The magnetic field at `O` is

Options

A. `(mu_(0)I)/(2R)sqrt((1/4+1/(pi^(2))))`
B. `(mu_(0)I)/(2R)sqrt((1/2+1/(pi^(2))))`
C. `(mu_(0)I)/(R)sqrt((1/4+1/(pi^(2))))`
D. `(mu_(0)I)/(R)sqrt((1/4+1/(pi)))`

Answer

A

Explanation

Magnetic field at `O:`
Wire `①` is in `x-y` plane
`B_(1)=(mu_(0)I)/(4piR)`, along `-z`axis
Wire `②` is in `y-z` plane
`B_(2)=(mu_(0)I)/(4R)`, along `-x`axis
Wire `③` is in `x-y` plane
`B_(3)=(mu_(0)I)/(4piR)`, along `-z`axis
`vec(B)_(0)=-(B_(1)+B_(2))hatk-B_(2)hati`
`|vec(B)_(o)|=sqrt((B_(1)+B_(3))^(2)+B_(2)^(2))=sqrt((mu_(0I)/(2piR))^(2)+((mu_(0)I)/(4R))^(2))`
`=(mu_(0)I)/(2R)sqrt(1/(pi^(2))+1/4)`