Quizzes · Computational EM & Plasma
Integrative questions across the course; each concept page carries its own quiz too.
Easy
E1 (MCQ). Which term of the Lorentz force can change a particle's kinetic energy?
- (a) \(q\mathbf{E}\) (b) \(q\mathbf{v}\times\mathbf{B}\) (c) both (d) neither
Answer
(a). The magnetic force is always ⊥ velocity — zero work, ever.
E2 (computational). Forward Euler on \(\dot y = 2t\), \(y(0) = 0\), \(\Delta t = 0.5\): compute \(y(1)\) and the exact value.
Answer
\(y_1 = 0 + 0.5\times0 = 0\); \(y_2 = 0 + 0.5\times1 = 0.5\). Exact: \(t^2 = 1\). Error 0.5 — first-order pain at coarse steps.
E3 (conceptual). Why does the course benchmark every integrator on a problem with a known exact solution?
Answer
Only an exact reference converts output into measured error, letting you verify the convergence order and catch implementation bugs — see convergence & error.
E4 (MCQ). In the PIC cycle, the step "deposit" moves information from:
- (a) grid → particles (b) particles → grid (c) grid → grid (d) particles → particles
Answer
(b). Charge assignment onto mesh points; "interpolate" is the reverse trip.
Medium
M1 (computational). An electron (\(q/m = 1.76\times10^{11}\) C/kg) in \(B = 0.01\) T: find the cyclotron frequency and period.
Answer
\(\omega_c = (q/m)B = 1.76\times10^9\) rad/s → \(f = 280\) MHz, \(T = 3.6\) ns. Your simulation time step must resolve this, whatever slow physics you care about — the tyranny of the fastest scale.
M2 (conceptual). Two simulations of the same magnetized orbit: one shows the radius slowly growing, the other shows it perfectly constant but the gyration phase drifting. Identify the likely integrators.
Answer
Growing radius = energy gain = forward Euler (or any non-symplectic explicit method). Constant radius with phase error = leapfrog — symplectic schemes preserve the orbit geometry but not the clock.
M3 (computational). A 2-D FDTD run uses \(N = 256\), \(\Delta x = 1/256\), \(c = 1\), \(\Delta t = \frac{\sqrt2}{2}\Delta x\). How many steps to cross the box once?
Answer
Crossing time \(= 1/c = 1\); steps \(= 1/\Delta t = 256\sqrt2 \approx 362\).
M4 (conceptual). Why does a purely toroidal magnetic field fail to confine a plasma?
Answer
\(B \propto 1/r\) ⇒ grad-B + curvature drifts are vertical and charge-dependent → charge separation → vertical E → E×B drives the whole plasma outward. Tokamaks add poloidal field so field lines wind around and short-circuit the separation.
M5 (MCQ). The Gauss–Seidel method solves the discrete Poisson equation by:
- (a) matrix inversion (b) repeated local averaging until convergence
- (c) Fourier transform (d) time marching with CFL limit
Answer
(b). Each point relaxes toward the mean of its neighbors (+ source); convergence is guaranteed but slow. → Poisson solvers
Hard
H1 (computational). A trapped particle in a mirror with \(B_\text{min} = 1\) T has pitch angle 60° at the midplane. At what field strength does it reflect?
Answer
\(\mu\) const: reflection where \(v_\perp = v\), i.e. \(B_\text{ref} = B_\text{min}/\sin^2 60° = 1/0.75 = 1.33\) T.
H2 (conceptual). In the two-stream simulation, doubling the particle count at fixed cells lowers the apparent early-time field energy. Why, and does the physical growth rate change?
Answer
The noise floor is shot noise: \(\mathcal{E}_\text{noise} \propto 1/N\). More particles = quieter start (the seed the instability grows from is smaller), so saturation arrives later — but the slope \(2\gamma\) of the linear phase is set by the physics (\(v_b\), densities), not by \(N\).
H3 (computational). Estimate the cost ratio of direct N-body vs PIC for \(N = 10^6\) particles on \(M = 10^3\) cells (per step, order-of-magnitude).
Answer
Direct: \(\sim N^2 = 10^{12}\) pair operations. PIC: deposit + push \(\sim N = 10^6\), field solve \(\sim M = 10^3\) → \(\sim 10^6\) total. Ratio \(\sim 10^6\) — the grid is six orders of magnitude cheaper.
H4 (conceptual). The leapfrog scheme applied to the Lorentz force is implicit, yet needs no iterative solver. What structural feature makes this possible, and where else in the course does the same trick appear?
Answer
The implicit unknown enters linearly (magnetic force ∝ \(\mathbf{v}\)), so \(\mathbf{v}_\text{new} + \mathbf{A}\times\mathbf{v}_\text{new} = \mathbf{C}\) has a closed-form vector solution. The same "implicitness made cheap by linearity" appears in FDTD's staggered updates and in the direct (tridiagonal) Poisson solve — linear structure is the algorithmicist's best friend.