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

Eight wires cut the page perpendicularly at the points shown in Fig. $30-40 .$ A wire labeled with the integer $k(k=1,2, \ldots,$, 8) carries the current $k i$. For those 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 in the direction shown.

Eight wires cut the page perpendicularly at the points shown in Fig. $30-40 .$ A wire labeled with the integer $k(k=1,2, \ldots,$,
8) carries the current $k i$. For those 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 in the direction shown.

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Key Concepts

Ampère's Circuital Law

This fundamental law of electromagnetism states that the line integral of the magnetic field around any closed loop is equal to the permeability constant multiplied by the net current enclosed by the loop. It provides a powerful method for relating magnetic fields to the currents that produce them and is used to compute magnetic effects in systems with high symmetry.

Right-Hand Rule and Current Sign Convention

The right-hand rule is used to determine the direction of the magnetic field produced by electrical currents. In conjunction with conventional current sign conventions, where currents directed out of the page are considered positive and those into the page negative (or vice versa as required), it allows one to correctly account for the contributions of multiple current elements when evaluating the net magnetic effects.

Superposition Principle in Magnetism

The superposition principle states that the total magnetic field created by multiple sources is the vector sum of the individual magnetic fields produced by each source independently. This principle is essential in problems where several currents contribute to the overall magnetic field within a region, enabling one to add their effects to determine the net field or the net current enclosed by a closed path.

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