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The positive electron mobility relates the drift velocity of conduction electrons to an electric field: ${\mathbf{v}}_d=-{\mu }_e\mathbf{E}$. In the isotropic Drude model, ${\mu }_e=e\tau /m^{\ast }$, where $\tau$ is the momentum relaxation time and $m^{\ast }$ the band effective mass. The minus sign describes negative charge; the electron mobility itself is positive.

Electron optics studies how electric fields and magnetic fields guide electrons and form images. The Lorentz force supplies a trajectory description, while the de Broglie wavelength supplies the wave scale needed for diffraction. Both descriptions matter: steering a narrow beam and resolving two nearby specimen features are different achievements.

For a magnetic electron lens,Here $e>0$ is the elementary charge, $\mathbf{v}$ the electron velocity, and $\mathbf{B}$ the magnetic field. The second equality follows because a cross product is perpendicular to each factor: a static magnetic field bends the trajectory without increasing the kinetic energy. Acceleration instead requires work by an electric field. This distinction lets an electron microscope adjust imaging with magnetic fields while an accelerating voltage sets the electron momentum.

The electro-optic Kerr effect produces linear birefringence proportional to the square of a static applied electric field. Defining the Kerr effect coefficient $K$ by $\Delta n={\lambda }_{0}K{E}^2$, a length $L$ produces phase retardation $\delta =2\pi KL{E}^2$. A switchable quarter-wave plate requires $|\delta |=\pi /2$ modulo $2\pi$, with fixed principal axes. This differs from the Faraday effect, whose eigenpolarizations are circular polarizations.

A device uses an applied electric field to change an optical medium's refractive index and hence the phase, amplitude or polarization of light. A phase change followed by suitable polarization analysis can provide an intensity transmission gate for active mode locking.

Electro-osmosis is fluid motion driven by an applied electric field acting on mobile electric charge near a surface. A charged interface attracts a diffuse population of counterions in an electric double layer. A field component tangent to the interface exerts a force density ${\rho }_e\mathbf{E}$ on that mobile charge; momentum transfer to the surrounding solvent drives flow opposed by viscous stress. In a capillary electrophoresis measurement, this flow advects the sample as well as its electrical drift relative to the fluid. Locally, the measured axial velocity is $v_{\mathrm{observed}}=u_{\mathrm{eo}}+{\mu }_{\mathrm{ep}}E$, where $u_{\mathrm{eo}}$ is the electroosmotic fluid velocity and ${\mu }_{\mathrm{ep}}$ is the electrophoretic mobility. ^[ref]

Electrophoresis is motion of charged dispersed species relative to a surrounding medium under an applied electric field. The drift depends on electrical driving and friction, including the surrounding ionic atmosphere. Gel electrophoresis adds a porous matrix that makes migration sensitive to molecular size and conformation.

The electrophoretic mobility $\mu$ is the proportionality coefficient between drift velocity and a weak applied electric field. Its sign specifies motion relative to the field. When only travel distances are used, the mobility magnitude is often intended. For long linear DNA in a weak-field reptation regime, $|\mu |$ can scale inversely with chain length, whereas free-solution mobility is approximately length-independent. Neither law supplies a universal gel electrophoresis size calibration for arbitrary gel concentration, field strength or molecular topology.

For a fixed electric charge $q$, a time-independent electric potential $\Phi$ and negligible collisions or radiation losses, conservation of energy givesHere $V$ is the signed potential drop along the trajectory. A proton starting nearly at rest at $+V$ and arriving at ground gains $eV$: a chosen $500\mathrm{kV}$ gives $500\mathrm{keV}$, by the definition of the electronvolt. Several intermediate electrodes can distribute that drop along a tube, but the gains telescope to the same endpoint difference. Passing through more electrodes does not multiply the available kinetic energy beyond $qV$.

This is the distinction from a cyclotron. In a static electric field, a closed trip gives zero net work. A cyclotron changes the gap electric field between successive visits, so correctly timed particles repeatedly gain kinetic energy. Its limit involves the cyclotron resonance condition and relativistic detuning of a cyclotron; a single-pass electrostatic accelerator instead requires the full accelerating electric potential difference to be sustained at once. Neither machine's beam energy by itself specifies the energy released by a nuclear reaction.

