What does the analog (similarity) of bar magnet’s and solenoid’s magnetic field lines suggest ?

A bar magnet of magnetic moment $10^4\,J/T$ is free to rotate in a horizontal plane. The work done in rotating the magnet slowly from a direction parallel to a horizontal magnetic field of $4×10^{-5}\, T$ to a direction $ 60° $ from the field will be…..$ J$

Verify the Ampere’s law for magnetic field of a point dipole of dipole moment ${\rm{\vec M = M\hat k}}$. Take $\mathrm{C}$ as the closed curve running clockwise along : the $\mathrm{z}$ – axis from $\mathrm{z} = \mathrm{a} \,>\, 0$ to $\mathrm{z = R}$;

A short bar magnet of magnetic movement $5.25 \times 10^{-2} \;J\, T ^{-1}$ is placed with its axis perpendicular to the earth's field direction. At what distance from the centre of the magnet, the resultant field is inclined at $45^{\circ}$ with earth's field on

$(a)$ its normal bisector and

$(b)$ its axis.

Magnitude of the earth's field at the place is given to be $0.42 \;G$. Ignore the length of the magnet in comparison to the distances involved.

Give some of the commonly known ideas regarding magnetism.