Figure 29-68 shows two closed paths wrapped around two...

Figure $29-68$ shows two closed paths wrapped around two conducting loops carrying currents $i_{1}=5.0 \mathrm{~A}$ and $i_{2}=3.0$ A. What is the value of the integral $\oint \vec{B} \cdot d \vec{s}$ for (a) path 1 and (b) path $2 ?$

Key Concepts

Enclosed Current

Enclosed current refers to the algebraic sum of currents that pass through the area bounded by the chosen closed path. When applying Ampere’s Law, you consider only those currents that are enclosed by the loop, taking into account their directions; currents flowing in opposite directions can cancel each other out. This concept is key in determining the net current used in the integral calculation for the magnetic field.

Ampere’s Law

Ampere’s Law connects the magnetic field around a closed loop to the net current passing through the surface enclosed by that loop. It states that the line integral of the magnetic field, taken around any closed path, is equal to the permeability of free space multiplied by the total current that the path encloses. This principle is fundamental for calculating magnetic fields in systems with symmetrical current distributions and for understanding the underlying relationship between electricity and magnetism.

Line Integral of the Magnetic Field

The line integral of the magnetic field along a closed path represents the cumulative effect of the magnetic field along that path, taking into account both its magnitude and direction. This path integral is central in magnetostatics as it forms the basis of Ampere’s Law, linking the physical circulation of the magnetic field to the currents that generate it. It allows for the evaluation of the magnetic field in various geometries by integrating over appropriately chosen paths.

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