Consider a loop of wire, forming a simple closed plane curve. We let the plane of the loop be the *xy* plane. Then let us parametrize the curve via (*x*(*u*),*y*(*u*)), 0≤*u*≤*u _{max}*with counterclockwise orientation (so that the normal vector to the loop via the right-hand rule is in the positive

*z*direction).

Suppose we then have a uniform magnetic field of magnitude

*B*directed at an angle

*α*from the positive

*z*axis; we can choose our

*x*axis so that is in the

*xz*plane with positive

*x*component.

Now, let us have a current

*I*in the loop (positive

*I*indicates counterclockwise current). What then, is the force on an element

*du*at

(

*x*(

*u*),

*y*(

*u*)), and what is the net effect of this force on the loop?

The magnetic force due to field on a length of wire carrying a current

*I*is . Thus, for an element

*du*, the length vector is . We also have . Thus, we have force

.

To find the total force, we integrate over

*u*:

(here we have used the fact that , as our curve is closed, and similarly for ).

So there is no net force on the loop. However, let us pick a point (

*x*,

_{c}*y*,0) in the plane of the loop, and find the torque about that point. For an element of the curve, the torque element is

_{c}.

Integrating this, and using again the fact that , we get

.

Now,

,

and similarly, , so we see that the *x* and *z* components of the torque are zero, and

. Note that the torque is independent of the plane point (*x _{c}*,

*y*,0) chosen, so we can choose it to be the center of mass of the loop.

_{c}For the remaining integral in our torque, we note that

, and to this integral over our closed curve (

*c*), we apply Green’s Theorem (see also here), which says

, where

*D*is the plane region bound by the simple closed curve

*∂D*. Here,

*f*(

*x*,

*y*)=0,

*g*(

*x*,

*y*)=x, so

, and so

,

where

*A*is the area enclosed by our loop. Our torque is thus:

.

Now, using the area vector , we see that , so

.

Noting that the magnetic moment of an object can be defined by the torque it experiences from an external magnetic field via

, we see that the magnetic moment of any planar current loop is

. Lastly, notice that the direction of our torque is such that the loop rotates to align the magnetic moment (and thus the area vector) with the magnetic field.