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For a planar surface contacting a symmetric monovalent ideal electrolyte on one side, the Poisson-Boltzmann equation is ${\psi }^{\prime \prime }=(2e{n}_{\infty }/\epsilon )\sinh (e\psi /(k_{B}T))$. Multiplication by ${\psi }^{\prime }$ and integration using the zero field and zero potential in the bulk givesThe solution decays towards the bulk, and the boundary condition $\sigma =-\epsilon {\psi }^{\prime }(0^{+})$ therefore yields the displayed charge-potential relation, with ${\kappa }^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Its small-potential limit is the planar Debye-Hückel double layer. It is nonlinear but still a mean-field model with point-like ions and constant permittivity; it does not include a compact interfacial layer or finite-ion-size saturation.

A guard electrode of a capacitor surrounds an electrically isolated sensing electrode and is held at the same electric potential. The small insulating slot separates their readout paths. The guard extends the nearly planar equipotential, moving the main outer edge away from the sensing region; it does not abolish the fringing field of a capacitor everywhere. In a capacitance fixture, fringe electric current is routed through the guard instead of the sensing readout. Keysight describes this three-electrode method in Materials Measurement: Dielectric Materials (Solution, 16451B fixture).^[ref]

Measuring the sensing and guard electric currents together would include the diverted response again. Grounding an arbitrary surrounding electrode also does not reproduce a guard unless its electric potential matches the sensing electrode. Finite guard width, the insulating slot, plate misalignment and specimen air gaps limit the approximation. Use capacitance measurement calibration and the geometry appropriate to the sensing region when inferring permittivity. The source also warns that guarding changes the electromagnetic field unfavourably at high frequency, so the low-frequency electrostatic picture has a restricted range.^[ref]

A shallow donor impurity in an isotropic semiconductor can be approximated by an electron of effective mass $m^{\ast }$ in a Coulomb potential screened by relative permittivity ${ϵ}_r$. Its ground-state binding energy and radius areThe donor level lies $E_B$ below the conduction band. These are magnitudes: the bound electron's energy relative to the band edge is negative. Band anisotropy, short-distance impurity structure and screening by mobile carriers can invalidate the simple Bohr model rescaling.

For a point electric charge $q$ in a homogeneous medium of permittivity ${ϵ}_1$ beside a plane interface with permittivity ${ϵ}_2$, its potential on the source side is reproduced by a mirrored image charge $q^{\prime }=q({ϵ}_1-{ϵ}_2)/({ϵ}_1+{ϵ}_2)$. The interface has no free surface charge; continuity of tangential electric field and normal electric displacement field fixes this coefficient.

A linear anisotropic medium has constitutive relations $D_i={\epsilon }_{ij}{E}_j$ and $B_i={\mu }_{ij}{H}_j$. Symmetric, time-independent tensors make the field-energy density a quadratic form.

Here permittivity is a tensor, so the electric displacement need not be parallel to the electric field.

For ideal coaxial conductors of radii $a<b$ filled with a homogeneous permittivity $ϵ$ and magnetic permeability $\mu$, Gauss's law gives $E_r=\lambda /(2\pi ϵr)$ and hence $C^{\prime }=2\pi ϵ/\ln (b/a)$. The magnetic field $H_{\phi }=I/(2\pi r)$ has energy per unit length $L^{\prime }{I}^2/2$, giving $L^{\prime }=\mu \ln (b/a)/(2\pi )$. These are the external-field parameters when internal magnetic penetration into the conductors is neglected. Their product is $L^{\prime }{C}^{\prime }=\mu ϵ$, and the characteristic impedance is $\sqrt{L^{\prime }/C^{\prime }}=\sqrt{\mu /ϵ}\ln (b/a)/(2\pi )$.

For fully dissociated electrolytes at the same temperature and solvent permittivity, linear screening theory gives ${\lambda }_D^{-2}\propto {\sum }_i{c}_i{z}_i^2=2I$. A divalent symmetric salt of formula concentration $c$ has $I=4c$, not $c$ or $2c$. ion association and nonideal concentrated-solution effects limit this comparison.

