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Steady current $I$ is flowing an elliptical path given by $r= \frac{r_{0}}{1- \frac{cosθ}{2}}$ in polar form, with one focus as the the origin of coordinate system. Magnitude of magnetic field at same focus due to the current is $η (\frac{μ_{0} I}{r_{0}})$ Find $η$ .

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Detailed Solution

$B = \frac{μ_{0} I}{4 π} \int_{i}^{f} \frac{d θ}{r}$

$= \frac{μ_{0} I}{4 π} \int_{0}^{2 π} \frac{(1 - e \cos θ) d θ}{r_{0}}$

$= \frac{μ_{0} I}{4 π r_{0}} {[θ - e \sin θ]}_{0}^{2 π} = \frac{μ_{0} I}{2 r_{0}}$

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