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= Magnetic Field Calculations for a Curved Wire =
Sidney Demings Fall 2023

== Background ==
Magnetic fields are a crucial aspect of electromagnetism, encompassing the study of forces and fields generated by moving electric charges. The focus here is on the magnetic fields produced by current-carrying wires, specifically when these wires are curved, as in electromagnets and electric motors.

== History ==
The exploration of magnetic fields and their relationship with electric currents began in the 19th century. Key figures like Hans Christian Ørsted and André-Marie Ampère made pioneering discoveries, laying the groundwork for modern electromagnetic theory.

== Equations and Descriptions ==

=== Biot-Savart Law ===
The Biot-Savart Law is used to calculate the magnetic field generated by a small segment of current-carrying wire:

: <math>\vec{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\vec{l} \times \hat{r}}{r^2}</math>

Where:
* '''\(\vec{B}\)''' is the magnetic field vector at a point in space.
* '''\(\mu_0\)''' is the permeability of free space, a constant representing how much resistance the vacuum of space offers to the formation of a magnetic field.
* '''\(I\)''' is the current through the wire.
* '''\(d\vec{l}\)''' is the differential length vector of the wire, representing a small segment of the wire.
* '''\(\hat{r}\)''' is the unit vector pointing from the wire segment to the point in space where the magnetic field is being calculated.
* '''\(r\)''' is the distance from the wire segment to the point in space.

=== Ampère's Law ===
Ampère's Law relates the magnetic field around a current-carrying conductor to the current it carries. It's particularly useful in symmetrical situations:

: <math>\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}</math>

Where:
* '''\(\vec{B}\)''' is the magnetic field.
* '''\(d\vec{l}\)''' represents a differential element of the path around the conductor.
* '''\(\mu_0\)''' is the permeability of free space.
* '''\(I_{\text{enc}}\)''' is the total current enclosed by the path.

== Practice Problems ==

=== Easy Problem: Straight Wire Segment ===
'''Problem Statement:'''
Calculate the magnetic field at point P located a distance \( d \) from a straight wire segment of length \( L \) carrying current \( I \).

'''Solution Approach:''' 
Utilize the simplified version of the Biot-Savart Law for a straight wire. The symmetry of the problem simplifies the integration process.

=== Medium Problem: Circular Wire Loop ===
'''Problem Statement:'''
Determine the magnetic field at the center of a circular loop with radius \( R \) carrying a current \( I \).

'''Solution Approach:''' 
Apply the Biot-Savart Law for a circular loop. Due to the symmetry, the integration over the loop's circumference simplifies, as all magnetic field contributions point in the same direction.

=== Hard Problem: Semicircular Wire with Straight Segments ===
'''Problem Statement:'''
For a wire bent into a semicircle of radius \( R \) with two straight segments of length \( L \), calculate the magnetic field at the semicircle's center. The wire carries current \( I \).

'''Solution Approach:''' 
This requires using the Biot-Savart Law for both the semicircular and the straight segments, calculating each segment's magnetic field contribution, and then summing them vectorially.


== Conclusion ==
Understanding the magnetic fields produced by curved wires is vital in electromagnetism. These calculations demonstrate the intricate interplay between electric currents and the resultant magnetic fields, highlighting a fundamental aspect of electromagnetic theory.