Worksheet 8: Magnetic Field of a Ring

Problem Formulation

Consider current I flowing in a circular ring of radius R=1 unit. Also assume that $K=\frac{\mu I}{4\pi }=1$ in these units. Choose the ring so that it is centered at (0,0,0), and that it lies in the xy plane. Use the Biot-Savart law to find the magnetic field due to this current ring at an arbitrary point (x,0,z) in the xz plane (i.e. a plane perpendicular to the plane of the ring). Note that this gives the field at any point since there is cylindrical symmetry about the z axis.

A few reminders:

The contribution from a small directed element of length $d\vec{l}$ of a wire carrying a current I to the magnetic field at a point $\vec{r}$ is given by Bio-Savart's law:

$$d\vec{B}=\frac{\mu I}{4\pi }\frac{d\vec{l}\times \vec{r}}{r^{3}}$$ (1)

Thus, to find the total magnetic field at a given point $\vec{r}% _{point}=(x,0,z)$ caused by a current-carrying ring, one needs to integrate the expression in equation (separately for each component, of course) over elements of the ring at position $\vec{r}_{ring}$, $\vec{r}=% \vec{r}_{point}-\vec{r}_{ring}$ (draw a figure to help you understand this). For a unit-radius ring in the xy plane, we can take the integration to be over the polar angle $\theta$, so that $\vec{r}_{ring}=\left( \cos \theta ,\sin \theta ,0\right)$. Also, for such a ring, $d\vec{l}=\left( \hat{z}% \times \vec{r}_{ring}\right) d\theta$, where $\hat{z}$ is the unit vector in the z direction (i.e. perpendicular to the plane of the ring).

Now you have all the expressions you need to write down (in a few lines using Mathematica--read next section if you wish) an expression giving $d\vec{B}$ from the element of wire at a polar angle $\theta$. This should have the form $d\vec{B}=\vec{B}_{\theta }\cdot d\theta$. Show your expression for $\vec{B}_{\theta }$ to your professor or TA before working on the rest of this worksheet. Then, the total magnetic field at the point $\vec{r}_{point}$ is simply the integral:

$$\vec{B}=\int_{\theta =0}^{2\pi }\vec{B}_{\theta }\cdot d\theta$$ (2)

About this document ...

This document was generated using the LaTeX2HTML translator Version 98.1p1 release (March 2nd, 1998)

Copyright © 1993, 1994, 1995, 1996, 1997, Nikos Drakos, Computer Based Learning Unit, University of Leeds.

The command line arguments were:
latex2html -split 0 worksheet8Help_00.tex.

The translation was initiated by Phil Duxbury on 2000-10-30