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Magnetic Field at Focus of Parabola

Let's solve the problem based on Magnetic Field at Focus of Parabola and learn how to solve it. First, try it yourself, then check your solution.

The Problem:

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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