# Determining Radius from Magnetic Field of a Single-Wire Loop

• frankifur
In summary, the conversation discusses a problem involving a single-turn wire loop producing a magnetic field at its center and on its axis. The relevant equations are mentioned and there is a question about using the sine function. The conversation also suggests simplifying the equation by taking the cubic root.
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.

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.

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?

frankifur
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.

Last edited:
frankifur
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.

## 1. How can I determine the radius of a single-wire loop from its magnetic field?

To determine the radius of a single-wire loop from its magnetic field, you can use the formula for the magnetic field at the center of a circular loop: $$B = \frac{\mu_0 I}{2R}$$, where $$B$$ is the magnetic field, $$\mu_0$$ is the permeability of free space, $$I$$ is the current, and $$R$$ is the radius. By rearranging the formula, you can solve for the radius $$R$$: $$R = \frac{\mu_0 I}{2B}$$.

## 2. What is the role of the current in determining the radius from the magnetic field?

The current $$I$$ flowing through the wire is directly proportional to the magnetic field $$B$$ generated at the center of the loop. To determine the radius $$R$$, you need to know the current because it affects the magnitude of the magnetic field. The radius can be calculated using the formula $$R = \frac{\mu_0 I}{2B}$$, indicating that a higher current results in a larger magnetic field for a given radius.

## 3. What units should be used for the magnetic field, current, and radius in the formula?

The standard units for the magnetic field $$B$$ are teslas (T), for the current $$I$$ are amperes (A), and for the radius $$R$$ are meters (m). The permeability of free space $$\mu_0$$ has a value of $$4\pi \times 10^{-7}$$ T·m/A in these units.

## 4. Can the formula be used for any point in the loop or just at the center?

The formula $$B = \frac{\mu_0 I}{2R}$$ specifically calculates the magnetic field at the center of a circular loop. The magnetic field at other points on or around the loop will be different and would require more complex calculations involving the Biot-Savart law or Ampère's law.

## 5. What assumptions are made in deriving the formula for the magnetic field at the center of a loop?

The primary assumptions are that the loop is perfectly circular, the current is uniformly distributed along the wire, and the loop is in a vacuum or air (where the permeability is $$\mu_0$$). Additionally, the formula assumes the loop is thin, meaning the wire's thickness is negligible compared to the radius of the loop.

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