An electrostatic accelerator gives a charged particle kinetic energy by taking it through a sustained electric potential difference. Its ion source, accelerating tube and target form a particularly direct chain: prepare a charge, let an electric field do work, and measure what arrives or what the target emits. A vacuum reduces collisions with gas before the beam reaches its target. The Cockcroft-Walton generator supplies the potential by accumulating electric charge on capacitors.

The electrostatic boundary conditions at a conductor relate the one-sided total electric fields at a smooth metal–vacuum interface. Choose a unit normal $\mathbf{n}$ from metal into the adjacent vacuum, let ${\mathbf{E}}_{\mathrm{metal}}$ and ${\mathbf{E}}_{\mathrm{vac}}$ be the two limiting electric fields, and let $\sigma$ be the signed surface charge density. A thin Gaussian pillbox with cap area $A$ has outward cap normals $+\mathbf{n}$ on the vacuum side and $-\mathbf{n}$ on the metal side. As its height tends to zero, its side electric flux tends to zero for bounded local electric fields, leavingThe minus sign comes from the second outward cap normal, not from the sign of the electric charge. A shrinking rectangular loop across the interface and the zero closed line integral in electrostatics give continuity of the tangential component:For a conductor in electrostatic equilibrium, ${\mathbf{E}}_{\mathrm{metal}}=0$, henceThus positive surface charge density gives an electric field away from metal and negative surface charge density gives an electric field toward metal. The derivative uses the same chosen unit normal, from the vacuum side. The ideal surface layer has no unique electric field value on the layer itself; these are limiting values. Ordinary surface electric charge preserves continuity of electric potential; an ideal electric dipole sheet would require a different model. ^[ref].

Here “exterior” means just outside the metal, including the vacuum within a cavity in a conductor. On a spherical inner wall of radius $a$, its unit normal from metal into the cavity in a conductor is $-\hat{\mathbf{r}}$. A central positive point charge $q$ gives ${\mathbf{E}}_{\mathrm{vac}}=q\hat{\mathbf{r}}/(4\pi {ϵ}_0{a}^2)$ there, soThe outward normal of the cavity in a conductor as a volume is instead $+\hat{\mathbf{r}}$; confusing these two unit normals reverses the inferred sign. For a stipulated spherical enclosure with inner radius $a$, outer radius $b$, no external sources and total isolated metal electric charge $Q_c$, spherical symmetry and Gauss's law give the analytical modelIts outer surface charge density is $(Q_c+q)/(4\pi b^2)$. Grounding to $V_c=0$ with zero electric potential at infinity instead gives $Q_c=-q$ and zero exterior electric field, while preserving the same nonzero electric field inside the charged cavity in a conductor.

As a numerical sign check, stipulate $q=1\mathrm{nC}$, $a=0.10\mathrm{m}$, $b=0.20\mathrm{m}$ and $1/(4\pi {ϵ}_0)\simeq 8.99\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$. The inner electric field is approximately $899N/C$ along $+\hat{\mathbf{r}}$, but ${\mathbf{E}}_{\mathrm{vac}}\cdot \mathbf{n}=-899N/C$ and ${\sigma }_{\mathrm{inner}}\simeq -7.96nC/{\mathrm{m}}^2$. An isolated neutral enclosure has $+1\mathrm{nC}$ on the outer wall, approximately $+1.99nC/{\mathrm{m}}^2$ and $225N/C$ just outside. Multiplying each surface charge density by its spherical surface area recovers the signed wall electric charges.

The local result uses the total electric field, including all induced and imposed sources. The factor $\sigma /{ϵ}_0$ follows from the zero metal-side electric field; a symmetric isolated sheet in vacuum instead has magnitude $|\sigma |/(2{ϵ}_0)$ on each side. ^[ref]. At sharp corners, limiting electric fields can be singular and the smooth local pillbox argument needs care. In material outside the metal, use the normal free-charge jump of the electric displacement field and the material's permittivity, as in electrostatic interface boundary conditions. ^[ref].

An electron accelerated through voltage $U_a$ crosses deflecting plates of length $\ell$ and separation $d$. If their electric field is uniform and their centre is distance $D$ from a screen, constant transverse acceleration inside the plates followed by free drift gives the displayed screen displacement. It includes the displacement accumulated inside the plates. The plate-exit displacement is $x\ell /(2D)$, which must be smaller than the half-gap for a centrally injected beam to pass.

