The current-carrying wire loop in Fig. 29-60 a lies all in...

The current-carrying wire loop in Fig. $29-60 a$ lies all in one plane and consists of a semicircle of radius 10.0 $\mathrm{cm},$ a smaller semicircle with the same center, and two radial lengths. The smaller semicircle is rotated out of that plane by angle $\theta$ , until it is perpendicular to the plane (Fig. $29-60 b )$ . Figure $29-60 c$ gives the magnitude of the net magnetic field at the center of curvature versus angle $\theta$ . The vertical scale is set by $B_{a}=10.0 \mu$ and $B_{b}=12.0 \mu$ . What is the radius of the smaller semicircle?

The current-carrying wire loop in Fig. $29-60 a$ lies all in one plane
and consists of a semicircle of radius 10.0
$\mathrm{cm},$ a smaller semicircle with the same
center, and two radial lengths. The
smaller semicircle is rotated out of that
plane by angle $\theta$ , until it is perpendicular to the plane (Fig. $29-60 b )$ . Figure $29-60 c$ gives the magnitude of the
net magnetic field at the center of curvature versus angle $\theta$ . The
vertical scale is set by $B_{a}=10.0 \mu$ and $B_{b}=12.0 \mu$ . What is
the radius of the smaller semicircle?

Key Concepts

Geometric Dependence of Magnetic Fields

The orientation and shape of current-carrying elements critically affect the magnetic fields they produce. When parts of a current loop are rotated or displaced out of the original plane, the direction and magnitude of their magnetic field contributions change. Mastery of this concept is essential for understanding how geometrical modifications lead to variations in the net magnetic field.

Superposition Principle

The net magnetic field produced by multiple current segments is the vector sum of the fields generated by each individual segment. This principle allows for the analysis of complex current geometries by breaking them into simpler parts, calculating their individual effects, and then summing these effects to determine the overall field.

Biot-Savart Law

This fundamental law relates the magnetic field produced at a point in space to the current elements that generate it. It is used to calculate the contribution of each current segment, whether straight, curved, or in an arc, to the overall magnetic field. Understanding this law is crucial when analyzing magnetic field distributions produced by various current-carrying geometries.

Magnetic Field of a Circular Arc

The magnetic field generated at the center of a circular arc segment by a steady current has a well?defined expression that depends on the current, the arc’s radius, and the subtended angle. This concept is pivotal for analyzing partial loops, such as semicircular or smaller arcs, and helps in determining the contributions of different segments to the net magnetic field at a central point.

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