Magnetic Field of a Spinning Ring

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Section 17.5 Magnetic Field of a Spinning Ring

Recall that for a thin current-carrying loop we have

\begin{gather*} \BB(\rr) = {\mu_0\over 4\pi} \int {\II(\rrp)\times(\rr-\rrp)\,ds'\over|\rr-\rrp|^3} = - {\mu_0 I\over 4\pi} \int {(\rr-\rrp)\times d\rrp\over|\rr-\rrp|^3} \end{gather*}

Activity 17.5.1.

Find an expression for the magnetic field for a spinning circular ring with radius \(R\text{,}\) charge \(Q\text{,}\) and period \(T\text{.}\) The expression should be valid everywhere in space and simplified enough that it could be evaluated by a computer algebra programs such as Mathematica or Maple.

Recall that for a thin current-carrying loop we have

\begin{gather*} \BB(\rr) = {\mu_0\over 4\pi} \int {\II(\rrp)\times(\rr-\rrp)\,ds'\over|\rr-\rrp|^3} = - {\mu_0 I\over 4\pi} \int {(\rr-\rrp)\times d\rrp\over|\rr-\rrp|^3} \end{gather*}

For a circular ring of current, we have

\begin{gather*} |\rr-\rrp|= \sqrt{s^2 - 2\, sR \cos(\phi-\phi') + R^2 + z^2} \end{gather*}

Figure 17.5.1. The magnetic field \(\BB\text{,}\) shown in a vertical plane with the \(z\)-axis at the left, superimposed on top of the level curves of \(A_\phi\text{.}\)

Make sure you can use this expression to approximate the magnetic field in the various regimes considered in the activity. You can check your work using this Mathematica notebook ¹. A graphical representation of the result is shown in Figure 17.5.1.

math.oregonstate.edu/bridge/paradigms/vfbring.nb