A parallel-plate capacitor consists of two parallel electrical conductors with overlapping area $A$ and separation $d$. If the gap is small compared with both plate widths and is filled by a uniform material of positive permittivity $ϵ$, its capacitance is approximatelyThis neglects the fringing field of a capacitor near the plate edges. The interior electric field points from the plate of positive electric charge to the plate of negative electric charge, and is approximately uniform. ^[ref].

For a worked vacuum example, stipulate $A=0.01{\mathrm{m}}^2$, $d=0.001\mathrm{m}$, and vacuum permittivity ${ϵ}_0\approx 8.854\times 10^{-12}F/m$. This rounds the value in Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others. Within the approximation,At terminal electric potential difference $V=10\mathrm{V}$, the electric field magnitude is $E\simeq V/d=10^{4}V/m$, the plate electric charge magnitude is $Q=CV=0.8854\mathrm{nC}$, and the capacitor energy is $U=C{V}^2/2=4.427\mathrm{nJ}$. These numbers use the stipulated rounded vacuum permittivity; finite plates also have corrections from the fringing field of a capacitor.

The formula makes design changes easy to predict: doubling the overlapping area doubles the capacitance, and halving the gap doubles the capacitance, with permittivity fixed and the fringing field of a capacitor neglected. Doubling permittivity also doubles capacitance if the material uniformly fills the gap and remains linear. This lets a compact device store more separated electric charge at the same terminal electric potential difference.

The short derivation uses surface charge density $\sigma =Q/A$. In the interior, Gauss's law gives electric field magnitude $E\simeq \sigma /ϵ=Q/(ϵA)$. The electric potential difference is $V\simeq Ed=Qd/(ϵA)$, so dividing $Q$ by $V$ gives $C\simeq ϵA/d$. A narrow curved capacitor gap has the same leading local approximation when its width is much less than the curvature radius.

Two perfectly conducting wide strips of width $a$ and separation $d\ll a$, filled with a homogeneous nonmagnetic dielectric of absolute permittivity $\epsilon$, support a transverse electromagnetic mode. Ignoring edge fields, their distributed capacitance is $C^{\prime }=\epsilon a/d$ and distributed external inductance is $L^{\prime }={\mu }_{0}d/a$. The magnetic-energy integral ${\mu }_0(I/a{)}^{2}ad/2=L^{\prime }{I}^2/2$ gives the latter. Their characteristic impedance is $Z_0=(d/a)\sqrt{{\mu }_0/\epsilon }$ and propagation speed is $1/\sqrt{{\mu }_0\epsilon }$. These parameters are per unit length, distinct from a lumped capacitor and inductor.

The static Lindhard dielectric function describes the real, zero-frequency linear response of a uniform free electron gas. Here ${ϵ}_1=\mathrm{Re}ϵ$, $\mathbf{r}$ is the position operator, and $\mathbf{q}$ is the perturbation wavevector, $q=|\mathbf{q}|$, $V$ the normalization volume, $e>0$ the elementary charge, and ${ϵ}_0$ the vacuum permittivity. The normalized one-electron states are ${\psi }_{\mathbf{k}}=V^{-1/2}{e}^{i\mathbf{k}\cdot \mathbf{r}}$ with $E_{\mathbf{k}}={\hslash }^2{k}^2/(2m_e)$; $f(E)$ is their Fermi-Dirac distribution occupation. Both momentum sums exclude spin; the prefactor two counts the two equal spin populations. The density-operator matrix element is ${\delta }_{{\mathbf{k}}^{\prime },\mathbf{k}+\mathbf{q}}$. Coincident energy denominators are interpreted by the static limit or Cauchy principal value.

A weak total electric potential ${\varphi }_{\mathrm{tot}}$ perturbs an electron by $-e{\varphi }_{\mathrm{tot}}$. First-order state mixing produces $\delta n=e{\Pi }_0(q){\varphi }_{\mathrm{tot}}$, whereThe induced charge is $-e\delta n$. Self-consistent Poisson equation screening then gives ${\varphi }_{\mathrm{ext}}=ϵ{\varphi }_{\mathrm{tot}}$, with $ϵ=1+e^2{\Pi }_0/({ϵ}_0{q}^2)$. This is the random phase approximation: use the independent-electron response to the self-consistent total field.^[ref]

At zero temperature $f(E_{\mathbf{k}})=\Theta (k_F-k)$. Shifting the integration variable in the second occupation term givesThe two angular integrals are equal. Using the supplied integral, with $x=q/(2k_F)$, givesThe limits are $F(0)=1$, $F(1)=1/2$ and $F(x)\sim 1/(3x^2)$ for large $x$. The first recovers Thomas-Fermi screening; the logarithm produces the Kohn singularity of static Lindhard screening at $q=2k_F$.

