A current is set up in a wire loop consisting of a...

A current is set up in a wire loop consisting of a semicircle of radius $4.00 \mathrm{~cm},$ a smaller concentric semicircle, and two radial straight lengths, all in the same plane. Figure $29-47 a$ shows the arrangement but is not drawn to scale. The magnitude of the magnetic field produced at the center of curvature is $47.25 \mu \mathrm{T}$. The smaller semicircle is then flipped over (rotated) until the loop is again entirely in the same plane (Fig. $29-47 b$ ). The magnetic field produced at the (same) center of curvature now has magnitude $15.75 \mu \mathrm{T},$ and its direction is reversed from the initial magnetic field. What is the radius of the smaller semicircle?

Key Concepts

Algebraic Determination of Unknown Geometric Parameters

This involves setting up equations based on the known expressions for magnetic fields (such as those from semicircular arcs) and applying conditions from the problem – for instance, differences in the net magnetic field magnitude and direction after a change in geometry. By comparing the contributions from different parts of the loop and using algebra to isolate the unknown parameter (in this case, the radius of the smaller semicircle), one can solve for the desired quantity.

Magnetic Field from a Current?Carrying Arc

This concept involves calculating the magnetic field produced at a point by a segment of a current?carrying wire that forms part of a circle. The Biot–Savart law is applied to derive the contribution from an arc. For a circular arc, the magnitude of the magnetic field depends on the current, the radius of the arc, and the angular span of the arc. Understanding this allows one to compute the field due to semicircular segments by setting the angular span to ? radians.

Superposition Principle in Magnetostatics

The superposition principle states that the net magnetic field created by multiple current elements is the vector sum of the fields produced by the individual elements. In problems involving loops, semicircular arcs, and straight wires, each segment contributes a magnetic field at a point, and these contributions must be added (or subtracted if they are in opposite directions) to obtain the total field. This principle is key when analyzing how flipping a segment of the current loop alters the overall field.

Right?Hand Rule for Magnetic Fields

The right?hand rule is used to determine the direction of the magnetic field created by a current. By pointing the thumb in the direction of the current, the curl of the fingers gives the direction of the magnetic field loops. This concept is crucial when determining how the orientation or rotation of a segment (such as flipping the smaller semicircle) affects the direction of its field contribution relative to the other parts of the loop.

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