The figure shows a circular loop made of a wire of radius R. The resistivity of the material varies as a function of θ such that ρ$$=$$ρ$$0sin2$$⁡θ. The positions of the Jockey such that the magnetic field at the centre (O) due to the current in the loop is zero, will be

Question

The figure shows a circular loop made of a wire of radius R. The resistivity of the material varies as a function of  θ such that  ρ$$=$$ρ$$0sin2$$⁡θ. The positions of the Jockey such that the magnetic field at the centre (O) due to the current in the loop is zero, will be

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Consider two sectors one of  α and other of (2$$π−α$$)B$$ at O will be zero if $$I1$$α$$=I2(2$$π−α$$) ⇒$$I1=ER1$$ ,⇒ $$I2=ER2$$ So α$$R1=2$$π−α$$R2$$ ⇒$$R1=$$∫$$0$$αρ$$0sin2$$⁡θA Rd θ $$=$$ρ$$0RA$$α$$-sin2$$α$$2$$∵∫$$sin2$$θdθ$$=12x-$$$$sin$$θ$$cos$$θ$$+CSimilarly$$  we can get  $$R2=$$∫α$$2$$πρ$$0sin2$$⁡θA Rd θ $$=$$ρ$$0RA2$$π$$-$$α$$+sin2$$α$$2$$ ∴α$$R2=2$$π$$-$$α$$R1$$⇒αρ$$0RA2$$π$$-$$α$$+sin2$$α$$2=2$$π$$-$$αρ$$0RA$$α$$-sin2$$α$$2$$ ⇒$$sin2$$α$$=0Onsolvingweget$$  α$$=$$π$$2$$,π,$$3$$$$π$$$$2$$

The figure shows a circular loop made of a wire of radius R. The resistivity of the material varies as a function of θ such that ρ=ρ0sin2⁡θ. The positions of the Jockey such that the magnetic field at the centre (O) due to the current in the loop is zero, will be

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