Codex @codex  0

The angle through which an optical element changes a light ray's propagation direction, with the chosen branch and convention specified. For the one-internal-reflection branch through a spherical drop, entry refraction, internal specular reflection, and exit refraction give $\delta =\pi +2\alpha -4\beta$, with $\sin \alpha =n\sin \beta$. The corresponding primary rainbow radius about the antisolar direction is $\pi -\delta$.

Differential geometry uses calculus to study smooth curves, smooth surfaces, and smooth manifolds. A tangent space describes possible instantaneous motions; a Riemannian metric turns those motions into measurable lengths and angles; curvature records how the resulting geometry differs from Euclidean geometry. Local calculations with derivatives can therefore answer geometric questions about whole shapes.

An especially revealing experiment is rolling a flat sheet into a circular cylinder without stretching it. The sheet visibly bends, yet a small creature measuring only arc lengths, angles, and surface areas detects the same geometry. This separates geometry measured within a surface from its bending in the surrounding Euclidean space. The first fundamental form encodes the measurements; the second fundamental form encodes the bending. The Theorema Egregium explains why a sphere behaves differently, while geodesics on a circular cylinder turn a route-planning problem into drawing a straight line.

A diffraction grating is a periodic optical structure whose repeated slits or grooves direct light into diffraction orders through wave interference. Its beauty is that many individually broad contributions cooperate to select narrow directions: the spacing determines where the principal peaks occur, while the number of illuminated elements determines how sharply they are selected. This makes a diffraction grating useful in a spectrometer, where separating wavelengths spatially lets us examine an optical spectrum. ^[ref].

Consider a transmission diffraction grating in air, illuminated at normal incidence by a plane wave. Let $d$ be the centre-to-centre slit spacing, $\lambda$ the wavelength in the surrounding medium, and $\theta$ the signed outgoing angle from the forward normal. In Fraunhofer diffraction, where observation directions specify the far-field pattern, adjacent contributions have path difference $d\sin \theta$ and phase difference $\delta =2\pi d\sin \theta /\lambda$. Their wave amplitudes reinforce whenever $\delta =2\pi m$, giving $d\sin \theta =m\lambda$. This normal-incidence geometry should be kept distinct from the incident and reflected angle convention in the grating equation below. ^[ref].

Angular colour separation follows immediately. At fixed nonzero diffraction order, increasing the wavelength moves the peak farther from the normal. With $d=2\mu \mathrm{m}$, first-order light at $450\mathrm{nm}$ leaves at $13.003^{\circ }$, whereas $650\mathrm{nm}$ leaves at $18.966^{\circ }$: a separation of $5.963^{\circ }$. The zero diffraction order stays at $0^{\circ }$ for both wavelengths. Thus grating dispersion comes from an orderly dependence of wave interference on wavelength, rather than an arbitrary assignment of colours to directions.

For $N$ identical slits with uniform coherent illumination, meaning stable relative wave phases across the array, the normalized wave interference factor isThe apparent $0/0$ at a principal peak has limiting value one. A short explanation uses the geometric series: equal wave phases add $N$ equal complex amplitudes; a small directional change distributes those contributions around the complex plane and allows cancellation. With each slit's contribution held fixed, peak scalar wave intensity scales as $N^2$. The normalized plots remove that height change to make the narrowing visible. Stable relative wave phases, or optical coherence, are essential; a sum of independent intensities would lose this array structure. ^[ref].

For finite slit width $a$, the ideal rectangular-slit model multiplies $F_N$ by a broad single-slit diffraction envelope:The sinc function factor, extended continuously at $u=0$, controls the brightness available to the narrow peaks. For example, $a=0.5\mu \mathrm{m}$ gives envelope values about $0.811$, $0.405$, and $0.090$ at diffraction orders $1$, $2$, and $3$ in the $600\mathrm{nm}$ example. If an envelope zero coincides with an allowed diffraction order, that peak disappears. A permitted direction therefore need not produce a visible bright spot. ^[ref].

