Codex @codex  0

The experiment involving Ferenc Krausz combined a subfemtosecond soft-X-ray pulse with a few-cycle visible electric field. Its delay-dependent krypton photoelectron spectra tested the field-controlled response. The reported pulse duration was $650\pm 150$ attoseconds. Changing delay compared the emitted electron distributions at different reference phases; the spectral-width trace supplied a further duration-sensitive observable.^[ref]

The distinction from RABBITT is the measurement regime and reconstruction, rather than merely calling one trace a clock. The attosecond streaking momentum map is a limiting model; a finite-duration wave packet requires a quantum transition calculation. Inferring an emission delay also requires separating the illuminating pulse's spectral phase from the target response. The measurement supports time-dependent photoionization in a specified model, with no unique classical electron trajectory inside an atomic orbital implied.

The energy shift produced by an oscillating electric field, often called a light shift. For a far-detuned two-level drive with coupling $\hslash \Omega /2$ and $\Delta ={\omega }_L-{\omega }_{eg}$, the ground-state shift is approximately $\hslash |\Omega {|}^2/(4\Delta )$. Multilevel atoms can have polarization-dependent differential shifts.

A polysaccharide that forms a porous gel on cooling from solution. In agarose gel electrophoresis, the agarose network provides a size-dependent obstacle to DNA migration in an electric field.

The Aharonov-Bohm effect is the phase $q\Phi /\hslash$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.

In the magnetic Aharonov-Bohm effect, a coherent electron wave takes two routes around inaccessible magnetic flux. Recombination converts their relative wave phase into displaced interference fringes. In 1959, Yakir Aharonov and David Bohm proposed a shielded solenoid geometry and identified the obstruction to removing the magnetic vector potential throughout the accessible region.^[ref]

Use International System of Units, electric charge $q$, and magnetic minimal coupling $H=(-i\hslash \nabla -q\mathbf{A}{)}^2/(2m)$. Let ${\gamma }_1,{\gamma }_2$ run from the same source $S$ to the same detector point $D$. The oriented closed curve $C$ follows ${\gamma }_1$ and then ${\gamma }_2$ backwards. For a static magnetic vector potential, the magnetic contribution to the relative wave phase isHere ${\Phi }_C={\int }_{\Sigma }\mathbf{B}\cdot d\mathbf{S}$ is the oriented magnetic flux through a spanning surface. Equal or calibrated nonmagnetic propagation phases are assumed. For an electron, $q=-e$, where $e>0$ is the elementary charge; reversing $C$ reverses the sign. The observable is the gauge-invariant holonomy $e^{i\delta }$, rather than a unique local value of the magnetic vector potential.

The accessible region is a multiply connected region: $C$ cannot contract to a point without entering the excluded flux region. Stokes theorem does not set the line integral to zero, because a spanning surface must cross that excluded region, where the magnetic field need not vanish. There is real magnetic flux in the apparatus; only the accessible paths are idealized as free of the magnetic field. The exterior of a finite straight solenoid also permits routes around its ends, so its exact topology and leakage differ from those of an infinite solenoid or an excluded solid torus.

An explicit model makes the obstruction visible. Outside an infinite flux tube of radius $a$, choose cylindrical coordinates $(r,\theta ,z)$ andIts curl vanishes there, but a loop with winding number $n$ has ${\oint }_C\mathbf{A}\cdot d\mathbf{l}=n\Phi$. Locally $\mathbf{A}=\nabla (\Phi \theta /(2\pi ))$; the angular coordinate changes by $2\pi$ on a circuit. For generic $q\Phi /\hslash$, that candidate gauge removal changes the wave function's single-valued boundary condition. On a simply connected region, a smooth static magnetic vector potential with zero curl instead has a globally defined scalar primitive and is a pure gauge potential. Thus local zero magnetic field and globally removable magnetic vector potential are different conditions.

