loop to be:
~m=N I(ab)nˆ (455)
in which case the torque takes the familiar form:
~τ=~m×B~ (456)which looksjust likethe torque on anelectricdipole,~τ=~p×E~! In fact, since the force on an electric
dipole also vanished in a uniform field, we caninstantly adopt(reasoning by algebraic analogy or
formally rederiving it all as we prefer) all of the results in table 3 below. But first, let’s generalize
our expression for the magnetic dipole moment a bit and consider a more general plane current loop
instead of just a rectangle.
Example 6.3.2: The Magnetic Moment of anArbitraryPlane Current Loop
∆x∆yxy−Fx FxFupFdownpivotN turns
Current I eachr.h.n (out)Figure 73: Arbitrary plane loop of current can be broken into small pieces that are aligned with or
perpendicular to torque axis.
In figure 73 we see a golf-putting-green shaped loop of current carrying wires in a plane. As
before, there areNturns carrying a currentI, and I’ve drawn an arbitrary rotation axis/pivot that
is perpendicular to theB~field that the loop will be in and located at the end of the (each) loop
rectangle for convenience.
As you can see, one can take the curve and break it up into perpendicular segments that approx-
imate the curve arbitrarily closely as the ∆xand ∆ysegments are made smaller and smaller. If one
considers justonesuch opposing pair of segments each (the shaded/textured areas in the figure),
the forcesFxbetween the ∆yparts of the curve are equal and opposite and along a common line
parallel to the axis of torque. They contribute no force and no torque in a uniform field so we don’t
even bother to sum over them, we just ignore them.