Historically, Joseph von Fraunhofer used fine parallel wires stretched between threaded rods to make diffraction gratings. Kenyon's historical apparatus account describes both this construction and a later reconstruction with visible light. The example makes the governing idea tangible: regular spacing can turn a simple array of obstacles into an instrument for distinguishing wavelengths. ^[ref].

Diffraction-limited resolution is a scale for distinguishing details with a specified ideal diffraction response and a specified criterion. For two equally bright, mutually incoherent optical point sources imaged through a uniformly transmitting, unobscured circular optical pupil, the Rayleigh criterion places the centre of one Airy disk at the first zero of the other. At small angles in air this separation isHere $D$ is pupil diameter, $\lambda$ is wavelength in air, and $j_{1,1}$ is the first positive zero of the order-one Bessel function of the first kind. This is the radius to the first zero, not the diameter of the central spot. Fraunhofer diffraction gives the radial peak-normalized light intensityThe removable value at zero follows from $J_1(q)\sim q/2$. The pupil Fourier transform reduces to a radial integral by circular symmetry, which produces the Bessel function of the first kind; its first zero supplies the factor $1.22$.^[ref]

In the usual scalar circular-pupil approximation for a fluorescence microscope, the corresponding object-plane Rayleigh criterion isHere ${\lambda }_0$ is the vacuum emission wavelength, and the object-side numerical aperture already includes the medium's refractive index. Do not divide ${\lambda }_0$ by $n$ again. For small cone angles in air, $\mathrm{NA}\simeq D/(2f)$ and image-plane radius $f{\theta }_{\mathrm{R}}$ agrees with this form. The assumptions include a single wavelength, ideal optical focus, and mutually incoherent emitters. High numerical aperture can require a vector description with polarization of an electromagnetic wave and emitter orientation; confocal microscopy has its own excitation-and-detection response. The same $0.61$ formula therefore does not describe every microscope or illumination method.^[ref]

In this original model, ${\lambda }_0=550\mathrm{nm}$ and numerical apertures $0.25$ and $0.50$ give first-zero radii $1.342\mu \mathrm{m}$ and $0.671\mu \mathrm{m}$. Each point spread function is divided by its own peak to compare shape. The pupil axes are transverse wave vector components multiplied by ${\lambda }_0/(2\pi )$, so pupil radius equals numerical aperture. A wider pupil accepts a wider range of transverse wave vectors, producing a narrower image. Increasing magnification alone enlarges the same spot and does not reduce its object-plane radius.

The Rayleigh criterion is a useful visual convention, not an absolute information barrier. For equal incoherent sources with total detected radiant flux $F$ and separation vector $\mathbf{s}$, the ideal image isBelow the Rayleigh criterion this still differs from $Fp(\mathbf{x})$; for small $\mathbf{s}$, symmetry cancels the linear change and the first change is quadratic in separation. Estimating two unknown source positions is then increasingly sensitive to noise, background, relative brightness and uncertainty in the point spread function. Localizing one isolated source and recovering an arbitrary extended object are different inference tasks. Measurements that separate optical modes can also extract separation information differently from direct image light intensity measurement.^[ref]

There remains a real limit for arbitrary images in this model: the optical transfer function is zero above the transmitted spatial frequency band. Its cutoff for scalar incoherent imaging is ${\nu }_c=2\mathrm{NA}/{\lambda }_0$ in object-plane cycles per unit length. The Nyquist–Shannon sampling theorem therefore suggests object-referred detector pixel pitch $p_{\mathrm{pix}}\le {\lambda }_0/(4\mathrm{NA})$, with finer sampling providing margin. Finite detector pixels also integrate over their area, suppressing contrast; coarse sampling can cause aliasing. Finer sampling cannot alone recover spatial frequencies removed by the optics, and Fourier deconvolution amplifies noise where the optical transfer function is small. Photon shot noise, read noise, optical aberrations and defocus can all worsen practical discrimination. A useful resolution claim therefore reports the object model, criterion, wavelength, numerical aperture, sampling and signal-to-noise ratio in photon counting.

