Question

An otherwise infinite, straight wire has two concentric loops of radii $$a$$ and $$b$$ carrying equal currents in opposite directions as shown in figure. The magnetic field at the common center is zero for:

A
ab=π1π
B
ab=ππ+1
C
ab=π1π+1
D
ab=π+1π1

Solution

The correct option is B $$\cfrac { a }{ b } =\cfrac { \pi } { \pi +1 }$$$${ B }_{ centre }=0$$$$\cfrac { { \mu }_{ 0 }I }{ 4\pi b } \bigodot +\cfrac { { \mu }_{ 0 }I }{ 4\pi b } \bigodot +\cfrac { { \mu }_{ 0 }I }{ 2b } \bigodot +\cfrac { { \mu }_{ 0 }I }{ 2a } \bigotimes =0$$$$\Rightarrow \cfrac { { \mu }_{ 0 }I }{ 2\pi b } +\cfrac { { \mu }_{ 0 }I }{ 2b } -\cfrac { { \mu }_{ 0 }I }{ 2a } \Rightarrow \cfrac { 1 }{ 2\pi b } +\cfrac { 1 }{ 2b } =\cfrac { 1 }{ 2a }$$$$\Rightarrow \cfrac { a }{ b } =\cfrac { \pi }{ \pi +1 }$$PhysicsNCERTStandard XII

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