A current I = 20 ampere flows through a wire shaped in the form of an infinite parabola of   latus rectum $4a(a=1mm)$ . 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:
$\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\int \frac{Id\overrightarrow{l}\times \overrightarrow{r}}{r^3}$ ,
Where $\overrightarrow{r}$  and $d{\overrightarrow{l}}_0$ 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 
$\frac{|d\overrightarrow{l}\times \overrightarrow{r}|}{r^3}=\frac{rdl\sin \alpha }{r^3}=\frac{rd\theta }{r^2}=\frac{d\theta }{r}$ .
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   $B=\frac{{\mu }_{0}I}{4\pi }\underset{0}{\overset{2x}{\int }}\frac{(1-cos\theta )}{2a}d\theta =\frac{{\mu }_{0}I}{4a}=\frac{4\times \pi \times {10}^{-7}\times 20}{4\times {10}^{-3}}=2\pi \times {10}^{-3}$ tesla