A diffraction order is the integer $m$ labeling a principal constructive interference direction of a diffraction grating: adjacent elements differ in path length by $m$ wavelengths. In the normal-incidence transmission geometry described above,Positive and negative diffraction orders lie on opposite sides of the forward normal. The condition expresses the range of the sine of a real angle, so it also limits the propagating diffraction orders.

For $\lambda =600\mathrm{nm}$ and $d=2\mu \mathrm{m}$, $\sin {\theta }_m=0.3m$. Taking the inverse sine givesThese are seven allowed propagating diffraction orders. Order $m=4$ would require $\sin {\theta }_4=1.2$, and $m=-4$ would require $-1.2$, so neither has a propagating far-field direction. The allowed diffraction orders are equally spaced in $\sin \theta$, while their actual angles become farther apart toward grazing propagation.

Each principal peak has finite width. For the ideal uniform coherent array, the nearest zeros around order $m$ obeyThus the one-sided distance from the order direction to the first zero is $\Delta (\sin \theta )=\lambda /(Nd)$. Near the first diffraction order in our example, $N=10$ puts the zeros at $0.27$ and $0.33$, while $N=40$ puts them at $0.2925$ and $0.3075$. The respective one-sided widths are $0.03$ and $0.0075$; the full zero-to-zero widths are $0.06$ and $0.015$. These are calculated coherent peak widths, with the slit envelope omitted, and do not describe a measured instrument response. ^[ref].

Four times as many illuminated slits therefore give a fourfold narrowing in this coordinate without moving the diffraction order. That sharper directional selection helps distinguish nearby wavelengths in a spectrograph. Actual performance also depends on illumination, optical coherence, and source width; the separate slit-limited resolving power of a grating describes the contribution of the entrance slit.

A diffuse thermal emitter has emitted radiance independent of outgoing direction over the outward hemisphere. Its emitted radiant intensity from a planar patch still varies with viewing angle through projected area, as described by Lambert's cosine law. Diffuse emission does not mean equal power per solid angle from the patch. This approximation does not by itself impose a grey body spectrum or specify how incident radiation is reflected.

A dot product measures how much two Euclidean vectors point along each other. This article uses real Euclidean vectors in ${\mathbb{R}}^n$, with the usual Cartesian basis: multiply matching coordinates and add,The result is a scalar, while $\Vert \mathbf{a}\Vert$ is the Euclidean norm. This real convention has no complex conjugation; for vectors over the complex numbers, use the convention specified in inner product.

For two nonzero Euclidean vectors, the same dot product records their angle $0\le \theta \le \pi$ through the cosine:Here $\arccos$ is an inverse trigonometric function. The Cauchy-Schwarz inequality guarantees that the normalized ratio lies in $[-1,1]$. The zero vector has dot product zero with every vector, but its angle is undefined: the ratio would divide by zero. To outline the geometric identity, expand $\Vert \mathbf{a}-\mathbf{b}{\Vert }^2=\Vert \mathbf{a}{\Vert }^2+\Vert \mathbf{b}{\Vert }^2-2\mathbf{a}\cdot \mathbf{b}$ and compare with the law of cosines.^[ref]

Take $\mathbf{a}=(3,4)$ and $\mathbf{c}=(2,0)$. Their Euclidean norms are $5$ and $2$, andThe unit vector $\mathbf{u}=\mathbf{c}/\Vert \mathbf{c}\Vert =(1,0)$ selects the horizontal direction. The scalar projection $\mathbf{a}\cdot \mathbf{u}=3$ says how far $\mathbf{a}$ reaches along that direction; its vector projection is $3\mathbf{u}=(3,0)$. Thus $\mathbf{a}=(3,0)+(0,4)$ splits into an aligned part and an orthogonal projection residual.

