Answer
We can rank the work done on the dipole by the agent:
$(2 \to 1) = (2 \to 4) \gt (2 \to 3)$
Work Step by Step
We can write a general expression for the potential energy:
$U(\theta) = -\mu \cdot B$
In orientation 2, $U_2 = -\mu~B~cos~\theta$
In orientation 1, $U_1 = \mu~B~cos~\theta$
We can find an expression for the work done on the dipole for the rotation $2 \to 1$:
$W = U_1-U_2$
$W = (\mu~B~cos~\theta)-(-\mu~B~cos~\theta)$
$W = 2\mu~B~cos~\theta$
In orientation 4, $U_4 = \mu~B~cos~\theta$
We can find an expression for the work done on the dipole for the rotation $2 \to 4$:
$W = U_4-U_2$
$W = (\mu~B~cos~\theta)-(-\mu~B~cos~\theta)$
$W = 2\mu~B~cos~\theta$
In orientation 3, $U_3 = -\mu~B~cos~\theta$
We can find an expression for the work done on the dipole for the rotation $2 \to 3$:
$W = U_3-U_2$
$W = (-\mu~B~cos~\theta)-(-\mu~B~cos~\theta)$
$W = 0$
We can rank the work done on the dipole by the agent:
$(2 \to 1) = (2 \to 4) \gt (2 \to 3)$
