Each of the eight conductors in Fig. 29-69 carries 2.0 A of current into or out of the page. Tw

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Each of the eight conductors in Fig. 29-69 carries 2.0 A of...

Each of the eight conductors in Fig. $29-69$ carries $2.0 \mathrm{~A}$ of current into or out of the page. Two paths are indicated for the line integral $\oint \vec{B} \cdot d \vec{s}$. What is the value of the integral for (a) path 1 and
(b) path 2 ?

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Key Concepts

Closed Line Integral in a Magnetic Field

Evaluating a closed line integral in the context of a magnetic field involves integrating the magnetic field vectors along a specified closed path. The result of this integral depends solely on the net current enclosed by the loop, regardless of the specific shape of the path, illustrating the path independence of the integral when Ampere’s Law applies.

Ampere's Law

Ampere's Law is a fundamental principle in electromagnetism that relates the line integral of the magnetic field around a closed loop to the total current enclosed by that loop. It is typically written as ?B·ds = ??I_enc, where ?? is the permeability of free space and I_enc is the net current passing through the area enclosed by the loop.

Enclosed Current

The concept of enclosed current involves determining the net current that passes through the surface bounded by a closed integration path. In problems involving multiple currents, each current contributes to the total enclosed current, and their algebraic sum (taking into account the direction of current flow) is used in applying Ampere’s Law.

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