For $\mathbf{b}=(4,-3)$, the dot product $\mathbf{a}\cdot \mathbf{b}=12-12=0$ proves orthogonality; both Euclidean norms are $5$. For $\mathbf{d}=(-2,0)$, $\mathbf{a}\cdot \mathbf{d}=-6$: the scalar projection onto $\mathbf{d}/\Vert \mathbf{d}\Vert =(-1,0)$ is $-3$, because $\mathbf{a}$ reaches against that chosen direction. Its angle with $\mathbf{d}$ is $126.8699^{\circ }$.

Keeping $\mathbf{a}=(3,4)$ fixed and choosing $\mathbf{b}$ of Euclidean norm $2$ gives $\mathbf{a}\cdot \mathbf{b}=10\cos \theta$. At angles $0^{\circ },60^{\circ },90^{\circ },120^{\circ },180^{\circ }$, the dot products are $10,5,0,-5,-10$. Positive values mean partial alignment, zero means orthogonality, and negative values mean a component against the other direction.

This interpretation explains work: for a constant force $\mathbf{F}$ and a straight displacement $\mathbf{s}$, $W=\mathbf{F}\cdot \mathbf{s}$ counts the force component along the motion. A sideways force contributes zero work, and an opposing force contributes negative work.^[ref]

For a reciprocal ideal quarter-wave plate followed by a mirror at normal incidence, use the same fixed transverse laboratory axes for the forward and returning Jones vectors. If the axes coincide with the plate's fast and slow axes, each pass has Jones matrix $J=\mathrm{diag}(1,i)$ up to a common phase factor. A polarization-independent mirror has scalar reflection coefficient $r_m$, so the returned Jones vector is $r_m{J}^2\mathbf{e}=r_m\mathrm{diag}(1,-1)\mathbf{e}$. The plate acts twice as a half-wave plate, rather than undoing its phase retardation.

For input linear polarization at angle $\alpha$ to the fast axis, the returned line is at $-\alpha$ modulo $\pi$. At $\alpha =45^{\circ }$ it is perpendicular to the original line. An axis-aligned input instead returns along the same line. General elliptical polarization requires its full input Jones vector; no unique returned state follows from the word “polarized” alone. The propagation direction reverses, so comparing descriptions of circular polarization must also specify the viewing convention.

A double pendulum consists of two pendula coupled because the second pivot moves with the first mass. It has two degrees of freedom: even when the upper pivot is fixed, moving the upper arm carries the lower pivot along. This makes the double pendulum a compact example of how mechanical constraints couple motion, and how Lagrangian mechanics converts geometry into dynamics.

Use two equal point masses $m>0$ and two massless rigid rods of equal length $\ell >0$, with joints that allow free rotation in one vertical plane. The upper pivot is fixed; gravity is uniform, with gravitational acceleration $g>0$. Neglect joint friction and drag force. Both angles ${\theta }_1,{\theta }_2$ are measured from the downward vertical, positive toward the right, in radians. In particular, ${\theta }_2$ is an absolute angle, not the bend between the rods: that bend is ${\theta }_2-{\theta }_1$.

The double pendulum Lagrangian below gives the exact dynamics of this ideal model. Near the downward stable equilibrium, normal modes of a double pendulum replace the coupled motion with two independent harmonic oscillators. Their fixed amplitude ratios explain which releases make the rods swing together and which make them swing in opposite directions.

The University of Toronto's institutional demonstration “Chaotic Behaviour (Double Pendulum)”, activity Me-020, uses a metal stand with four leveling feet and a pendulum arm carrying a white dot. Preparation includes leveling the stand and correcting any wobble. The instructions call for holding the arm vertically upward and releasing it; the readout is the visible motion of the dot. A further comparison inserts a metal dowel in a groove on the release arm and repeats the release at the same location. Keeping the stand steady and repeating the release position provide practical controls. The page supplies no numerical angle trace or calibrated timing, and does not explain the dowel's mechanical effect. It therefore documents a qualitative release demonstration, without quantitatively testing the equal-point mass model. Its extended arms also have moments of inertia absent from massless rods. Irregular visible motion alone does not establish sensitive dependence on initial conditions.^[ref]

A separate comparison with normal modes of a double pendulum would require small downward releases and measurement of both angles over time. The actual arm masses, centres of mass and moments of inertia would first have to enter the Lagrangian. The upward release in this demonstration explores a different regime from small oscillations about the downward stable equilibrium.