Use the Fourier transform convention $\rho (\mathbf{q})=\int d^{3}r{e}^{-i\mathbf{q}\cdot \mathbf{r}}\rho (\mathbf{r})$. A point charge $Ze$ has total screened charge $Ze/ϵ(q)$ and induced charge $-Ze[1-ϵ(q{)}^{-1}]$. Angular inversion therefore gives the radial induced-charge screening kernelIn particular, ${\rho }_{\mathrm{ind}}(\mathbf{q}=0)=-Ze$: the total screening charge is negative although its local density can oscillate in sign.

Integrate the sine transform by parts twice; $g(0)=g(\infty )=0$ removes the boundary terms. For the Friedel density tail from a derivative singularity, write $Q=2k_F$ and $g^{\prime \prime }(q)\sim C/(q-Q)$. ThenIndeed, near $Q$ the principal value transform is ${\int }_{-\infty }^{\infty }\sin [(Q+s)r]ds/s=\pi \cos (Qr)$ for $r>0$. Smooth terms do not remove this leading oscillatory long-range contribution. Thus ${\rho }_{\mathrm{ind}}\propto \cos (2k_{F}r)/r^3$.

For a three-dimensional, single-valley gas with two spin states, the Friedel oscillation wavelength from carrier density isLow-temperature scanning tunneling spectroscopy near exposed dopants can map the oscillatory local density of states, with a Fourier feature near $2k_F$ at the Fermi energy. An energy-resolved tunnelling image measures local spectral weight, not directly the bulk energy-integrated charge: comparing occupied-bias maps and accounting for the surface geometry is necessary. In actual GaAs, the background relative permittivity and band effective mass change screening strengths; they do not change this wavelength at the stated density.

Write $e=q>0$ for the elementary charge and $ϵ={ϵ}_0{ϵ}_r$ for the permittivity. Taking the bulk electric potential as zero, the Boltzmann distribution of point ions gives $c_i(x)=c_i^{\infty }\exp [-e{z}_i\varphi (x)/(k_{B}T)]$. Here $c_i$ is a number density, not a molar concentration. Combining this with Poisson's equation gives the planar Poisson-Boltzmann equationMultiplying by ${\varphi }^{\prime }$ and integrating from the neutral bulk, where $\varphi ={\varphi }^{\prime }=0$, yieldsFor the solution occupying $x>0$, with the field on the other side neglected, Gauss's law gives $\sigma =-ϵ{\varphi }^{\prime }(0)$. On the branch that decays to zero without crossing it,Source sign qualification: the unsigned square root printed in the paper gives $|\sigma |$. A negative wall electric potential, such as the next part's $-73\mathrm{mV}$, has ${\varphi }^{\prime }(0)>0$ and hence $\sigma <0$, providing a concrete counterexample to interpreting that printed expression as signed surface charge density. This is the planar Grahame equation, with an explicit sign convention.

Full dissociation gives bulk concentrations in $\mathrm{mM}$ of $155$ for sodium, $5$ for potassium, $2$ for calcium, $114$ for chloride, and $25$ for hydrogen phosphate. The positive and negative charge concentrations are both $164\mathrm{mM}$, so the bulk is electrically neutral. At $T=298.15\mathrm{K}$,The Boltzmann distribution then givesFor completeness the coion concentrations are $114e^{-\nu }=6.65196\mathrm{mM}$ and $25e^{-2\nu }=0.0851195\mathrm{mM}$. They are depleted, not zero. Using $1\mathrm{mM}=1\mathrm{mol}{\mathrm{m}}^{-3}$ and multiplying molar concentrations by the Avogadro constant in the Grahame equation givesThis calculation retains every species. Since $|{\varphi }_0|/V_T=2.84$, a linear Debye–Hückel approximation is not justified. The large accumulation, especially of divalent counterions, follows from the valence in the Boltzmann factor. However, multi-molar local concentrations also warn that point-ion mean-field theory neglects finite ion size, hydration, specific adsorption, correlations, and changes in local permittivity. These are predictions of the stated idealized model, not precise measured interfacial concentrations.

