Spotlight #53 – Optics, Statistics & Electrochemistry

🔮 Optics — Light Polarization

Natural light oscillates in all transverse directions, but polarization selects a single direction. The new Light Polarization simulation puts four classic experiments at your fingertips:

Malus’s Law: three polarizers in series. Dragging the analyzer angle demonstrates I = I₀ cos²θ — first verified by Étienne-Louis Malus in 1808.
Brewster’s Angle: Fresnel equations (Rs and Rp) plotted against incidence angle. At Brewster’s angle the reflected beam is fully s-polarized; rotating the polarizer extinguishes it completely.
Wave View: five polarization types animated — linear, RCP, LCP, elliptical, and unpolarized — shown as Ex/Ey component traces and 2D Lissajous figures.
Birefringence: a wave plate with adjustable retardation Γ and fast-axis angle converts polarization states. Stokes S3 (circular component) is plotted to show quarter-wave and half-wave plate effects.

Related simulations in the optics category include Blackbody Radiation, Colors of Light, and Wave Interference.

🔮 Open Light Polarization →

📊 Statistics — Student’s t-Test

The t-test, introduced by William Sealy Gosset writing as “Student” in 1908, remains one of the most used statistical tests in science. The challenge is that its key outputs — p-value, confidence interval, power — are abstract without visual grounding.

The new simulator provides three test types on a single page:

One-sample: is the sample mean different from a known value? Drag the null μ₀ line directly on the scatter plot to see p-values update live.
Two-sample independent: do two groups differ? The Group A (blue) and Group B (pink) dot plots sit on top/bottom tracks with means marked; an arrow shows the raw difference Δ.
Paired t-test: before/after design. Each pair is connected with a line coloured green (improvement) or red (regression) — making the direction of individual changes immediately visible.

The lower canvas shows the t(df) density with red rejection tails for the chosen α level and a blue-shaded area for the observed p-value. The width of the t-distribution narrows as n increases — drag the sample size slider to see the transition from heavy tails (small n) to near-normal (large n).

Key Concepts Illustrated

Effect size (Cohen’s d) is independent of n — a small effect can be significant with enough data.
Statistical power (1−β) increases with larger n, smaller σ, and larger effect size.
Welch correction adjusts df downward for unequal variances, widening the t-distribution slightly.

Related probability simulations: Bayesian Inference, Central Limit Theorem, Bootstrap Resampling.

📊 Open t-Test Simulator →

🔋 Energy — Battery Electrochemistry

The Li-ion battery is arguably the most important energy technology of the past 30 years. Its behaviour is governed by layers of electrochemistry that are rarely made tangible.

Charge & Discharge Curves

The simulation computes full CC-CV charge and CC discharge curves from an analytic NMC OCV model. The OCV curve (open circuit voltage vs State of Charge) is overlaid as a dashed reference, and the gap between OCV and terminal voltage grows with current — the ohmic overpotential η_IR = I · R_int.

C-Rate Effects

Switching to C-Rate mode overlays five curves — C/5 through 5C — on a single discharge plot and generates a Ragone plot (energy vs power density) from the area under each curve. The classic battery-vs-capacitor tradeoff becomes visible: high C-rates sacrifice energy for power.

Capacity Fade

Three mechanisms are modelled simultaneously and their contributions plotted:

SEI growth follows a parabolic law (Q_loss ∝ n^α) — characteristic of lithium-ion consumption at the anode surface.
Lithium plating adds rapid linear loss above a cycle threshold, mimicking fast-charge or low-temperature damage.
Structural degradation adds a gentle linear component — particle cracking, cathode dissolution over time.

The 80% capacity retention threshold — the standard End-of-Life definition — is marked with a yellow dot and cycle count.

Butler-Volmer Kinetics

The Butler-Volmer equation plots the electrode current as a function of overpotential η, with anodic and cathodic branches shown separately. The Nyquist impedance plot in the lower canvas sketches the ohmic resistance intercept and the charge-transfer resistance semicircle, both derived from the exchange current density i₀.

Related energy simulations: Carnot Cycle, Combustion, Solar Cell.

🔋 Open Battery Simulator →