A connection in which branches share both terminal nodes and therefore the same electric potential difference. Branch electric currents add. For sinusoidal signals, the total electrical admittance is the sum of branch electrical admittances; ideal capacitances add.

A two-port network relates voltage and current, or incoming and outgoing travelling-wave amplitudes, at two pairs of electrical terminals. Power-normalized scattering parameters obey $b_i={\sum }_j{S}_{ij}{a}_j$. In a reciprocal network with equal reference impedances, $S_{12}=S_{21}$. A load reflection at port 2 can return through both transmission coefficients to port 1.

An alternating-current electrical circuit comparing a parallel resistor-capacitor arm with a series resistor-capacitor arm. Let the upper-left arm have electrical impedance $Z_1=R_1/(1+i\omega R_1{C}_1)$, the lower-left arm $Z_2=R_2+1/(i\omega C_2)$, and the upper-right and lower-right arms have electrical resistances $R_3,R_4$. For input between the upper and lower nodes and an unloaded difference measurement between the arm junctions, voltage divider balance requires $Z_2/Z_1=R_4/R_3$. Separating real and imaginary parts givesThe balance angular frequency is positive for positive components. Equal $R_1=R_2=R$ and $C_1=C_2=C$ require $\omega =1/(RC)$ and $R_4=2R_3$. Reversing which reactive arm is upper changes the divider ratios, so the connections must always be specified.

A circuit node held close to zero electric potential by active feedback, without being directly connected to the reference conductor. In an ideal linear operational amplifier with its non-inverting input grounded and stable negative feedback, $V_{-}=V_{+}=0$ while neither input draws electric current. External branch currents meeting the inverting node therefore obey Kirchhoff's current law without flowing into the amplifier input. Equality of the input potentials fails when the output saturates or the feedback ceases to enforce linear operation.

An ideal open circuit permits no conduction electric current through a terminal pair, corresponding to infinite load electrical impedance. At the end of a lossless transmission line, its voltage-wave reflection coefficient is $+1$ and the terminal current is zero. Real open terminals may retain parasitic capacitance and radiation loss.

An ideal short circuit fixes zero electric potential difference across a terminal pair, corresponding to zero load electrical impedance. At the end of a lossless transmission line, its voltage-wave reflection coefficient is $-1$ and the terminal voltage is zero. The available current is still limited by source and circuit impedances; real short circuits can dissipate substantial Joule heating.

An oriented curve whose tangent points along the local electric field. Where the field is nonzero, choosing arc length $s$ gives $d\mathbf{r}/ds=\mathbf{E}/|\mathbf{E}|$. These curves visualize the direction of the force on a positive test charge; they are not generally particle trajectories, since a particle can have inertia and initial velocity. In electrostatics, they are normal to equipotential surfaces and cannot form a closed loop in a region with a single-valued potential that strictly decreases along the line. A field null can separate families of lines.

A charged particle carries nonzero electric charge and couples to an electromagnetic field. For a classical point particle, the Lorentz force is $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ in SI, distinguishing electric acceleration from velocity-dependent magnetic deflection.

An electron, a proton and an ion are examples with different masses and charge-to-mass ratios. A magnetic field can bend their tracks in a particle detector, while ionization along the path makes them observable in a cloud chamber or bubble chamber. A neutral particle can still interact electromagnetically through internal structure or moments; zero net charge does not imply total electromagnetic invisibility.

Electric charge distributed through a region, whose own electric field can alter the motion of further charged particles.

Charge invariance means that the total electric charge of a specified collection of particles is unchanged by a Lorentz transformation. It differs from conservation of electric charge, which compares the same isolated system at different times. For a convective charge distribution with rest density ${\rho }_0$, the four-current is $J^{\mu }={\rho }_0{U}^{\mu }$, where $U^{\mu }$ is its four-velocity; its measured charge density increases under a boost while the simultaneous spatial volume contracts, leaving total charge unchanged.

In macroscopic electromagnetism, free charge is the part of the charge density represented explicitly as ${\rho }_f$, separately from bound charge associated with the electric polarization of matter. The electric displacement field satisfies $\nabla \cdot \mathbf{D}={\rho }_f$. The word free does not imply zero force or zero energy: it distinguishes this source from the polarization response in the chosen material description.

Magnetoelastic coupling links a material's magnetic state to its strain and stress. Magnetostriction describes a resulting deformation; mechanical loading can also change the magnetic state.

An eigenstrain is a local preferred deformation in the absence of stress. Constraints or incompatibility can make the actual strain differ from it, generating stress. Magnetostrictive strain is one example.

In four dimensions, the contracted Bianchi identity gives $(1/2-\alpha ){\partial }_{\mu }R=-\kappa {\nabla }^{\nu }{T}_{\mu \nu }$ for the displayed equation. Its trace is $(1-4\alpha )R=-\kappa T$. For $\alpha \ne 1/4$,Compatibility with covariant stress-energy conservation for unrestricted conventional matter therefore requires $\alpha =1/2$, yielding the Einstein tensor. Other coefficients impose constant trace on conserved sources. The exceptional $\alpha =1/4$ equation instead requires $T=0$; conservation additionally makes $R$ constant. A trace-free equation sourced by $T_{\mu \nu }-g_{\mu \nu }T/4$ is a different system and does not impose vanishing matter trace.

Let $A_{21}$ be the spontaneous emission coefficient and let $B\rho$ be the induced transition rate per atom under a resonant radiation spectral energy density $\rho$. For nondegenerate levels the Einstein coefficients obey $B_{12}=B_{21}=B$. In a closed two-level system with no collisional or other population transfer, steady state requiresFor nonzero upper population and drive, division gives the ratio of total spontaneous and stimulated emission rates in the title. In particular $N_1>N_2$ when $A_{21}>0$; continuous driving of this closed model approaches equal populations but cannot create a population inversion. No thermal population law is assumed.

A liquid in rigid rotation about a vertical axis is at rest in its rotating reference frame. Constant surface pressure makes $gz-{\Omega }^2{r}^2/2$ constant along the free surface, giving the displayed paraboloid. Comparing with $z-z_0=r^2/(4f)$ gives the focal length $f$. The result assumes uniform gravitational acceleration, a vertical rotation axis and negligible surface-tension corrections. It explains how a rotating liquid can form a reflecting surface or a mould for a mirror.