Two infinitely long straight wires lie in the xy-plane along the lines `x =pm R`. The wire located at `x =+R` carries a constant current `I_(1)` and the wire located at ` x = -R` carries a constant current `I_(2)`. A circular loop of radius R is suspended with its centre at `(0,0, sqrt(3)R)` and in a plane parallel to the xy-plane. This loop carries a constant current ? in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the `+hat J` direction. Which of the following statements regarding the magnetic field ` vec B` is (are) true?

Options

A. If `I_(1) gt 0 , " then " vec B` cannot be equal to zero at the origin `(0,0,0)` .
B. If `I_(1) gt 0 " and " I_(2) lt 0 , " then " vec B` can be equal to zero at the origin `(0,0,0)` .
C. If `I_(1) lt 0 " and " I_(2) gt 0 , "then " vec B` can be equal to zero at the origin `(0,0,0)` .
D. If `I_(1) = I_(2)` , then the Z - component of the magnetic field at the centre of the loop is `(- (mu_(0)I)/(2R))`

Answer

a, b, d

Explanation

Current flowing in the loop is clockwise as seen from the top, hence magnetic field produced by the ring at the origin will be along negative Z-axis.
A: When same current is flowing through both the straight wires in the same direction, then magnetic fields due to them at the origin will be equal and opposite and will cancel each other. Hence net field at the origin will be due to loop only and cannot be zero. Hence option (a) is correct.
`B : I_(1) gt 0` means that a magnetic field at origin due to `x = + R` wire is along positive Z-axis, similarly `I_(2) lt 0` means that field at origin due to `x = - R` wire is also along positive Z-axis. But we know that field due to loop is along negative Z-axis, hence net field at the origin in this case can be equal to zero. Hence option (b) is also correct.
`C : I_(1) lt 0` means that field at origin due to `x = + R` wire is along negative Z-axis, similarly `I_(2) gt 0` means that field at origin due to `x = - R` wire is also along negative Z-axis. But we know that field due to loop is also along negative Z-axis, hence net field at the origin in this case cannot be equal to zero. Hence option (c) is wrong.
`D : ` In this case, the direction of magnetic fields is as shown in the figure.

`B_(1) = B_(2)` ( Fields due to straight wires) Magnetic field along Z-axis is only due to ring
`B = (mu_(0)i)/(2R) ` in - direction.
Hence option (d) is also correct.