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In pursuit of an empirically determined value for the electron charge-to-mass \(e/m\) ratio we must guide electrons in a variable circular path using a homogeneous magnetic field. The Helmholtz configuration of two current-carrying coils with radius *R* separated by a distance *d* as shown in Fig. \ref{fig1} provides the necessary field. To determine the magnetic field \(\vec{\textit{B}}\) we will first find the magnetic field due to a single current-carrying loop using the Biot-Savart Law.

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The Biot-Savart Law shows how *d*\(\vec{B}\) depends on each differential piece of the current-carrying loop *Id*\(\vec{l}\), the position vector \(\vec{\textit{r}}\) of the point where we’re finding the field, and the sine of the angle \(\theta\) between these two vectors with the equation [1]: \[\label{eq1}
\textit{d}\vec{B} = \frac{\mu_0}{4\pi} \frac{\textit{I}\textit{d}\vec{l}\times\hat{\textit{r}}} {\textit{r}^2}\] Where \(\vec{\textit{r}}\) has been decomposed into its magnitude *r* and its unit vector \(\hat{r}\). The contributions to the magnetic field from the loop trace out a circle (see Fig. \ref{fig2}) thus the symmetrical horizontal components of the magnetic field cancel leaving a total field \(\vec{B}\) pointing upward. Due to this cancellation we only want to integrate the upward components *d*\(B_z\).

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We can isolate the components using *d*\(\vec{B}_z = \textit{d}\vec{B}\cos{\theta}\), which comes from the right triangle in the top of Fig. \ref{fig2} formed by *d*\(\vec{B}\), \(\vec{B}\), and the radius of the circle. Thus, integrating only the \(\hat{z}\) components of Eq. (\ref{eq1}) to obtain the total magnetic field we have: \[\label{eq2}
\vec{B}=\frac{\mu_0}{4\pi}\int{\frac{\textit{I}\textit{d}\vec{l}\times\hat{\textit{r}}} {\textit{r}^2}\cos{\theta}}\] We can use Fig. \ref{fig2} to simplify this integral by substituting constant values. Using the right triangle formed by *z*, *R*, and *\(\vec{r}\)* we know the magnitude \(\textit{r}=\sqrt{\textit{R}^2+\textit{z}^2}\) and we can also determine \(\cos{\theta}=\textit{R}/{\textit{r}}=\textit{R}/{\sqrt{\textit{R}^2+\textit{z}^2}}\). Re-writing Eq. (\ref{eq2}) with these values and bringing the constant current *I* outside the integral yields: \[\label{eq3}
\vec{B}=\frac{\mu_0\textit{I}}{4\pi(\textit{R}^2+\textit{z}^2)}\frac{\textit{R}}{\sqrt{\textit{R}^2+\textit{z}^2}}\int{\textit{d}\vec{l}\times\hat{\textit{r}}}\] Fortunately,because \(\textit{d}\vec{l}\) and \(\hat{\textit{r}}\) are perpendicular \(\int{\textit{d}\vec{l}\times\hat{\textit{r}}}\) becomes \(\int{\textit{d}l}\), which is the circumference of the wire loop given by \(2\pi\textit{R}\). Adding this result to Eq. (\ref{eq3}) gives the total magnetic field due to a single loop of current-carrying wire, but we are interested in the field from two coils with many current-carrying loops. Let us say there are *N* turns of wire in each coil then the magnetic field from a single coil would be *N* times our result. Since we also wish to determine the field due to two coils, we must also multiply the result by two, which yields our final expression for the magnetic field due to two coils with N turns: \[\label{eq4}
B=\frac{\mu_0\textit{N}\textit{I}\textit{R}^2}{(\textit{R}^2+\textit{z}^2)^{\frac{3}{2}}}\] We can further simplify Eq. (\ref{eq4}) by making two careful choices about the experimental setup. First, we choose to investigate the point exactly halfway between the coils, as shown in Fig. \ref{fig1}, such that \(\textit{z}=\frac{\textit{d}}{2}\). Second, we set the coils such that the separation distance *d* is the same as the radius *R*. The denominator of Eq. (\ref{eq4}) then becomes \((\textit{R}^2+(\frac{\textit{R}}{2})^2)^\frac{3}{2}\). Substitutting back into Eq. (\ref{eq4}) and canceling we arrive at our final expression for the magnetic field at the center of the Helmholtz configuration: \[\label{eq5}
B=\frac{8\mu_0\textit{N}\textit{I}}{\sqrt{125}\textit{R}}\]