Eight wires cut the page perpendicularly at the points shown...

Eight wires cut the page perpendicularly at the points shown in Fig. $29-70 .$ A wire labeled with the integer $k(k=1,2, \ldots, 8)$ carries the current $k i,$ where $i=4.50 \mathrm{mA}$ For those wires with odd $k,$ the current is out of the page; for those with even $k,$ it is into the page. Evaluate $\oint \vec{B} \cdot d \vec{s}$ along the closed path indicated and in the direction shown.

Eight wires cut the page perpendicularly at the points shown in
Fig. $29-70 .$ A wire labeled with the
integer $k(k=1,2, \ldots, 8)$ carries
the current $k i,$ where $i=4.50 \mathrm{mA}$
For those wires with odd $k,$ the current is out of the page; for those with even $k,$ it is into the page.
Evaluate $\oint \vec{B} \cdot d \vec{s}$ along the closed
path indicated and in the direction
shown.

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Key Concepts

Current Direction and Sign Convention

When applying Ampere's Law, it is essential to assign appropriate signs to currents based on their directions relative to the chosen integration path. The algebraic sum of these signed currents (the enclosed current) determines the magnitude and direction of the net magnetic field circulation. This concept relies on consistent conventions, such as the right-hand rule, to distinguish between currents flowing into and out of the page or in other orientations.

Superposition Principle

The superposition principle allows for the analysis of multiple current sources by simply adding their individual contributions to determine the overall effect. In the context of magnetic fields and Ampere's Law, the net enclosed current is the sum of the individual currents, each considered with its appropriate sign. This principle is critical when dealing with multiple wires or current-carrying elements within the same closed path.

Ampere's Law

Ampere's Law states that the line integral of the magnetic field around a closed loop is directly proportional to the net current enclosed by that loop. This principle, expressed as ? B · ds = ?? I_enc, provides a powerful method for calculating the magnetic field in situations where high symmetry or well-defined current distributions allow for analytical solutions.

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