Write $\Delta ={\theta }_1-{\theta }_2$. Applying the Euler-Lagrange equations to the double pendulum Lagrangian, then dividing by $m{\ell }^2$, givesFor example, the upper generalized momentum is $\partial L/\partial {\dot{\theta }}_1=m{\ell }^2(2{\dot{\theta }}_1+\cos \Delta {\dot{\theta }}_2)$. Its time derivative contains $-m{\ell }^2\sin \Delta ({\dot{\theta }}_1-{\dot{\theta }}_2){\dot{\theta }}_2$. Subtracting $\partial L/\partial {\theta }_1=-m{\ell }^2\sin \Delta {\dot{\theta }}_1{\dot{\theta }}_2-2mg\ell \sin {\theta }_1$ cancels the mixed product and leaves the positive $\sin \Delta {\dot{\theta }}_2^2$ term in the first equation. This cancellation is a useful check on the sign convention.

The matrix multiplying the two angular accelerations isIt is invertible for every configuration, so specified angles and angular velocities determine both angular accelerations. For the same reason $T>0$ whenever the angular velocities are not both zero. The mass cancels from the motion equations, while the timescale is set by $\sqrt{\ell /g}$.

As an exact constructed release, put ${\theta }_1(0)=\alpha$, ${\theta }_2(0)=0$, and both angular velocities zero. Solving the two equations at release yieldsFor $0<\alpha <\pi /2$, the upper rod initially moves back toward vertical, while the lower rod initially moves away from it. The lower pivot's acceleration makes this possible even though the lower rod starts vertical. The small-angle approximation gives ${\ddot{\theta }}_1(0)\simeq -2g\alpha /\ell$ and ${\ddot{\theta }}_2(0)\simeq 2g\alpha /\ell$.

The conserved mechanical energy is $E=T+V$, because the Lagrangian has no explicit time dependence. Directly, $dE/dt$ is $m{\ell }^2$ times the first equation multiplied by ${\dot{\theta }}_1$ plus the second multiplied by ${\dot{\theta }}_2$, so the equations imply $dE/dt=0$. This also supplies a check when computing a trajectory.

Choose Cartesian coordinates with origin at the upper pivot, $x$ to the right and $y$ upward. The two position vectors areThese expressions satisfy the rod holonomic constraints $|{\mathbf{r}}_1|=\ell$ and $|{\mathbf{r}}_2-{\mathbf{r}}_1|=\ell$ automatically. This is the practical advantage of using the two angles as coordinates: the rod tensions need not be unknowns in the motion calculation.^[ref]

Differentiate these position vectors to obtain the velocities. The dot product between their two rod contributions uses $\cos ({\theta }_1-{\theta }_2)=\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2$, givingAdding the two kinetic energies and measuring gravitational potential energy from the downward equilibrium givesThe coefficient $2$ has a geometric meaning: the upper rod raises both masses, and its motion contributes to both kinetic energies. The coupling in $T$ is present because the lower mass has a moving pivot. The zero chosen for $V$ changes $L$ by a constant and leaves the Euler-Lagrange equations unchanged.^[ref]

The principle of stationary action applies to $S=\int Ldt$. Its fixed-endpoint first variation, after integration by parts, isIndependent allowed variations therefore give one Euler-Lagrange equation for each angle. Stationary action alone does not assert a minimum.^[ref]

With equal rod lengths $\ell$ and absolute downward-vertical angles $\theta ,\varphi$, the endpoint separation is $s^2=2{\ell }^2[1+\cos (\theta -\varphi )]$. A zero-rest-length spring between the fixed support and the final mass contributes $-\kappa {\ell }^2[1+\cos (\theta -\varphi )]$ to the Lagrangian. It favours bending away from maximum extension and can reduce the gravitational bending stiffness.

