Flower-like loop

The figure shows a circular loop of radius \(R\) (black curve) that has been harmonically deformed into a flower-like curve (blue curve) given by the equation in polar coordinates \[ r(\theta)= R (1+\varepsilon\cos(n \theta)) \quad \text{with} \quad \varepsilon=0.5 \quad \text{and} \quad n=5. \]

A current \(I_{0}\) flowing in the circular loop produces a magnetic field \(B=1~\text{mT}\) at the center O. Determine the magnitude of the magnetic field in mT at O when the same current, \(I_{0}\), flows in the flower-like loop. The following integral may be useful: \[\int_{0}^{2\pi}\frac{dx}{1+\varepsilon \cos(x)}=\frac{2\pi}{\sqrt{1-\varepsilon^{2}}}.\]

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