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Figue shows, in cross-section, several conductors that carry currents through the plane of the figure. The currents have the magnitudes $I_{1} = 4.0 A, I_{2} = 6.0 A,$ and $I_{3} = 2.0 A .$ In the directions shown. Four paths labelled $a$ to $d$ are shown. What is the line integral $\int B . d I$ for each path. Each integral involves going around the path in the counter-clokwise direction.
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(a) zero

(b) negative

(a) positive

(b) negative

(a) positive

(b) positive

(a) zero

(b) positive

answer is A.

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Detailed Solution

$\int B . d I = μ_{0} (i_{n e t})$

Along path (a), net current is enclosed by this path is zero. Hence, line integral $= 0$ .

Along path (b), $i_{n e t}$ is $\otimes$ . So, magnetic lines along this current is clockwise. But, we have to take line integral in counter clockwise direction. Hence, line integral will be negative.

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