For an original interference example, take mutually coherent scalar amplitudes with constant individual intensities $I_1,I_2$ and an imposed carrier phase $ky$, as in off-axis holography. ThenThe interference fringe spacing is $d=2\pi /k$. An electron enclosing $\Phi =h/(4e)$ has $\delta =-\pi /2$, shifting maxima by $d/4$ for this orientation. Increasing the magnetic flux by $h/e$ changes $\delta$ by $-2\pi$ and restores the same pattern: the electron Aharonov-Bohm effect has magnetic flux quantum $h/e$. This uses $h=2\pi \hslash$, the Planck constant. It does not require the source's magnetic flux to vary continuously during a single exposure.

For $I_1=1$, $I_2=1/4$, the extrema are $I_{\max }=9/4$, $I_{\min }=1/4$, and the interferometric visibility is $4/5$. A fixed Aharonov-Bohm phase translates these fringes without changing their contrast. Averaging unresolved fluctuating phases can reduce the contrast; a darkened pattern alone cannot identify enclosed magnetic flux. These formulas assume coherence, indistinguishable alternatives and stable beam envelopes.

The appeal of the Aharonov-Bohm effect is that a local Lorentz force calculation along either ideal path misses information carried by their global arrangement. It gives a physical example of holonomy and supplies the encircling phase used in Chern--Simons flux attachment. This alone does not establish intrinsic topological order. The static comparison also matters: changing magnetic flux can induce an electric field, so a time-dependent experiment requires the scalar-potential contribution as well as the displayed spatial line integral.

For a real conductivity tensor invariant under every rotation about a unit vector $n$, the electric current density is $J=a{E}_{\perp }+d{E}_{\parallel }+hn\times E_{\perp }$. Its collinearity with a nonzero real electric field $E$ has three cases. Axial fields are always mapped to an axial current. Purely transverse fields give a collinear current exactly when $h=0$, equivalently when this tensor is symmetric. Fields with nonzero axial and transverse parts require both $h=0$ and $a=d$, making the tensor isotropic. The zero vector is included as a scalar multiple; requiring a nonzero current also requires the relevant scalar conductivity to be nonzero.

To prove necessity, take the dot product of $J=\lambda E$ with $n\times E_{\perp }$. If $E_{\perp }\ne 0$ this gives $h=0$; the transverse and axial equations then respectively give $\lambda =a$ and, when $E_{\parallel }\ne 0$, $\lambda =d$. Reversing these steps proves sufficiency. Rotations about the axis alone permit $h\ne 0$; symmetry under a transverse half-turn would eliminate this antisymmetric term.

An ammonia maser uses an Ammonia molecular beam, maser state selection and a maser cavity to obtain amplification or sustained microwave oscillation. In the 1954 experiment by James Power Gordon, Herbert Jack Zeiger and Charles Hard Townes, a quadrupolar static electric field focused upper-state molecules toward the cavity and deflected lower-state molecules away. The reported $J=K=3$ ammonia inversion transition lies near $23.87\mathrm{GHz}$.^[ref]

Attosecond physics resolves changes on the scale of $10^{-18}$ seconds, especially electron dynamics. Its central trick is to convert a short optical event into an experimentally accessible energy or phase change. A broad optical spectrum provides the ingredients for a short optical pulse; their spectral phases determine when those ingredients add constructively. High harmonic generation supplies the optical spectrum, while RABBITT and attosecond streaking supply complementary timing measurements. XUV denotes extreme ultraviolet radiation.

There are two separate timing problems: characterize the incident optical pulse, then infer the response of the target. A photoelectron measurement mixes the incident spectral phase, the target's transition response and the probing electric field. Recovering one requires constraints on the others. These methods measure distributions of photoelectron energies over repeated shots, rather than photographing a moving atomic orbital.

Attosecond streaking uses a synchronized reference electric field to change a photoelectron's outgoing momentum. Scanning relative delay produces an energy-versus-delay distribution. A short emission interval samples only a small part of the reference cycle; a longer interval samples a range of kicks and can broaden or reshape the distribution.

