Figure E28.38 shows, in cross section, several conductors that carry currents through

step 1 So here in this problem we have three wire currents, i1 which is equal to 3 .5 umpairs that is p

Figure E28.38 shows, in cross section, several ...

Figure E28.38 shows, in $\begin{array}{l}\text { cross section, several } & \text { conductors }\end{array}$ that carry currents through the plane of the figure. The currents have the magnitudes $I_{1}=3.5 \mathrm{~A}, I_{2}=6.4 \mathrm{~A}$ and $I_{3}=1.7 \mathrm{~A},$ and the directions shown. Four paths, labeled $a$ through $d$, are shown. What is the line integral $\oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}$ for each path? Each integral involves going around the path in the counterclockwise direction. Explain your answers.

Figure E28.38 shows, in $\begin{array}{l}\text { cross section, several } & \text { conductors }\end{array}$ that carry currents through the plane of the figure. The currents have the magnitudes $I_{1}=3.5 \mathrm{~A}, I_{2}=6.4 \mathrm{~A}$
and $I_{3}=1.7 \mathrm{~A},$ and the directions shown. Four paths, labeled $a$ through $d$, are shown. What is the line integral $\oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}$ for each path? Each integral involves going around the path in the counterclockwise direction. Explain your answers.

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Key Concepts

Ampère’s Law

Ampère’s Law states that the line integral of the magnetic field around a closed loop equals the permeability of free space multiplied by the net current enclosed by that loop. This principle is foundational in magnetostatics and is used to relate the magnetic field in the vicinity of currents to the currents themselves.

Enclosed Current

The concept of enclosed current refers to the total current that passes through the surface bounded by the integration path. When applying Ampère’s Law, only the currents that penetrate the area enclosed by the path contribute to the line integral; currents outside the loop do not affect the result, regardless of their proximity or magnitude.

Path Orientation

The direction in which the integration path is traversed (in this case, counterclockwise) is critical because it determines the sign of the enclosed current when using the right-hand rule. This convention ensures consistency in the application of Ampère’s Law by correlating the direction of the magnetic field with the direction of the current flow.

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