a) In Checkpoint 5, if the dipole moment is rotated from orientation 2 to orientation 1 by an external agent, is the work done on the dipole by the agent positive, negative, or zero?

(b) Rank the work done on the dipole by the agent for these three rotations, greatest first.$\mathbf{}\mathbf{\text{2}}\mathbf{\to }\mathbf{\text{1,2}}\mathbf{\to }\mathbf{\text{4,2}}\mathbf{\to }\mathbf{\text{3}}$

Four orientations of magnetic dipole moments in the given magnetic field.

Use the relation of work done with the potential energy of a magnetic dipole and find the work done for different orientations of the dipole. From its sign, determine whether it is positive or negative.

The work done is given as-

$W=\Delta U=-\overrightarrow{\mu }.\overrightarrow{B}$

Where W is work done, U is potential energy,B is magnetic field, and 𝜇 is magnetic dipole moment.

The work done on the dipole appears as the change in potential energy. The work done in rotating magnetic dipole of magnitude$\mu ,$due to magnetic field of magnitude, from initial orientation${\theta }_i$to${\theta }_f$is given by,

$$\begin{gathered} W_a=\Delta U=U_f-U_i \\ {W}_a=-\mu B\cos {\theta }_f-(-\mu B\cos {\theta }_i) \end{gathered}$$

$W_a=\mu B\cos {\theta }_i-\mu B\cos {\theta }_f$

$W_a=\mu B(\cos {\theta }_i-\cos {\theta }_f)$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(1\right)$

In orientation 2,${\theta }_i=\theta$and in orientation 1

Substituting this values in 1),${\theta }_f=(180°-{\theta }_i)$

$W_a=\mu B(\cos \theta -\cos (180-\theta ))$

$W_a=\mu B(\cos \theta +\cos \theta )$

$W_a=2\mu B\cos \theta$

Since $\theta$is less than $90°$, $\cos \theta$is greater than zero.

So, the work done is positive.

Hence,the work done is positive.

Work done for three rotations $2\to 1,2\to 4,2\to 3$:

Work done in rotation$2\to 1$is$W_{2\to 1}=2\mu B\cos \theta$,

Work done in rotation from$2\to 4$is,

${\theta }_i=\theta$and${\theta }_f=(180+\theta )$

Substituting in equation 1),

$W_a=\mu B(\cos \theta -\cos (180+\theta ))$

$W_a=\mu B(\cos \theta -(-\cos \theta ))$

$W_a=2\mu B\cos \theta$

Work done in rotation from $2\to 3$is,

${\theta }_i=\theta$and${\theta }_f=(360-\theta )$

$W_a=\mu B(\cos \theta -\cos (360-\theta ))$

$\cos (360-\theta )=\cos 360*\cos \theta +\sin 360*\sin \theta$

$\cos (360-\theta )=\cos \theta$

Substituting in 1),

$W_a=\mu B(\cos \theta -\cos (360-\theta ))$

$W_a=\mu B(\cos \theta -\cos \theta )$

$W_a=0$

Hence,the ranking for the three cases is$W_{2\to 1}=W_{2\to 4}>W_{2\to 3}$.