A wire loop formed by joining two semicircular wires of radii R1 and R2 carries a current I as shown in the following diagram. The magnetic induction at the centre O is :
A
μ0Iπ4πR1
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B
μ0Iπ4πR2
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C
μ0Iπ4π(1R1−1R2)
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D
μ0Iπ4π(1R1+1R2)
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Solution
The correct option is Dμ0Iπ4π(1R1+1R2) Magnetic field at O due to segments FA and CD is equal to zero. Magnetic field at O due to semi-circle ABC:12μ0I2R2(−^k) Magnetic field at O due to semi-circle DEF:12μ0I2R1(−^k) →Bnet=12μ0I2R2(−^k)+12μ0I2R1(−^k)=μ0I4(1R1+1R2)(−^k)=μ0Iπ4π(1R1+1R2)(−^k)where ^k is out of the plane of the loop and perpendicular to the loop