6.A straight wire carrying a current of 12 A is bent into a semi-circular arc of radius 2.0 cm as shown in Fig. (a). Consider the magnetic field B at the centre of the arc.
a . What is the magnetic field due to the straight segments?
b. In what way the contribution to B from the semicircle differs from that of a circular loop and in what way does it resemble?
c. Would your answer be different if the wire were bent into a semi-circular arc of the same radius but in the opposite way as shown in Fig. (b)?

Explanation

(a) dl and r for each element of the straight segments are parallel. Therefore, dl × r = 0. Straight segments do not contribute to |B|.
(b) For all segments of the semicircular arc, dl × r are all parallel to each other (into the plane of the paper). All such contributions add up in magnitude. Hence direction of B for a semicircular arc is given by the right-hand rule and magnitude is half that of a circular loop. Thus B is `1.9 x× 10^(-4)` T normal to the plane of the paper going into it.
(c ) Same magnitude of B but opposite in direction to that in (b).