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 cur- rent 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 cur-
rent 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.

Key Concepts

Ampère's Law

Ampère's Law is a foundational principle in magnetostatics that relates the line integral of the magnetic field around a closed loop to the net current passing through the loop. It provides a powerful method for calculating magnetic fields, especially in systems with symmetry, by equating the circulation of the magnetic field to the total enclosed current multiplied by the permeability of free space.

Superposition Principle

The superposition principle in electromagnetism states that the net magnetic field created by multiple current sources is the vector sum of the individual fields produced by each current. This concept simplifies the analysis of complex problems by allowing each current’s contribution to be calculated separately and then combined to form the overall field.

Current Direction and Sign Convention

Establishing the correct sign for each current contribution is critical when evaluating magnetic effects. The direction of current (such as out of the page versus into the page) determines the sign of its contribution in the context of Ampère’s Law, and the application of the right-hand rule helps in consistently assigning these directions to ensure accurate results.

Line Integral of a Vector Field

The line integral of a vector field along a closed path is a mathematical tool used to sum the component of the field tangent to the path at every point along the curve. In electromagnetic theory, such integrals are employed to connect the magnetic field’s behavior around a circuit to the currents passing through the enclosed area.

Transcript

Please give Ace some feedback