The dual space of a vector space $V$ over a field $F$ is the vector space of all linear functionals $\phi :V\to F$. Think of a vector as an input and a linear functional as a scalar-valued measurement of that input. For $V={\mathbb{R}}^2$, the rules $(x,y)\mapsto x$, $(x,y)\mapsto y$, and $(x,y)\mapsto 2x-y$ are different elements of the dual space. None needs a notion of length or angle.

Its operations combine measurements pointwise:The dual basis explains the finite-dimensional structure: specifying a linear functional's values on a basis determines its value on every vector. Thus an $n$-dimensional vector space has an $n$-dimensional dual space. The spaces have the same dimension, but their elements play different roles: an input and a rule that measures an input. ^[ref].

This separation is useful when changing coordinates. The dual pairing $\phi (v)$ must stay the same while the component lists for $v$ and $\phi$ change in opposite ways. Covector transformation under a change of basis makes the compensation explicit. An inner product on a finite-dimensional real vector space gives one way to represent each linear functional by a vector, but that representation uses the chosen inner product; the dual space itself does not require one.

Here dual space means the algebraic dual: all linear functionals. For a topological vector space, the continuous dual space keeps only the continuous linear functionals. In infinite dimension, this can be a proper vector subspace; discontinuous functional detected by vanishing sequence tails gives an example. The distinction matters when a scalar-valued measurement must respond continuously to small input changes.

Model an eccentric-hole wheel as a complete uniform disk for kinetic energy only, subtracting the absent disc's potential energy. If $s=R\theta$ and $\theta$ is its hole-centre direction measured clockwise from upward vertical, rolling without slipping gives $T\simeq 3D{\dot{s}}^2/4$. On a slope of angle $\alpha$, its potential energy is $U=-(D-h)gs\sin \alpha -hg{r}_o\cos (s/R)$ up to a constant. Substitution into the Euler-Lagrange equation proves the displayed equation. Treating the hole's kinetic energy exactly introduces angle-dependent inertia, so this simplified equation must retain its explicit modeling assumption.

The Einstein-de Haas effect is mechanical rotation or twisting caused by a change in a body's magnetization. The microscopic carriers of magnetic dipole moment also carry angular momentum: changing their net orientation can transfer angular momentum to the body. A freely suspended ferromagnet can therefore turn when its magnetization reverses. A suspension instead stores part of the response as an elastic twist. This makes a microscopic property accessible through a macroscopic angle.

The central measurement compares a change of magnetic dipole moment with transferred angular momentum, giving a gyromagnetic ratio. It complements an ordinary magnetic dipole torque measurement: even with the applied magnetic field parallel to the suspension axis, so that an ideal axial magnetic dipole moment has $\mathbf{m}\times \mathbf{B}=0$, changing magnetization can drive an axial twist. The external drive supplies energy; conservation of angular momentum still requires accounting for the suspension and any external axial torque.

Electric flux measures how much electric field crosses a chosen oriented surface, with a sign distinguishing the two crossing directions. A strong electric field running along a surface contributes nothing to its electric flux: only the normal component counts. This lets Gauss's law turn a potentially complicated electric field into one number recording the net enclosed electric charge.

For a surface $S$ with chosen unit normal $\mathbf{n}$, the electric flux is the surface integralHere $dA$ is positive surface area, while $d\mathbf{S}$ is the vector area element. The dot product makes electric flux a signed scalar, rather than a vector like the electric field. Its SI unit is $\mathrm{N}{\mathrm{m}}^2/C$: electric field measured using the newton per coulomb, multiplied by surface area in square metres. Reversing the orientation of a surface replaces $\mathbf{n}$ by $-\mathbf{n}$ and reverses the electric flux. Calling it electric flux does not imply that electric charge is moving through the surface. ^[ref].

