A wire carrying current `I` has the shape as shown in the adjoining figure. Linear parts of the wire are very long and parallel to X-axis while semicicular portion of radius `R` is lying in `Y-Z` plane. Magnetic field at point `O` is

Options

A. `vec(B) = (mu_(0))/(4pi) (I)/(R) (pi hat(i) + 2 hat(k))`
B. `vec(B) = - (mu_(0))/(4pi) (I)/(R) (pi hat(i) - 2 hat(K))`
C. `vec(B) = -(mu_(0))/(4pi) (I)/(R) (pi hat(i) + 2 hat(k))`
D. `vec(B) = (mu_(0))/(4pi) (I)/(R) (pi hat(i) - 2 hat(K))`

Answer

C

Explanation

Magnetic field due to staright wire 1
`vec(B)_(1) = (mu_(0) I)/(4pi R) [sin 90^(@) +sin 0^(@)] (-vec(k))`
`=(-mu_(0)I)/(4pi R)(hatk)=vec(B)_(3)`
Magnetic field due to semicircular wire 2
`B_(2)= (-mu_(0) I)/(4R) (-hat(i)) = (-mu_(0) I)/(4pi R) (pi hat(i))`
Magnetic field `vec(B)` at centre, `vec(B)_(c) = vec(B)_(1) + vec(B)_(2) + vec(B)_(3)`
`implies vec(B)_(c) = (-mu_(0) I)/(4pi R) (pi hat(i) + 2 hat(k))`