Two isolated conducting drops of radius $a$ and equal charge merge at fixed total charge and volume. The new radius is $2^{1/3}a$, so its capacitance grows by only $2^{1/3}$ while total charge doubles. Relative to the sum of the initially separated self energies, the final electric field energy grows by $2^{2/3}$. Decreased surface energy and/or applied mechanical work can supply that increase. Sufficiently large like charges need not coalesce spontaneously; energy conservation is not a guarantee of merging. A maintained external voltage would define a different problem.

A first spherical shell with charge $q$ followed by shells of charges $2(-1{)}^{n+1}q$ leaves enclosed charge $(-1{)}^{n+1}q$ after each shell. Gauss's law therefore gives the same squared electric field $q^2/(16{\pi }^2{ϵ}_0^2{r}^4)$ in every gap outside the first radius. Integrating the electrostatic energy density over all space gives $W=q^2/(8\pi {ϵ}_0{r}_1)$, independent of the other shell radii.

A solid sphere of radius $R$ and uniform charge density has total electric charge $Q$. Gauss law gives the radial electric field $E_r=Qr/(4\pi {ϵ}_0{R}^3)$ inside and $E_r=Q/(4\pi {ϵ}_0{r}^2)$ outside. The electric potential vanishing at infinity is $Q(3-r^2/R^2)/(8\pi {ϵ}_{0}R)$ inside and $Q/(4\pi {ϵ}_{0}r)$ outside. Integrating the electrostatic energy density ${ϵ}_0{E}^2/2$ over both regions gives $U=3Q^2/(20\pi {ϵ}_{0}R)$. In a fixed-density family with $Q\propto Z$ and $R^3\propto Z$, the electrostatic energy scales as $Z^{5/3}$.

A state with a stationary electric charge distribution and no net drift of mobile charge driven by the electric field. A conductor in electrostatic equilibrium has zero interior electric field, while bound charges in an insulating material need not move to eliminate its interior field.

The electric force exerted by a time-independent electric field. For a point charge, the external field excludes its own singular self-field. The method of images evaluates conductor-induced forces by the image field at the physical charge's position; the fictitious images need not be physical charges in the conductor interior.

Consider lab-neutral parallel wires with stationary positive line density $+\lambda$ and electron line density $-\lambda$. Let wire 1's electrons move at $v_1$ along the wire and wire 2's at $v_2$. The conventional currents are $I_j=-\lambda v_j$. In the rest frame of wire 2's electrons, wire 1's transformed total linear charge density is ${\lambda }_1^{\prime }={\gamma }_2[0-v_2{I}_1/c^2]={\gamma }_2\lambda v_1{v}_2/c^2$. Its radial electric field is ${\lambda }_1^{\prime }/(2\pi {ϵ}_{0}d)$.

A wire-2 electron at rest there feels only an electric force. Transforming its transverse force back to the lab divides it by ${\gamma }_2$, and multiplying by the lab electron count per length $\lambda /e$ giveswith positive radial direction away from wire 1. The lab-stationary positive charges feel zero net electric force because wire 1 is neutral in their frame. Equal current directions attract; opposite directions repel. Transforming force on each carrier and then counting carriers in the specified frame avoids an incorrect extra length-contraction factor in force per unit length.

An interaction between electric charges mediated by an electric field. In a uniform dielectric the pair potential energy is $q_1{q}_2/(4\pi ϵr)$; mobile ions can modify it through electrostatic screening.

Across a charged dielectric interface the tangential electric field is continuous and the normal displacement jump is the fixed surface charge density. Without a dipole sheet the electrostatic potential is continuous. For a charge sheet with equal permittivity on both sides, its normal potential derivative has jump $-{\sigma }_{\mathrm{fixed}}/\epsilon$. Setting the field to zero on one side is an additional physical boundary assumption, as in a conductor or a negligible-exterior-admittance idealization.

A plasma wave dominated by a longitudinal electric field. The leading dispersion relation is $\mathbf{k}\cdot \mathbf{ϵ}\cdot \mathbf{k}=0$. In a cold magnetized plasma, this becomes $k_{\perp }^{2}S+k_{\parallel }^{2}P=0$. An exactly longitudinal finite-wavenumber solution must also satisfy the full Maxwell equations; the oblique electrostatic condition is usually an asymptotic large-refractive index approximation.