A charged biological molecule attracts counterions and repels ions of the same sign. Thermal diffusion opposes complete localization, producing a mobile ionic atmosphere whose charge partly cancels the molecule's electric field at larger distances. Solvent polarization also reduces the field through its permittivity, but that dielectric response is distinct from the mobile-ion screening cloud.

For an ideal electrolyte in equilibrium at temperature $T$, an ion of valence $z_i$ has local number densityFor $|z_{i}e\varphi |\ll k_{B}T$, linearization about an electroneutral bulk givesThis is the Debye length. Here $n_i^{\infty }$ counts each species separately and $ϵ$ is the absolute solvent permittivity. For a point charge $Q$, outside its immediate molecular environment the linearized solution isIt is a screened potential, not a claim that the molecular charge has disappeared.

In terms of ionic strength $I=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$ with $c_i$ in $\mathrm{mol}{\mathrm{L}}^{-1}$, the density sum is $2N_{A}10^{3}I$, soIn water near $298\mathrm{K}$, ${\lambda }_D\simeq 0.304\mathrm{nm}/\sqrt{I/(1\mathrm{mol}{\mathrm{L}}^{-1})}$: about $9.6\mathrm{nm}$ at $1\mathrm{mM}$ and $0.79\mathrm{nm}$ at $0.15\mathrm{M}$. Increasing ion density or valence shortens the range; temperature and solvent permittivity also matter. The displayed temperature dependence holds at fixed permittivity, which itself changes with temperature in water.

The derivation assumes dilute mobile ions, a continuum uniform dielectric and weak potentials. Near highly charged DNA or protein surfaces, nonlinear Poisson-Boltzmann equation behaviour, counterion condensation, finite ion sizes, specific binding and correlations can matter. At physiological salt the predicted length approaches molecular scales, so the formula is a useful far-field scale with limited microscopic accuracy, rather than a complete account of the first hydration layer.

The polymer contour length and magnitude of total electric charge areUse a uniform charged-sphere model for the packed DNA, a homogeneous dielectric with $ϵ={ϵ}_r{ϵ}_0$, and initially neglect electrostatic screening. Gauss's law gives $E(r)=Qr/(4\pi ϵR^3)$ inside the sphere and $E(r)=Q/(4\pi ϵr^2)$ outside. Integrating the electrostatic energy both inside and outside,For an illustrative aqueous relative permittivity ${ϵ}_r=80$ and $R=20\mathrm{nm}$,The question does not specify permittivity; the result scales as $80/{ϵ}_r$. Using vacuum instead would increase it eightyfold and would not describe an aqueous interior.

Strictly, the work of packaging is a difference of electrostatic energies, rather than the packed self-energy alone. An extended charged rod also has self-energy. With a microscopic cutoff $a$ of order the DNA radius, the thin-rod approximation isOne derivation integrates the pair Coulomb potential along the rod, replacing separation by $\sqrt{z^2+a^2}$: the exact regularized expression is ${\lambda }^2[L\mathrm{arsinh}(L/a)-\sqrt{L^2+a^2}+a]/(4\pi ϵ)$. The diagram's $1\mathrm{nm}$ transverse label suggests $a\simeq 0.5\mathrm{nm}$, giving $U_{\mathrm{rod}}\simeq 6.25\times 10^6\mathrm{pN}\mathrm{nm}$. Hence this explicit reference-state estimate is $\boxed{\Delta U\simeq 1.32\times 10^8\mathrm{pN}\mathrm{nm}}$. The cutoff and precise initial conformation are additional model assumptions, but their correction is small compared with the unscreened sphere value.

An ion channel supplies a water-accessible pathway through the otherwise strongly insulating, hydrophobic lipid bilayer. Moving a hydrated ion directly into that low-permittivity region is energetically expensive. The channel's polar pore and coordinating groups reduce the barrier while retaining selectivity. Transport is passive: the electrochemical potential difference drives net movement, and the channel does not itself pump an ion uphill. For an ideal species of charge number $z$,The net electric current of a pathway permeable only to that species reverses at its Nernst equilibrium potential. An open channel can carry zero net electric current at that voltage even while individual ions cross both ways.

