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Quizzes

Answers are hidden — work each one before opening it. Individual concept pages carry their own three-question sets; this page is for cross-cutting practice.

Conceptual Questions

Q1. What is the physical difference between Debye shielding in a plasma and screening in a metal?

Answer

In a plasma the shielders are a dilute, classical, Boltzmann-distributed cloud with \(N_D \gg 1\) particles inside a Debye sphere. In a metal the conduction electrons are degenerate and \(N_D < 1\) — the screening length (Thomas–Fermi) is smaller than the interparticle spacing, so the continuum shielding-cloud picture fails outright. See the parameter survey, where copper fails the test.

Q2. Why does the frozen-in theorem fail in Sweet–Parker reconnection even though the plasma is still highly conducting?

Answer

Frozen-in fails where the diffusion term beats the convection term, i.e. where the gradient scale is small enough that the local magnetic Reynolds number drops to order unity. In the Sweet–Parker sheet the thickness \(\delta = L/\sqrt{S}\) is tiny, so even a minute resistivity dominates there. Global \(S \gg 1\) does not prevent a thin local layer from being resistive.

Q3. How does magnetic-moment conservation explain the loss cone in Earth's radiation belts?

Answer

\(\mu = mv_\perp^2/2B\) is conserved, so a particle moving into stronger field converts parallel motion into perpendicular. It mirrors only if \(v_\perp\) can absorb all the energy before the field peaks, requiring \(\sin\alpha > 1/\sqrt{R_m}\). For the \(L=4\) line \(R_m = 115\) and \(\alpha_{\rm LC} = 5.3°\); anything inside that cone precipitates and makes aurora. Worked fully in the Van Allen example.

Q4. Why can a purely toroidal magnetic field not confine a plasma, and what fixes it?

Answer

\(B \propto 1/R\) has both curvature and a gradient, so ∇B and curvature drifts push ions up and electrons down. The resulting vertical \(\mathbf{E}\) drives an \(\mathbf{E}\times\mathbf{B}\) drift that is radially outward for both species. Adding a poloidal field twists each line so it samples both top and bottom, averaging the drift away. See magnetic confinement.

Q5. Ideal MHD has exactly two sources of instability. Name them and give one instability driven by each.

Answer

From the energy principle: the pressure–curvature term (drives interchange and ballooning modes) and the parallel-current term (drives kinks). Every other term — field bending, field compression, plasma compression — is positive-definite and stabilising.

Computational Questions

Q1. Calculate the Lundquist number for a tokamak plasma with \(n=10^{20}\) m⁻³, \(T=1\) keV, \(B=3\) T, \(L=10\) m, \(\eta=10^{-6}\) m²/s.

Answer

\(\rho = nm_i = 1.67\times10^{-7}\) kg/m³. \(v_A = B/\sqrt{\mu_0\rho} = 3/\sqrt{(1.2566\times10^{-6})(1.67\times10^{-7})} = 3/4.58\times10^{-7} \approx 6.5\times10^{6}\) m/s. With \(\eta\) given as a magnetic diffusivity, \(S = Lv_A/\eta = (10)(6.5\times10^6)/10^{-6} \approx 6.5\times10^{13}\) — enormous, which is exactly why Sweet–Parker is too slow.

Q2. For a Z-pinch with \(I=1\) MA and \(a=0.5\) m, compute \(B_\theta(a)\) and the plasma \(\beta\) if \(p = 2\) Pa.

Answer

\(B_\theta = \mu_0I/2\pi a = (1.2566\times10^{-6})(10^6)/(2\pi\times0.5) = 0.40\) T. \(B^2/2\mu_0 = 0.16/(2.513\times10^{-6}) = 6.4\times10^4\) Pa. \(\beta = 2/6.4\times10^4 = 3.1\times10^{-5}\) — a very low-beta pinch.

Q3. A mirror has \(R_m = 10\). What fraction of an isotropic plasma is in the loss cone?

