A current I = 20 ampere flows through a wire shaped in the form of an infinite parabola of latus rectum 4a (a = 1 mm). If the magnetic field at the focus of the parabola is  $\eta \pi \times {10}^{-\beta }$ tesla, where $\eta$ and $\beta$ are the positive integer, less than 9. Find the value of  $\eta \times \beta$.

The magnetic field at the focus P of the parabola is given by the Biot - Savart law:
 $\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\int \frac{Id\overrightarrow{\mathcal{l}}\times \overrightarrow{r}}{r^3}$,
Where $\overrightarrow{r}$ and $d{\overrightarrow{I}}_0$ are as shown in figure. The direction of $\overrightarrow{B}$  is perpendicular to the plane of the parabola and out of it. We note that
$\frac{|d\overrightarrow{\mathcal{l}}\times \overrightarrow{r}|}{r^3}=\frac{rd\mathcal{l}\sin \alpha }{r^3}=\frac{rd\theta }{r^2}=\frac{d\theta }{r}$

Therefore,  $B=({\mu }_{0}I/4\pi ){\int }_0^{2\pi }(d\theta /r)$. The equation of the parabola is  $r(1-\cos \theta )=2a$, so that

$B=\frac{{\mu }_{0}I}{4\pi }\underset{0}{\overset{2\pi }{\int }}\frac{(1-\cos \theta )}{2a}d\theta =\frac{{\mu }_{0}I}{4a}=\frac{4\times \pi \times {10}^{-7}\times 20}{4\times {10}^{-3}}=2\pi \times {10}^{-3}tesla$