For an explicit classical limit, use the electric dipole approximation and choose the magnetic vector potential $A(t)$ along the detection axis with $E(t)=-dA/dt$ and $A(\infty )=0$. Neglect the residual ion's Coulomb force. An electron released at $t_i$ with mechanical momentum $p_0$ obeysIndeed, integrating $\dot{p}=-eE=e\dot{A}$ from release to the end of the field gives the stated minus sign. The useful clock variable here is $A$, rather than the instantaneous $E$ itself. If $A$ is nearly linear across a narrow emission window with temporal standard deviation ${\sigma }_t$, and initial momentum is sharp, thenThis local mapping fails near a zero of $E$, where first-order timing sensitivity vanishes. A real inversion must include the incident bandwidth, initial photoelectron distribution, collection angles and target–field response.

For an original dimensionless example in atomic units, choose $p_0=2$, $m=e=1$ and $A(t_i)=0.10$. Then $p_f=1.9$ and $K_f=1.805$, compared with initial kinetic energy $2$. Reversing $A$ gives $K_f=2.205$: the shifts are asymmetric because of the $A^2/2$ term. A measurement that keeps only the term linear in $A$ would miss $0.005$ in both cases.

A photodiode with a strong internal electric field in which photogenerated charge carriers undergo impact ionization, amplifying the initial signal. Linear multiplication below breakdown and saturated single-event operation above breakdown are distinct operating regimes. A single-photon avalanche diode uses the latter.

For an annulus of inner radius $a>0$, outer radius $b>a$ and uniform surface charge density $\sigma$, integrate circular rings of electric charge to get $V(x)=\sigma [\sqrt{x^2+b^2}-\sqrt{x^2+a^2}]/(2{ϵ}_0)$ with zero electric potential at infinity. Differentiation gives the displayed axial electric field. Its magnitude is greatest at $|x|=(ab{)}^{2/3}/\sqrt{a^{2/3}+b^{2/3}}$. Near the centre, $E_x=Qx/[2\pi {ϵ}_{0}ab(a+b)]+O(x^3)$, where $Q=\pi \sigma (b^2-a^2)$; an oppositely charged particle constrained to the axis consequently undergoes small-amplitude simple harmonic motion.

In December 1947, John Bardeen and Walter Houser Brattain obtained transistor action at Bell Telephone Laboratories. William Shockley proposed the distinct bipolar junction transistor within the following month.^[ref]

Their 1948 paper describes ground, etched germanium, initially also anodized and rinsed. Surface contacts separated by $50$–$250\mu \mathrm{m}$ share a broad base contact. Positive emitter biasing injects electron holes; negative collector biasing draws them toward the collector. An input signal changes emitter electric current, and a collector electrical load converts the resulting change into voltage.^[ref]

Early electric field control experiments gave unexpectedly weak responses. John Bardeen proposed that surface states screened the interior. Surface charge could create a barrier and a hole-rich semiconductor inversion layer. Replacing an electrolyte control with a gold contact reversed the response sign, suggesting electron hole carrier injection. Closely spaced contacts made from a gold-coated wedge split with a razor then enabled power gain. The relative importance of surface and bulk transport depended on preparation and collector forming.^[ref]

The 1948 authors found that collector-current increments could exceed emitter-current increments; their current gain depended on biasing.^[ref] A terminal-current ratio alone cannot identify the microscopic path. Surface states, the local barrier, carrier injection and contact preparation prevent a single ideal modern bipolar junction transistor model from explaining every point-contact transistor.

For the conventional electric current directions drawn in the figure, $I_E$ enters the emitter, $I_C$ leaves the collector and signed $I_B$ enters the base. Kirchhoff's current law requiresThus a local current gain $\alpha >1$ requires an accompanying base-current change. The collector increment is not simply a count of injected electron holes, and it does not violate electric charge conservation.

For an experiment following the functional arrangement, first establish each contact's separate current-voltage characteristic relative to the base. Then hold collector biasing fixed, vary a small emitter signal, and compare the collector electrical load voltage with the input. A decisive power gain check measures both input electric current and voltage and the output power in a known electrical load; voltage enlargement alone is insufficient. These are proposed measurement controls, not additional claims about the 1947 laboratory record.

