A uniform rod $PQ$ of length $l$ and mass $m$ is placed on a smooth horizontal surface. It is hinged at one of its ends and touches the conducting circular loop at another end. The system is subjected to vertical magnetic field B₀. Rod is given angular velocity ${\omega }_0$ about the axis passing through $P$. If the time after which its angular velocity becomes half is $\frac{a}{b}\frac{mR\text{l}\mathrm{n}2}{B^2{l}^2}$ . Find the value of $\left(a+b\right)$, where $a$ and $b$ are smallest positive integers.