Magnetic Field of 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]\displaystyle{ \vec{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\vec{l} \times \hat{r}}{r^2} }[/math]

Where:

• [math]\displaystyle{ \vec{B} }[/math] is the magnetic field vector at a point in space.
• [math]\displaystyle{ \mu_0 }[/math] is the permeability of free space, a constant representing how much resistance the vacuum of space offers to the formation of a magnetic field.
• [math]\displaystyle{ I }[/math] is the current through the wire.
• [math]\displaystyle{ d\vec{l} }[/math] is the differential length vector of the wire, representing a small segment of the wire.
• [math]\displaystyle{ \hat{r} }[/math] is the unit vector pointing from the wire segment to the point in space where the magnetic field is being calculated.
• [math]\displaystyle{ r }[/math] 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]\displaystyle{ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} }[/math]

Where:

• [math]\displaystyle{ \vec{B} }[/math] is the magnetic field.
• [math]\displaystyle{ d\vec{l} }[/math] represents a differential element of the path around the conductor.
• [math]\displaystyle{ \mu_0 }[/math] is the permeability of free space.
• [math]\displaystyle{ I_{\text{enc}} }[/math] is the total current enclosed by the path.

Strategy for Solving Problems

To effectively solve problems involving the calculation of magnetic fields due to curved wires, the following general strategy can be adopted: 1. Identify the Geometry: Determine the shape of the wire (straight, circular, semicircular, etc.) and any symmetries that might simplify the problem. 2. Choose the Right Law: Decide whether to use the Biot-Savart Law or Ampère's Law based on the problem's symmetry and complexity. 3. Set Up the Integral: For the Biot-Savart Law, set up the integral by identifying the differential element [math]\displaystyle{ d\vec{l} }[/math] and the position vector [math]\displaystyle{ \hat{r} }[/math]. For Ampère's Law, identify the path of integration. 4. Perform the Calculations: Carry out the integration or apply Ampère's Law as appropriate, keeping track of the direction of the magnetic field. 5. Combine Contributions: In cases with multiple segments (like straight and curved wires), calculate each segment's contribution and then sum them vectorially.

Computational Model for Magnetic Field of a Curved Wire

File:Curvedwirea.png

Illustration of the Biot-Savart Law applied to a curved wire segment.

The diagram above visually explains the computational model based on the Biot-Savart Law, which describes the magnetic field generated by a steady current. It shows the magnetic field [math]\displaystyle{ dB }[/math] at a point in space (point O) due to a small current element [math]\displaystyle{ ds }[/math] as being directly proportional to the current [math]\displaystyle{ I }[/math], the length of the element [math]\displaystyle{ ds }[/math], and the sine of the angle [math]\displaystyle{ \theta }[/math] between the element and the position vector [math]\displaystyle{ \vec{r} }[/math], and inversely proportional to the square of the distance [math]\displaystyle{ a }[/math] from the element to the point O. The constants [math]\displaystyle{ \mu_0 }[/math] and [math]\displaystyle{ 4\pi }[/math] are related to the permeability of free space and the geometry of the space, respectively. The total magnetic field [math]\displaystyle{ B }[/math] at point O due to the entire current-carrying wire is calculated by integrating the contributions from all such elements along the wire.

Practice Problems

Easy Problem: Straight Wire Segment

Problem Statement: Calculate the magnetic field at point P located a distance [math]\displaystyle{ d }[/math] from a straight wire segment of length [math]\displaystyle{ L }[/math] carrying current [math]\displaystyle{ I }[/math].

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 [math]\displaystyle{ R }[/math] carrying a current [math]\displaystyle{ I }[/math].

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 [math]\displaystyle{ R }[/math] with two straight segments of length [math]\displaystyle{ L }[/math], calculate the magnetic field at the semicircle's center. The wire carries current [math]\displaystyle{ I }[/math].

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.