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Year 4 of the mathematical and theoretical physics course of the University of Oxford

Collisionless plasma physics course of the University of Oxford

Past exam of the collisionless plasma physics course of the University of Oxford

2025

The barrier <wavenumber> is $\kappa=\omega_{pe0}\sqrt{x_T^2-x^2}/(cL)$. The substitution $x=x_T\sin u$ gives
$$
\begin{aligned}
\eta&=\frac{\omega_{pe0}}{cL}\int_{-x_T}^{x_T}\sqrt{x_T^2-x^2}\,dx
=\frac{\omega_{pe0}x_T^2}{cL}\int_{-\pi/2}^{\pi/2}\cos^2u\,du\\
&=\boxed{\frac{\pi\omega_{pe0}L}{2c}\left(1-\frac{\omega^2}{\omega_{pe0}^2}\right).}
\end{aligned}
$$
Thus the leading <transmission coefficient> is
$$
\boxed{T\simeq\exp\!\left[-\frac{\pi\omega_{pe0}L}{c}\left(1-\frac{\omega^2}{\omega_{pe0}^2}\right)\right].}
$$
At fixed $\omega/\omega_{pe0}$, increasing the column width or peak <plasma frequency> suppresses transmission exponentially. Raising $\omega$ towards $\omega_{pe0}$ narrows the evanescent region and increases transmission. However, when $\eta$ becomes order one or smaller the two <wave turning points> interact; the opaque-barrier <WKB approximation> cannot be extrapolated to claim $T\to1$ at $\omega=\omega_{pe0}$. Smooth matching to an exterior is also implicit in the leading result; at $|x|=L$ the stated profile joins continuously to <vacuum>.

\Image[/past-exam-of-the-collisionless-plasma-physics-course-of-the-university-of-oxford/original2025-tunnelling.png]
{title=Local squared <wavenumber> and evanescent barrier for an <ordinary plasma wave>}

The shaded area in the lower panel is $\eta$ after restoring the factor $\omega_{pe0}L/c$.

Solution

Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 2 / h / Solution

Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 2 / h / Solution

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