By applying Ampère's law:
\[ \oint \vec{B} \cdot d\vec{\ell} = \mu_0 i_{\text{enc}} = 0 \]
Since the net enclosed current outside the cable is zero (equal and opposite currents in the central and outer conductors), the magnetic field outside the cable is zero.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: