 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2017/s7/1/a/solution

In cylindrical coordinates the orthogonal velocity components are

\overset{r}{˙}

r \dot{θ}

\overset{z}{˙}

. Their squared magnitudes add, and

\dot{r} \cdot A = \overset{z}{˙} A (r)

. Thus

L = m (\overset{r}{˙}^{2} + r^{2} \dot{θ}^{2} + \overset{z}{˙}^{2}) /2 + q \overset{z}{˙} A (r)

. Taking the curl gives

B_{θ} = - A^{'} (r)

, consistent with the prescribed field.

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