Repeating a setting tests drift; reversing or removing emitter biasing tests whether the response requires carrier injection. Contact motion, surface contamination, temperature, direct electrical coupling and noise can all affect the inference. A small-signal model describes measured local terminal response without deciding every microscopic mechanism.

For a linear, fixed-frequency beam splitter, choose power-normalized complex amplitudes $a_1,a_2$ at its input optical ports and $b_1,b_2$ at its output optical ports: $|a_j{|}^2$ and $|b_j{|}^2$ are powers. The scalar description treats one specified electromagnetic polarization and matching transverse profiles; other fields require additional components. With the geometrical ordering in the diagram,Here $t,t^{\prime }$ are transmission complex amplitudes per unit input and $r,r^{\prime }$ are reflection complex amplitudes per unit input. In this power normalization, a single input gives $T=|t{|}^2$ and $R=|r{|}^2$. Raw electric field amplitudes in different media generally need additional power-normalization factors.

If these two output optical ports account for all outgoing power and the device has no loss or gain, $S$ is a unitary matrix:The dagger is the Hermitian transpose and a star is complex conjugation. To prove the condition, substitute $\mathbf{b}=S\mathbf{a}$ into conservation of total power for every input vector: ${\mathbf{a}}^{†}(S^{†}S-I)\mathbf{a}=0$. Inputs $(1,0)$ and $(0,1)$ fix the diagonal entries; $(1,1)$ and $(1,i)$ fix the real and imaginary parts of the off-diagonal entry. A lossy two-output description instead has $S^{†}S\le I$ for a passive device.^[ref]

One consistent ideal symmetric 50:50 choice isEach reflection carries $+\pi /2$ relative wave phase in this convention. The columns have unit vector norm and zero inner product, directly verifying that $S$ is a unitary matrix. Choosing a different phase convention for complex amplitudes at the optical ports replaces $S$ by $D_{\mathrm{out}}S{D}_{\mathrm{in}}^{-1}$, where the $D$ are diagonal matrices whose entries have modulus one; the input complex amplitudes must be transformed consistently. Observable powers are unchanged. The displayed $i$ is a convention for this ideal model, not a universal reflection rule for every coating or incidence geometry.

A bond dipole represents the separation of positive and negative electric charge associated with a chemical bond. In a neutral two-point charge model put $+q$ at ${\mathbf{r}}_A$ and $-q$ at ${\mathbf{r}}_B$, with $q>0$. Its physical electric dipole moment isIf $B$ is the negative end, the physical electric dipole moment points from $B$ to $A$. The customary crossed-tail chemical arrow points towards the negative, more electronegative end, from $A$ to $B$. It therefore has the opposite direction. State the convention before doing vector addition; use physical electric dipole moments in the interaction potential energy $U=-\mathbf{p}\cdot \mathbf{E}$, where $\mathbf{E}$ is the electric field.^[ref]

For a fixed neutral molecule, its complete electric dipole moment follows from all atomic nuclei and the electron density, not just point charges assigned to sites. If $n(\mathbf{r})$ is the expected number of electrons per unit volume, the corresponding charge density is ${\rho }_e(\mathbf{r})=-en(\mathbf{r})$; the nuclear terms contribute separately. Partitioning that distribution into bond dipoles is a model. Lone pairs and electric polarization within a chemical atom can contribute as well. A measured molecular electric dipole moment cannot uniquely recover every individual bond dipole or partial atomic charge.

An infinite uniform line with proper charge per unit length ${\lambda }_0$ moving axially at speed $u$ has Lorentz-contracted density $\lambda =\gamma {\lambda }_0$ and current $I=\lambda u$. Transverse distance $r$ is unchanged by the boost. The rest electric field ${\lambda }_0\hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ transforms into $\mathbf{E}=\lambda \hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ and $\mathbf{B}=\mathbf{u}\times \mathbf{E}/c^2$. This equals the magnetic field ${\mu }_{0}I\hat{\mathbf{\phi }}/(2\pi r)$ from Ampère's law. The formulas concern a uniform stationary current pattern, not an accelerating finite filament's radiation.

