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Magnetic Field at Focus of Parabola

An infinite wire carrying current \(I\) is bent in the form of a parabola. Find the magnetic field at the focus of the parabola. Take the distance of the focus from the apex as \(a\).

Solution:

From Biot-Savart law, the magnetic field ar \(S\) is given by $$ \vec{B}= \frac{\mu_0}{4 \pi} \int\frac{I\vec{dl}\times\vec{r}}{r^3}$$
From the figure, we note that
$$ |\vec{dl}\times \vec{r}|$$=area of the parallelogram by \(\vec{dl}\) and \(\vec{r}\) $$ = 2\times1/2\times r.rd\theta$$$$=r^2d\theta$$
Hence, $$ \vec{B}=\frac{\mu_0I}{4 \pi}\int_{0}^{2\pi}\frac{d\theta}{r}$$ Using \(r(1-cos\theta)=2a\) as the equation to the parabola, we get $$ \vec{B}=\mu_0I/4a
$$

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