Determining Radius from Magnetic Field of a Single-Wire Loop

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Homework Help Overview

The discussion revolves around a problem involving a single-turn wire loop that produces a magnetic field at its center and along its axis. Participants are tasked with determining the radius of the loop and the current flowing through it, utilizing the Biot-Savart Law.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster expresses uncertainty about their approach and the algebra involved, suggesting they may be misinterpreting the problem. They question the use of sine for the axial component of the magnetic field. Other participants discuss the relationships between the magnetic field components and the geometry of the loop, with some suggesting alternative ways to manipulate the equations without expanding them.

Discussion Status

Participants are actively engaging with the problem, exploring different interpretations of the equations involved. Some guidance has been offered regarding the relationships between the magnetic field components, but there is no explicit consensus on the correct approach yet.

Contextual Notes

The original poster has shared their written work for review, indicating potential issues with their calculations. There is a focus on the definitions and conventions used in the equations, which may contribute to the confusion in the problem-solving process.

frankifur
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Homework Statement
A single-turn wire loop produces a magnetic field of 41.2 μT at its center, and 5.15 nT on its axis, at 26.0 cm from the loop center.

a. Find the radius

b. Find the current
Relevant Equations
Biot-Savart Law
So I thought I knew how to do this problem but I've run into some issues that make the algebra feel impossible and I am beginning to feel like I'm taking the wrong approach, I ended up rewriting it in a doc because I was concerned maybe my handwriting was the cause of my error so the work is attached.
 

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frankifur said:
Homework Statement: A single-turn wire loop produces a magnetic field of 41.2 μT at its center, and 5.15 nT on its axis, at 26.0 cm from the loop center.

a. Find the radius

b. Find the current
Relevant Equations: Biot-Savart Law

So I thought I knew how to do this problem but I've run into some issues that make the algebra feel impossible and I am beginning to feel like I'm taking the wrong approach, I ended up rewriting it in a doc because I was concerned maybe my handwriting was the cause of my error so the work is attached.
sin for the axial component? Are you sure?
 
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haruspex said:
sin for the axial component? Are you sure?
Looks OK to me. Angle ##\theta##, indicated by an arc in the small upper triangle, is equal to the angle indicated by an arc in the larger triangle. The symbols used by the OP to define the sine as ##R/x## are a bit unconventional.
Screen Shot 2023-04-06 at 7.40.00 AM.png
 
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To @frankifur:
Note that $$B_{axis}=\frac{B_{center}R^3}{\left[R^2+z^2 \right]^{3/2}}=\frac{B_{center}\cancel{R^3}}{\cancel{R^3}\left[1+(z/R)^2 \right]^{3/2}}.$$Does this help?
 
You don't need to expand the paranthesis. Just take the cubic root of both sides and you have an eqution in R2. Or, if you rearange it as suggested by Kuruman, move the Bcenter
back to the left hand side and take the root of order 3/2. The field values are given numbers.
 

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