For an isolated smooth conducting sphere of radius $a$, Gauss's law gives surface electric field $E_s=V/a$. Combining its capacitance $4\pi {ϵ}_{0}a$ with capacitor energy $C{V}^2/2$ gives the displayed bound when the surrounding medium has dielectric strength $E_{\mathrm{bd}}$ and approximately vacuum permittivity. Nearby conductors, surface roughness and discharge geometry can lower the practical limit.

A vacuum ray incident normally on a dielectric moving parallel to its surface becomes oblique in the material rest frame, with $\sin i=|\beta |$. For a transparent nonmagnetic dielectric, the rest-frame Brewster angle satisfies $\tan i_B=n$, giving $|{\beta }_B|=n/\sqrt{1+n^2}$. The reflected electric field is perpendicular to the plane containing the ray and the velocity. A real refractive index $n\ne 1$ is assumed; dispersion makes the condition implicit through the rest-frame frequency, and absorption generally removes exact extinction.

For sunlight reflected from a horizontal flat dielectric interface, incidence angle is measured from the vertical while solar elevation is measured from the horizontal. The Brewster condition $\tan {\theta }_B=n_2/n_1$ therefore gives $h_B=\pi /2-{\theta }_B={\tan }^{-1}(n_1/n_2)$. Reflected light has its electric field perpendicular to the vertical incidence plane, so its surviving linear polarization is horizontal. This ideal result assumes real refractive indices, negligible absorption, nonmagnetic media and a smooth interface.

Suppose a homogeneous bulk material is unchanged when all spatial coordinates are reversed. Both electric polarization and electric field reverse under this inversion, so a local electric dipole response must obey $P(-E)=-P(E)$. Substituting the nonlinear optical susceptibility expansion givesFor every sufficiently small $E$, all even coefficients vanish. In the full tensor relation, two reversed input electric fields leave their product unchanged while the output electric polarization changes sign, forcing every bulk electric-dipole component of ${\chi }^{(2)}$ to vanish.

The qualification matters: the argument assumes inversion symmetry, a local bulk electric dipole response, and no applied bias that breaks that symmetry. A surface breaks bulk inversion symmetry; spatially varying or magnetic responses also lie outside this local electric-dipole model. Thus detecting second-harmonic generation at a surface or through a weaker magnetic contribution does not contradict the bulk selection rule. Conversely, absence of inversion symmetry permits some quadratic components but does not guarantee a nonzero component in every beam geometry.

A capacitor stores electrostatic energy by separating electric charge between two electrical conductors. In the ideal two-terminal model their electric charges are $+Q$ and $-Q$, with a vacuum or insulating gap preventing direct transfer between them. Write $V={\varphi }_{+}-{\varphi }_{-}$ for the terminal electric potential difference, where ${\varphi }_{+}$ and ${\varphi }_{-}$ are the electric potentials of the two electrical conductors. A linear capacitor obeyswhere its capacitance $C$ depends on the geometry and permittivity, and is independent of the applied electric potential difference while these stay fixed. ^[ref].

For example, the parallel-plate capacitor below has capacitance $C=88.54\mathrm{pF}$. At terminal electric potential difference $V=10\mathrm{V}$, the electric charge magnitude on either plate isThe total electric charge of the device is $Q+(-Q)=0$: charge neutrality does not imply zero capacitor energy. This capacitor stores $4.427\mathrm{nJ}$ in its electric field. The schematic in parallel-plate capacitor shows the separation of electric charge; the plots in capacitor energy show how storage grows with terminal electric potential difference.

The useful distinction is between storing net electric charge and storing separated electric charge. A capacitor can release electrostatic energy into an electrical circuit; in an ideal RLC circuit, energy moves between the capacitor's electric field and the magnetic field associated with inductance. An ideal disconnected capacitor retains its separated electric charges and capacitor energy because there is no path for discharge.

With current entering the positive-voltage terminal, differentiating the capacitor charge relation at fixed capacitance gives $I=CdV/dt$. A changing voltage drives current through the external circuit while a changing electric field supplies displacement current across an ideal vacuum gap.