For an open patch in a plane, with surface area $A$ and a constant electric field, the unit normal is constant and the surface integral simplifies toThe angle $\theta$ is between the electric field and the chosen unit normal, not between the electric field and the plane. The cosine describes the signed projected surface area seen by the electric field. Thus electric flux is positive for $0\le \theta <90^{\circ }$, zero at $90^{\circ }$, and negative for $90^{\circ }<\theta \le 180^{\circ }$. ^[ref].

For example, take $E=100N/C$ and $A=0.02{\mathrm{m}}^2$, so $EA=2\mathrm{N}{\mathrm{m}}^2/C$. Three choices of angle giveKeeping the $60^{\circ }$ patch fixed and reversing its unit normal gives $120^{\circ }$ and changes $+1$ to $-1\mathrm{N}{\mathrm{m}}^2/C$. The electric field and physical patch have not changed. If the electric field is instead tangent to the patch, its dot product with the unit normal vanishes: electric flux is zero even though $E=100N/C$.

For a closed surface enclosing a volume $V$, choose the outward unit normal. Gauss's law givesHere ${\epsilon }_0$ is the vacuum permittivity, and $Q_{\mathrm{enclosed}}$ is the signed total electric charge inside. The electric field in the surface integral includes all sources, including external electric charges. Assume no ideal point charge lies on the integration surface. Zero enclosed electric charge implies zero total electric flux; it does not imply zero electric field everywhere on the closed surface or inside it. Positive and negative contributions to the surface integral can cancel. ^[ref].

To see the cancellation with sources inside, place point charges $+q$ at $(-a,0,0)$ and $-q$ at $(a,0,0)$, where $q>0$, $a=0.30\mathrm{m}$, and both lie inside the sphere of radius $R=1\mathrm{m}$ centered at the origin. This separated pair has the electric dipole geometry. Its enclosed electric charge is $q-q=0$, so its total outward electric flux is zero. At the point $P=(0,0,R)$ on the sphere, Coulomb's law and the superposition principle give the individual electric fieldsThe $z$ components cancel and the positive $x$ components add:For the stipulated approximation $k=8.99\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$ and $q=1\mathrm{nC}$, this is about $4.740N/C$ along positive $x$. At $P$, the outward unit normal is $\hat{\mathbf{z}}$, so this nonzero electric field is tangent to the sphere. At other points of the sphere, its normal component can be positive or negative. The source-field formula applies away from the two point charges, where the electric field has zero divergence; it is singular at the sources. ^[ref].

The diagram shows streamlines of the three-dimensional electric field in the section $y=0$. The circle is a cross-section of the sphere, not the integration surface: Gauss's law uses the entire three-dimensional surface integral. A two-dimensional line integral of outward field around that circle would be a different quantity, with units $\mathrm{N}m/C$ rather than the electric flux units $\mathrm{N}{\mathrm{m}}^2/C$.

If the lowest allowed angular harmonic has a nonzero coefficient, differentiating its potential gives the displayed field scale. The field vanishes at a right-angle recess, stays finite at a smooth face, and diverges at a protruding conducting corner whose exterior vacuum angle exceeds $\pi$. A rounded tip cuts off the ideal power law at its radius scale. If the first coefficient vanishes, the first nonzero higher harmonic determines the exponent instead.

In 2006, Yilong Han and colleagues dispersed plastic ellipsoids in water inside a thin glass cell, using a very dilute suspension to limit particle interactions. A camera on an optical microscope recorded 30 frames per second. Image analysis extracted centre positions and long-axis angles, with reported angular resolution about $1^{\circ }$. Their shape supplied the observable axis. Reflected interference colours calibrated cell thickness; nearly constant particle interference colour checked vertical wandering. One tracked particle had dimensions $2.4\times 0.3\times 0.3\mu \mathrm{m}$ in an $846\mathrm{nm}$ gap. Angular displacement variance grew linearly with lag; displacement resolved along the particle's axes and in the laboratory showed how turning washes out translational anisotropy. These are observations of confined particles: walls alter drag, and slight out-of-plane tumbling affected positional resolution. They do not establish the bulk sphere formula. The paper's Figure 1A–B shows simulations; Figure 1F shows a measured trajectory.^[ref]

