Step 1: Model the dipole as two point poles.
A bar magnet in a field \(\vec{B}\) behaves like two magnetic poles \(+q_m\) (north) and \(-q_m\) (south) separated by the magnetic length \(2l\). Its moment is defined as \(m=q_m(2l)\).
Step 2: Write the pole forces when the axis is tilted by \(\theta\).
Each pole experiences a force of size \(q_mB\); the north-pole force points along \(\vec{B}\) and the south-pole force points opposite. The lines of action are separated, so there is no net translation, only a turning effect.
Step 3: Use the moment-of-a-couple definition directly.
For a couple, torque \(=\) magnitude of one force \(\times\) arm. The perpendicular arm between the two lines of action is \(2l\sin\theta\), hence
\[\tau = (q_mB)(2l\sin\theta) = (q_m\,2l)\,B\sin\theta.\]
Step 4: Replace \(q_m\,2l\) by \(m\).
\[\tau = mB\sin\theta,\qquad \vec{\tau}=\vec{m}\times\vec{B}.\]
By the right-hand rule for \(\vec{m}\times\vec{B}\), the torque tends to rotate the dipole until \(\vec{m}\) aligns with \(\vec{B}\).
Step 5: Operational definition of \(m\).
Setting \(B=1\) unit and \(\theta=90^\circ\) gives \(\tau=m\). Therefore the magnetic dipole moment is numerically the torque required to hold the dipole at right angles to a unit magnetic field.
\[\boxed{\tau = mB\sin\theta}\]