In an E1cb elimination, deprotonation first produces a conjugate-base intermediate, followed by leaving-group loss. Dehydration of an appropriate aldol product can form a conjugated enone.

An elimination mechanism involving ionization to a carbocation before beta deprotonation forms the alkene. Its limiting unimolecular rate law assumes ionization controls the rate.

A concerted elimination in which a base removes a beta hydrogen as a leaving group departs and a pi bond forms. Orbital alignment and substrate structure affect the allowed pathway.

Loss of a hydrogen halide to form a carbon–carbon multiple bond. Along a chlorinated polymer backbone, repeated elimination can create unsaturated or conjugated segments rather than exclusively changing ester side groups.

A charge cloud $\rho =A/r$ for $0<r<a$ encloses $Q(r)=2\pi A{r}^2$. With no central point charge and potential zero at infinity, Gauss's law gives $\varphi =A(2a-r)/(2{ϵ}_0)$ inside and $A{a}^2/(2{ϵ}_{0}r)$ outside. The density singularity is integrable, and the potential is finite at the origin.

A spherical electrical conductor has constant electric potential throughout its connected interior in electrostatic equilibrium. For an isolated sphere in a homogeneous medium, spherical symmetry and Gauss's law give the exterior field of a point electric charge at its center.

An equipotential sphere centred in a flat interface between vacuum and a linear dielectric admits the same potential $C/r$ in both media. The radial field is tangential at the interface, so tangential E and normal D boundary conditions are satisfied. Displacement flux through the hemispheres gives $Q=2\pi {ϵ}_0(1+{ϵ}_r)C$, yielding the displayed sphere potential. E has equal magnitude at equal radius; D and free surface charge are larger by ${ϵ}_r$ on the dielectric side.

Integrating electrostatic pressure gives a force toward the dielectric of magnitude $Q^2({ϵ}_r-1)/[8\pi {ϵ}_0(1+{ϵ}_r{)}^2{a}^2]$. A deforming liquid interface or surface tension needs additional mechanical conditions beyond this ideal flat-interface solution.

The outward normal electric traction on an equipotential conductor adjoining a linear isotropic dielectric, for the ideal normal-field configuration, is $p_e=ϵE_n^2/2=\mathbf{E}\cdot \mathbf{D}/2$. Integrating this normal traction over a surface gives the electric force. It is a field-stress contribution; mechanical pressure and dielectric boundary conditions must be included when determining an actual fluid-interface equilibrium.

In a charge-free wedge $0<\theta <\alpha$ bounded by equal-potential conductor faces, separation of Laplace's equation gives the displayed local harmonic expansion. Bounded potential at a finite corner removes negative powers. Outer boundary conditions determine the remaining coefficients; the conductor potential alone does not determine a nonconstant solution in an unbounded domain.

Mobile electric charges redistribute in response to an imposed electric potential, partly compensating its field. The induced density response and the Poisson equation must be solved self-consistently. Classical dilute mobile ions lead to the Debye–Hückel approximation; a degenerate electron system instead has a response set by its electronic compressibility and interactions. A high charge response can pin a partly occupied Landau level near the Fermi energy, whereas an incompressible gap suppresses that response. electrostatic screening is not a claim that every material perfectly excludes every static field.

Detection of a dot's charge through its capacitive influence on a nearby quantum point contact or single-electron transistor. Discrete charge changes shift the sensor response, allowing individual tunnel events to be resolved even when direct current through the dot is too small to measure.

A measurement method in which spin-dependent tunnelling or allowed charge transfer maps a spin outcome onto a distinguishable dot charge state. Quantum-dot charge sensing then records that charge. The measurement axis or singlet/triplet observable must be specified.

Two capacitively and tunnel-coupled electron islands with separately tunable occupations. Detuning controls which dot holds an electron and whether two-electron charge states such as $(1,1)$ and $(0,2)$ are accessible.

A single confined electron's two spin states form a qubit when initialization, coherent rotations and measurement are controlled. Local driving rotates one spin; gate-controlled exchange can entangle neighboring dot spins. Spin-to-charge conversion permits electrical readout. Observing charge blockade alone does not demonstrate coherent qubit operation.

Solid residue formed or collected during anodic dissolution in electrorefining. It can contain noble metals, precipitated salts and undissolved inclusions. Slime can retain valuable impurities, block transport or contaminate a deposit by entrainment.

The thin-double-layer approximation ${\mu }_e=ϵ\zeta /\eta$, neglecting important relaxation and surface-conduction corrections.