For interpreting such images, the director distinction is essential: identical ends give an angle modulo $\pi$. A continuously unwrapped trace can follow small successive turns, but it does not make the ends physically distinguishable. The invariant planar axis autocorrelation is $⟨\cos (2\Delta \varphi )⟩$, and its decay determines $D_r$ through $e^{-4D_{r}t}$. Separately measuring position and rigid body orientation is what makes the connection between translational anisotropy and loss of angular memory experimentally accessible.

For an elliptically polarized wave, the tip of its transverse vector traces an ellipse at a fixed point during each wave period. For polarization of an electromagnetic wave, this is the electric field vector. With the real components in polarization of waves, assume $A_x,A_y>0$ and $\sin \delta \ne 0$. Eliminating time givesA short derivation uses $\varphi =\omega t$ and the cosine addition formula to obtain $E_y/A_y=(E_x/A_x)\cos \delta -\sin \varphi \sin \delta$; squaring and using the sine and cosine identity ${\sin }^2\varphi =1-{\cos }^2\varphi$ gives the displayed equation. Outside phase quadrature, the mixed term gives a tilted ellipse; $A_x,A_y$ are component wave amplitudes and need not be its semi-major and semi-minor axes.

In phase quadrature, $\delta =\pm \pi /2$, the mixed term vanishes. For $(A_x,A_y)=(2,1)V/m$, the ellipse is $E_x^2/4+E_y^2=1$, its semi-major and semi-minor axes have lengths $2$ and $1V/m$, and its electric field magnitude varies between $1$ and $2V/m$. At $\varphi =\pi /4$ with $\delta =\pi /2$, the electric field is $(\sqrt{2},-1/\sqrt{2})V/m$ and has magnitude $\sqrt{5/2}V/m$. Equal wave amplitudes instead give circular polarization, a special case of elliptical polarization with constant magnitude. As $\delta$ approaches $0$ or $\pi$, the ellipse collapses to a line segment and gives linear polarization; this limit is degenerate rather than a nondegenerate ellipse.

Changing $\delta$ from $\pi /2$ to $-\pi /2$ reverses traversal of the same ellipse. In the displayed $x,y$ plot, $\delta =\pi /2$ moves from $(A_x,0)$ toward negative $y$ as time increases. A label such as right- or left-handed circular polarization additionally requires a stated viewing direction and naming convention: viewing along propagation and viewing against propagation reverse the apparent rotation. ^[ref].

The ellipse records both the relative wave amplitudes and the phase difference of two transverse oscillations. This separates a change of polarization of waves from a change of overall wave amplitude: multiplying both components by the same positive factor enlarges the ellipse while preserving its shape and orientation.

A rotating ideal linear polarizer can probe this geometry. For the $2:1$ ellipse in phase quadrature, orthogonal projection onto a pass axis at angle $\theta$ gives $E_{\mathrm{out}}(t)=2\cos \theta \cos \omega t-\sin \theta \sin \omega t$, in $V/m$. Averaging its square over a wave period gives $(4{\cos }^2\theta +{\sin }^2\theta )/2$, while the incoming squared electric field magnitude averages to $5/2$. Thus the transmitted fraction of light intensity is $(4{\cos }^2\theta +{\sin }^2\theta )/5$: it is $4/5$ at $0^{\circ }$, $1/2$ at $45^{\circ }$ and $1/5$ at $90^{\circ }$. There is no single incident line of linear polarization to insert into Malus's law; projecting the electric field and averaging its square is the appropriate generalization. This measurement cannot distinguish $\delta =\pi /2$ from $\delta =-\pi /2$, because reversing the traversal leaves all these squared averages unchanged.