Question

Find electric field at the focus of a parabola carring line charge density lambda

Answer 1

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)=2aas the equation to the parabola, we get