Determining Radius from Magnetic Field of a Single-Wire Loop

[frankifur]

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.

 

[haruspex]

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?

 

[kuruman]

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.

 

[kuruman]

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?

 

[nasu]

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 B_center
back to the left hand side and take the root of order 3/2. The field values are given numbers.