The selectivity filter balances loss of the hydration shell against favorable coordination within the pore. In a typical potassium channel, backbone carbonyl oxygen atoms create appropriately arranged binding sites for a substantially dehydrated potassium ion. The smaller sodium ion is not automatically preferred simply because it fits geometrically: its hydration is stronger and the filter does not compensate its dehydration in the same way. Geometry, flexibility, coordination chemistry and ion occupancy all matter. Sodium-selective and calcium-selective channels use different arrangements, often involving charged side chains; a single rigid-sieve explanation is inadequate.

Rapid passage coexists with transient binding because several nearby sites have comparable free energies and relatively low barriers between them. Multiple ions occupying a pore influence one another through electrostatic interactions, enabling correlated exchange and displacement rather than irreversible trapping at one deep site. High throughput therefore requires favorable passage kinetics as well as strong discrimination against unwanted species. A more tightly bound species can even block a pathway instead of being its fastest permeant.

Gating and selectivity are distinct. A voltage-sensing domain couples membrane potential to activation-gate conformation; ligand-gated ion channels couple binding to opening; mechanosensitive ion channels couple membrane stress to opening. Ion-channel inactivation introduces additional unavailable states. None of these mechanisms alone identifies which ion the open pore prefers.

For $N$ similar independent channels, a local Ohmic approximation with outward-positive electric current isHere open probability controls availability, single-channel electrical conductance controls current per open pore, and $V-E_{\mathrm{rev}}$ is the electrical driving factor in this approximation. Nonlinear current-voltage relations and mixed-ion permeation may require a fuller model. Single-channel recording resolves discrete openings; reversal shifts when ionic concentrations change test selectivity; gating electric currents test sensor motion; structural evidence identifies the pore and its coordinating groups. Together these distinguish rapid selective permeation from the separate process of opening the gate.

At normal incidence from vacuum the reflectivity iswith the passive branch of the complex refractive index. Below the optical resonance the real permittivity rises towards its pole and reflectance increases. Between the TO and LO frequencies, the lossless permittivity is negative; the wave is evanescent and the reflectivity is unity in that idealization. Finite damping rounds and reduces this reststrahlen band. Above the LO edge, positive permittivity returns and reflectivity can become very small when the refractive index is near one, before rising towards its electronic-background value.

Reading the surviving source graph to its available precision givesThese are band-edge estimates, not a high-precision damping fit. The minimum immediately above the band is not exactly the LO zero: for zero damping ${ϵ}_r=1$ occurs at ${\omega }^2=({ϵ}_{\infty }{\omega }_{\mathrm{LO}}^2-{\omega }_{\mathrm{TO}}^2)/({ϵ}_{\infty }-1)>{\omega }_{\mathrm{LO}}^2$.

The high-energy plotted tail near $90\mathrm{meV}$ has $R\simeq 0.22$. Ignoring residual phonon dispersion gives $n\simeq (1+\sqrt{R})/(1-\sqrt{R})=2.77$ and $ϵ\simeq 7.65$. Correcting with the finite-energy oscillator ratio givesThus a background of order $8$–$9$ is supported; equating the still-dispersive tail exactly to the asymptotic background would slightly underestimate it. The original model curves below use these approximate energies, without copying or fitting individual source data points.

Write $\mu ={\mu }_0{\mu }_r$ and $ϵ={ϵ}_0{ϵ}_r$ for homogeneous scalar material constants, and use $\mathbf{J}=\sigma \mathbf{E}$. Taking the curl of Faraday's law of induction and substituting the Ampere-Maxwell law givesThe vector calculus identity for a double curl therefore yieldsThe second term represents electric current response, rather than displacement current. A dispersive $\sigma (\omega )$ is used in the frequency domain; its general time-domain counterpart is a temporal response convolution.

