A hint for my previous problem.

Level pending

I saw that a lot of people have got my recently posed problem Rotating rod in magnetic field wrong. I believe they have answered \(0\). One gets \(0\) if he assumes magnetic field constant in whole region inside metallic ring(in red) having radius \(a << r\) . The same image is uploaded again :

Actually, the magnetic field changes its value, and its expression can be obtained by aproximations. The magnetic field at a distance x from center (\(x <<r\)) can be written as \(B \approx B_{0} \bigg(1+ \dfrac{k}{2} \bigg(\frac{x}{r} \bigg)^n \bigg)\), where \(B_0\) is magnetic field at center(\(O\)) of the circular current carrying loop. Find \(k+n\).

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