Answer

\(\sin\alpha_{\rm LC} = 1/\sqrt{10} = 0.316\), so \(\alpha_{\rm LC} = 18.4°\). Lost fraction \(= 1 - \cos 18.4° = 0.051\), i.e. 5.1% immediately — and collisions keep refilling it.

Q4. An ionosonde measures a critical frequency of 7 MHz. What is the peak electron density, and will a 100 MHz FM signal reflect?

Answer

\(\sqrt{n_e[\text{cm}^{-3}]} = 7\times10^6/8980 = 780\), so \(n_e = 6.1\times10^{5}\) cm⁻³ \(= 6.1\times10^{11}\) m⁻³. At 100 MHz the signal is far above cutoff and passes straight through — vertically. Even at 80° incidence the maximum usable frequency is only \(7\times\sec 80° \approx 40\) MHz. See the ionosphere example.

Q5. Derive the growth rate of the Buneman instability from the two-stream dispersion relation.

Answer

Start from \(1 = \omega_{pe}^2/(\omega - kv_0)^2 + \omega_{pi}^2/\omega^2\). In the Buneman limit (cold electrons drifting through cold ions) the fastest growth occurs near \(\omega \approx kv_0\), and expanding gives a cubic whose complex roots yield \(\gamma_{\max} \approx \frac{\sqrt3}{2}\left(\frac{m_e}{2m_i}\right)^{1/3}\omega_{pe}\) at \(kv_0 \approx \omega_{pe}\). Note the mass-ratio scaling — growth is fast but not \(\omega_{pe}\)-fast.

Multiple-Choice Questions

  1. Plasma frequency — A) \(\omega_{pe}\propto\sqrt{n_e}\) · B) \(\propto n_e\) · C) \(\propto\sqrt{T_e}\) · D) \(\propto T_e\) ??? success "Answer" A. \(\omega_{pe} = \sqrt{n_ee^2/\epsilon_0m_e}\) — no temperature dependence at all.

  2. Debye length — A) grows with temperature · B) shrinks with density · C) is independent of charge · D) both A and B ??? success "Answer" D. \(\lambda_D \propto \sqrt{T/n}\).

  3. Frozen-in theorem — A) always true · B) true when \(\eta = 0\) · C) requires \(T_i = T_e\) · D) applies only to unmagnetised flows ??? success "Answer" B. It is exact in ideal MHD; any resistivity breaks it, most consequentially in thin current sheets.

  4. Sweet–Parker rate — A) \(M_A\sim S^{1/2}\) · B) \(\sim1/S\) · C) \(\sim S^{-1/2}\) · D) constant ??? success "Answer" C. And the sheet aspect ratio \(\delta/L\) has the same scaling.

  5. Landau damping — A) needs collisions · B) strongest when \(k\lambda_D\gg1\) · C) depends on \(\partial f_0/\partial v\) at resonance · D) only for ions ??? success "Answer" C. It is collisionless, and its sign is set by the slope of the distribution at the phase velocity — more slow particles than fast means net energy to the particles.

  6. E×B drift is the same for ions and electrons because — A) both have the same mass · B) the drift \(\mathbf{E}\times\mathbf{B}/B^2\) contains no charge · C) electrons are much faster · D) it is not the same ??? success "Answer" B. Hence it drives no current — unlike the ∇B, curvature and gravitational drifts, which all carry \(1/q\). Verify it in the drift orbit lab.

  7. The O-mode is used for tokamak interferometry because — A) it has the highest frequency · B) \(n^2 = P\), so its cutoff depends only on density, not on \(B\) · C) it cannot be reflected · D) it resonates at \(\omega_{ce}\) ??? success "Answer" B. With \(\mathbf{E}\parallel\mathbf{B}_0\) the electrons never feel the field, so the phase shift is a clean measure of \(\int n_e\,dl\).

  8. The virial theorem shows that — A) plasmas cannot be confined magnetically · B) no static isolated plasma can confine itself with its own field · C) \(\beta\) must exceed 1 · D) only toroidal devices work ??? success "Answer" B. External currents — coils, walls — or gravity are mandatory. Stars evade it via gravity; every laboratory device pays for coils.

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