Conductor of length l has shape of a semi-cylinder of radius $R(<<\mathcal{l})$. Cross section of the conductor is shown in Fig. Thickness of the conductor is $t(<<R)$ and conductivity of its material varies with angle $\theta$ only, according to the law  $\sigma ={\sigma }_0\text{\hspace{0.17em}cos}\theta$. If a battery of emf V and of negligible internal resistance is connected across its end faces, calculate magnetic induction at mid point O of the axis of the semi-cylinder is $\frac{{\mu }_0{\sigma }_0^b{V}^c{t}^d}{a\cdot \mathcal{l}}$ find  $a^2+b^2+c^2+d^2$