A long wire bent as shown in fig. carries current I. If the radius of the semicircular portion is a, the magnetic field at centre C is

Options

A. `(mu_0I)/(4r)`
B. `(mu_0I)/(4pir)sqrt(pi^2+4)`
C. `(mu_0I)/(4pi)+(mu_0I)/(2pir)`
D. `(mu_0I)/(4pir)sqrt(pi^2-4)`

Answer

B

Explanation

Total magnetic field induction at P is
`vecB=vecB_(ab)+vecB_(bc)+vecB_(cd)`
`=(mu_0)/(4pi)I/r(-hatj)+(mu_0)/(4pi)(piI)/(r)(-hatk)+(mu_0)/(4pi)I/r(-hatj)`
`=(mu_0)/(4pi)(2I)/(r)(-hatj)+(mu_0)/(4pi)(piI)/(r)(-hatk)`
`:. |vecB|=[((mu_0)/(4pi)(2I)/(r))^2+((mu_0)/(4pi)(piI)/(r))^2]^(1//2)`
`=(mu_0)/(4pi)I/rsqrt(4+pi^2)`