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    PHYS321 Derivation

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    Motivating the Derivation

    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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    \label{fig1} The Helmholtz configuration of two coaxial current-carrying wire coils of equal radii. Figure based on Ref. [2] Fig. 5.59.

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    The Biot-Savart Law

    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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    \label{fig2}The magnetic field due to a single current-carrying loop of wire. Figure based on Ref. [2] Fig. 5.21.