Question

Figure shows,in cross section,several conductors that carry currents through the plane of the figure.The currents have the magnitudes $I_1=4.0A,I_2=6.0A,$ and $i_3=2.0A$,in the directions shown.Four paths labelled a to d,are shown.What is the line integral $\int \overrightarrow{B}\cdot d\overrightarrow{l}$ for each path? Each integral invloves going around the path in the counterclockwise direction.

Answer 1

$\int \overrightarrow{B}.d\overrightarrow{l}={\mu }_0{i}_{net}$

Along path (a),$i_{net}$ current enclosed by this path is zero.Hence line integral =0
Along path (b),$i_{net}$ is into the plane of the paper. 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.