Configuration corresponding to lowest potential energy [closed]

Question

Figure shows a small magnetised needle P placed at a point O. The arrow shows the direction of its magnetic moment. The other arrows show different positions (and orientations of the magnetic moment) of another identical magnetised needle. I am asked to find the configuration corresponding to the lowest potential energy among all the configurations shown.

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This is till where I could reach

Since the direction of the magnetic moment of the needle placed at d is visible I imagine a small magnet at that place having the north pole at the head of the arrow, with that I get the idea of the magnetic field lines. Now I know that when the direction of external magnetic field lines are in the direction of the magnetic moment no torque is applied. Therefore, the magnetized needle when placed at Q3 and Q6 is in stable equilibrium but the answer in my book shows placement of the second magnetic needle at Q6 has lowest potential energy. Please explain where am I going wrong.

Answer 1

Given a magnetic moment, you can roughly imagine the current generating it as a circular current loop with axis parallel to the moment and right hand rule determining flow of current. This in turn gives you some idea of the magnetic field.

The potential energy is $-\vec{\mu}\cdot\vec{B}$. This achieves a minimum when the dot product is at a positive maximum, i.e. when the field and the moment are aligned. You get zero potential if they are perpendicular and a maximum when they point in exactly opposite directions.

The moments are parallel to the field only at Q3 and Q6. Wherever the field is stronger between the two will be the minimum energy. That should be Q6 for a realistic dipole since it's closer to either current source. I'd have to crunch the numbers to verify that for an ideal dipole.