In the given figure net magnetic field at O will be

Options

A. `(2mu_(0)i)/(3pia)sqrt(4 - pi^(2))`
B. `(mu_(0)i)/(3pia)sqrt(4 + pi^(2))`
C. `(2mu_(0)i)/(3pia^(2))sqrt(4 + pi^(2))`
D. `(2mu_(0)i)/(3pia)sqrt(4 - pi^(2))`

Answer

B

Explanation

Magnetic field at 0 due to
Part (1) : `B_(1) = 0`
Part (2): `B_(2) = (mu_(0))/(4pi)(pi i)/((a//2))ox` along - Z- axis)
Part (3) : `B_(3) (mu_(0))/(4pi).(i)/((a//2))(darr)` (along - Y - axis)
Part (4) : `B_(4) = (mu_(0))/(4pi).(pi)/((3a//2))odot` (along + Z axis)
Part (5) : `B_(5) = (mu_(0))/(4pi)*(i)/((3a//2))(darr)` (along - Y - axis)
`B_(2) - B_(4) = (mu_(0))/(pi) ,(pii)/(a) (2 - (2)/(3)) = (mu_(0)i)/(3a) ox` (along - Z axis)
`B_(3) + B_(5) = mu_(0)/(4pi) .(1)/(a) (2 + (2)/(3) ) = (8 mu_(0)i)/(12 pi a) (darr)` (along - Y - axis)
Hence net magnetic field
`B_(nst) = sqrt((B_(2) - B_(4)^(2) )+ (B_(3) + B_(5))^(2)) = ((mu_(0)i)/(3pia)sqrt(pi^(2) + 4)`