For the convention $e^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, the plane wave equation isDecompose the electric field into components parallel and perpendicular to $\mathbf{k}$. With $S=\mu ϵ{\omega }^2+i\mu \sigma \omega$, the longitudinal eigenvalue is $S$ and each transverse eigenvalue is $S-k^2$; thus the determinant is $S(S-k^2{)}^2$. For a transverse electromagnetic wave, $\mathbf{k}\cdot {\mathbf{E}}_0=0$ andwhen ${\mu }_r=1$ and the background ${ϵ}_r\simeq 1$. A real dissipative electrical conductivity generally makes $k$ complex, with the decaying branch chosen for a passive medium.

For a cold, collisionless, unmagnetized plasma, treat ions as stationary and write $e>0$ for the magnitude of the elementary charge. The electron equation of motion is $-i\omega m_e\mathbf{v}=-e\mathbf{E}$, soThis reactive response includes no collision-induced Joule heating. Substitution into the transverse dispersion relation givesThe quantity ${\omega }_p$ is the electron plasma frequency. The refractive index is real above this cutoff; below it, the wave is evanescent. The electron response is represented here by $\sigma (\omega )$, so it must not also be counted a second time in the background permittivity.

For a nonzero longitudinal component, taking the dot product of the matrix equation with $\mathbf{k}$ instead gives $S(\mathbf{k}\cdot {\mathbf{E}}_0)=0$. The same cold plasma response makes $S=({\omega }^2-{\omega }_p^2)/c^2$, henceAt nonzero $k$, the remaining equation forces ${\mathbf{E}}_0$ to be parallel to $\mathbf{k}$, and Faraday's law of induction gives ${\mathbf{B}}_0=0$. These are electrostatic Langmuir waves, with no $k$ dependence of angular frequency in the zero-temperature model. Equilibrium neutrality does not forbid their perturbed charge density; Gauss's law requires such a density when $\mathbf{k}\cdot {\mathbf{E}}_0\ne 0$.

Use the constitutive relations $\mathbf{D}=ϵ\mathbf{E}$ and $\mathbf{H}=\mathbf{B}/\mu$, where $\mathbf{D}$ is the electric displacement field, $\mathbf{H}$ the magnetic field strength, $ϵ$ the permittivity, and $\mu$ the magnetic permeability. Let ${\rho }_f$ be the charge density of free charge and ${\mathbf{J}}_f$ its current density. In this homogeneous medium, Gauss's law and the magnetostatic Ampère law are, respectively,Here $W$ is a volume with closed boundary $\partial W$, and $S$ is an oriented spanning surface whose boundary orientation follows the right-hand rule. Thus the electric field form is $\nabla \cdot \mathbf{E}={\rho }_f/ϵ$ and ${\oint }_{\partial W}\mathbf{E}\cdot \mathbf{n}dS=Q_f/ϵ$; the magnetic field form is $\nabla \times \mathbf{B}=\mu {\mathbf{J}}_f$ and ${\oint }_{\partial S}\mathbf{B}\cdot d\mathbf{\ell }=\mu I_f$.

The remaining static Maxwell equations areThese express Gauss's law for magnetism and the static limit of Faraday's law. The volume and circulation forms follow from the divergence theorem and Stokes theorem. The electric charge and electric current on the right of the macroscopic laws are free sources; material electric polarization and magnetization are already included in $\mathbf{D}$ and $\mathbf{H}$.

For identical thicknesses, compare $d/\kappa$ using the supplied coefficients:for barium titanate, lead zirconate titanate and quartz respectively. Thus barium titanate, then lead zirconate titanate, then quartz is the increasing voltage sensitivity of a piezoelectric sensor order. A large charge coefficient alone does not guarantee a large voltage: the large permittivity of a ferroelectric also increases its capacitance.

An optically isotropic film changes the phase of both transverse polarization components equally, so crossed polarizers still extinguish the transmitted light. The etched rod-and-gap composite has form birefringence: its effective permittivity depends on the direction of the electric field, even when the constituent material is intrinsically isotropic. In a simple subwavelength lamellar model,The precise effective-medium formula changes with rod geometry, but the anisotropy remains. Resolving incident light along the two principal directions produces a phase retardation. An ideal lossless specimen between crossed polarizers transmitswhere $I_P$ is the intensity after the first polarizer. Extinction occurs when a principal direction follows either polarizer; transmission is strongest near $45^{\circ }$.