Figure E2842 shows in cross section several conductors that

Figure E28.42 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, and the directions shown. Four paths, labeled a through d, are shown. What is the line integral for each path? Each integral involves going around the path in the counterclockwise direction. Explain your answers.

Solution

path a

No current is inside that path, thus line integral = 0

path b

Line integral for path b = mu₀ * -I₁ = 4 pi * 10^-7 * 4

= 5.03 * 10^-6 T.m

path c

Line integral for path c = mu₀ * (-I₁ + I₂)= 4 pi * 10^-7 * (-4 + 6)

= 2.51 * 10^-6 T.m

path d

Line integral for path c = mu₀ * (-I₁ + I₂ + I3)= 4 pi * 10^-7 * (-4 + 6 + 2)

= 5.03 * 10^-6 T.m