91影视

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-

$W=螖U=鈭�\stackrel{鈫�}{渭}.\stackrel{鈫�}{B}$

Where W is work done, U is potential energy,B is magnetic field, and 饾渿 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$渭,$due to magnetic field of magnitude, from initial orientation${胃}_i$to${胃}_f$is given by,

$$\begin{gathered} W_a=螖U=U_f鈭�U_i \\ {W}_a=鈭�渭B\cos {胃}_f鈭�(鈭�渭B\cos {胃}_i) \end{gathered}$$

$W_a=渭B\cos {胃}_i鈭�渭B\cos {胃}_f$

$W_a=渭B(\cos {胃}_i鈭�\cos {胃}_f)$$鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�.\left(1\right)$

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

Substituting this values in 1),${胃}_f=(180掳鈭�{胃}_i)$

$W_a=渭B(\cos 胃鈭�\cos (180鈭�胃))$

$W_a=渭B(\cos 胃+\cos 胃)$

$W_a=2渭B\cos 胃$

Since $胃$is less than $90掳$, $\cos 胃$is greater than zero.

So, the work done is positive.

Hence,the work done is positive.

Work done for three rotations $2鈫�1,2鈫�4,2鈫�3$:

Work done in rotation$2鈫�1$is$W_{2鈫�1}=2渭B\cos 胃$,

Work done in rotation from$2鈫�4$is,

${胃}_i=胃$and${胃}_f=(180+胃)$

Substituting in equation 1),

$W_a=渭B(\cos 胃鈭�\cos (180+胃))$

$W_a=渭B(\cos 胃鈭�(鈭�\cos 胃))$

$W_a=2渭B\cos 胃$

Work done in rotation from $2鈫�3$is,

${胃}_i=胃$and${胃}_f=(360鈭�胃)$

$W_a=渭B(\cos 胃鈭�\cos (360鈭�胃))$

$\cos (360鈭�胃)=\cos 360*\cos 胃+\sin 360*\sin 胃$

$\cos (360鈭�胃)=\cos 胃$

Substituting in 1),

$W_a=渭B(\cos 胃鈭�\cos (360鈭�胃))$

$W_a=渭B(\cos 胃鈭�\cos 胃)$

$W_a=0$

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