In Fig. 29-41, a wire forms a semicircle of radius R=9.26 cm and two (radial) straight segments

step 1 So in this problem, we have been given a current carrying wire that is divided into two segments

In Fig. 29-41, a wire forms a semicircle of radius R=9.26...

In Fig. 29-41, a wire forms a semicircle of radius $R=9.26 \mathrm{~cm}$ and two (radial) straight segments each of length $L=13.1 \mathrm{~cm}$. The wire carries current $i=34.8 \mathrm{~mA}$. What are the (a) magnitude and (b) direction (into or out of the page) of the net magnetic field at the semicircle's center of curvature $C ?$

Key Concepts

Biot–Savart Law

The Biot–Savart Law provides the fundamental basis for calculating the magnetic field produced by a small current element. In problems involving current distributions, it is used to integrate over the current path to determine the field at a specified point. This law is especially useful for non?uniform or curved current configurations and underlies the derivation of formulas for the fields of wires, loops, and arcs.

Magnetic Field of a Circular Arc

For a circular arc carrying a steady current, the magnetic field at the center of curvature can be derived via integration of the Biot–Savart Law. The result shows that the magnitude of the field is proportional to the angle subtended by the arc. In particular, for an arc subtending an angle ? (in radians), the field is given by (??I?)/(4?R). When the arc is a semicircle (? = ?), this formula simplifies accordingly.

Superposition Principle

The superposition principle states that the net magnetic field at any point due to several current elements or segments is the vector sum of the fields produced by each segment individually. In problems that involve different parts of a current-carrying conductor (such as curved arcs and straight segments), this principle allows us to calculate each contribution separately and then add them to find the overall field.

Right-Hand Rule

The right-hand rule is a simple and effective mnemonic for determining the direction of the magnetic field produced by a current-carrying conductor. By orienting the thumb in the direction of the conventional current flow, the curling of the fingers indicates the direction of the magnetic lines of force around the conductor. This rule is essential for understanding the directional aspects of magnetic fields in various geometries.

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