If a current-carrying loop of wire is placed in a uniform magnetic
field, the net force on the loop is zero.
The magnetic field may, however, exert a torque (or twist) on the loop.
The general formula for the torque on an object is
where 
is the vector that points from the center of rotation
to the point where the force 
is applied.
If this torque formula is integrated around a loop of current to
find the net torque on the loop due to the uniform magnetic field, the
result is
where the quantity 
is called
the
magnetic dipole moment of the loop, and is defined
as follows.
The magnitude of the dipole moment is
where
I is the current flowing in the wire,
and where A is the area of the loop (assuming that the loop is flat),
and where N is the number of turns of wire in the loop.
The direction of is determined by using the right-hand rule on the
direction of current flow: curl your fingers in the direction of the
current, and your thumb points in the direction of .
We could also write
where 
is the area vector for the area bounded by the loop.
Its magnitude is the area and its direction is perpendicular to the surface,
just as we defined it when we discussed flux.
The ambiguity in the direction
of this vector is resolved by using
the right-hand rule on the current, as discussed above.
The torque formula above shows that the loop will be in equilibrium (zero torque)
only if is either parallel with 
or opposite to .
The opposite position is unstable (like trying to balance
a pencil point-down on your finger), so a free current loop tries to
twist until its dipole moment vector is
parallel with the magnetic field.
To see what happens to a current loop in a non-uniform magnetic field, we
need another formula.
The formula for the potential energy of a current loop in a magnetic field,
uniform or nonuniform, is
Note that the potential energy will be lowest (most negative) when
is parallel to , so the aligned position preferred by the loop
is the position of lowest potential energy, as expected.
Suppose, now, that the loop has its dipole moment aligned with ,
but that is non-uniform.
Does a force act on the loop, and if so, in what direction does it point?
We decide this question, in the usual way, by thinking about the potential
energy.
Objects tend toward positions of lower potential energy; since the potential
energy is already negative, lower potential energy would have to mean
larger magnitudes of negative potential energy.
This can be achieved if the loop moves to places where is larger.
We may now see what a free current loop will do in a general magnetic field. First, it will tend to align itself with the magnetic field, and then it will be attracted to the place where the magnetic field is strongest. If you have played with magnets on a table, you may have noticed that this is exactly what they do. If one magnet is moved close to a second one, the free magnet will flip around, then rush toward the strong magnetic field at one of the poles of the first one.
Now we have a connection between electricity and magnets: magnets behave just like current loops. In fact, the dipole moment vector of a bar magnet points from its south end toward its north end, and the torque and potential energy formulas given above apply to bar magnets as well as to current loops. In Chapter 30 we will discover that this similarity in behavior is caused by strong atomic currents in bar magnets.