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Conductive-paper electric potential mapping uses a resistive sheet as a two-dimensional analog of an electrostatic boundary-value problem. Painted or taped conducting electrodes impose selected electric potentials. One voltmeter contact stays at a reference electrode while a movable contact locates points at the same electric potential; joining those points gives an equipotential map.

The analogy follows from steady electric charge conservation and local Ohm's law. In a source-free sheet of uniform isotropic electrical conductivity $\sigma$,The measured potential therefore satisfies Laplace's equation. In-plane electric field components follow from $\mathbf{E}=-\nabla V$; more closely spaced equal-potential contours indicate a larger local field magnitude. Nonuniform conductivity, electrode contacts and voltmeter loading can alter the inferred solution. The apparatus models a two-dimensional geometry, not every finite three-dimensional electrode arrangement. ^[ref]

Section 9.7 of Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher. Whole-book authorship applies. web.mit.edu/6.013_book/www/chapter9/9.7.html. Example 9.7.1 explicitly assumes a gap small compared with transverse dimensions and approximately uniform fields in circuit sections. The subsequent discussion distinguishes those estimates from a full magnetic scalar-potential solution.

Hermann Anton Haus was an electrical engineer whose research ranged from quantum optics to optical communication systems and ultrashort laser pulses. His work on optical solitons helped connect nonlinear wave propagation with the transmission of information through optical fibers.^[ref]

With James R. Melcher, he wrote Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher, a text that links field equations with concrete devices and demonstrations. Its magnetic circuit analysis, for example, shows how geometry and constitutive assumptions turn an electromagnetic field problem into an approximate engineering model.^[ref]

James R. Melcher was an electrical engineer whose work helped define continuum electromechanics: the interaction of electromagnetic fields with deformable materials and flowing fluids. His applications included energy conversion, flow measurement and the control of industrial processes.^[ref]

With Hermann Anton Haus, he wrote Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher. A central engineering theme is the relation between a full field description and a justified approximation. In a magnetic circuit, for example, geometry and permeability assumptions determine when a small set of circuit-like variables can represent the spatial fields.^[ref]

In the small-sphere oscillatory-field limit, surface current density excludes the changing magnetic field. Adding a central magnetic dipole to a uniform applied field and imposing zero normal field at the sphere gives $\mathbf{m}=-2\pi a^3{\mathbf{H}}_{\mathrm{inc}}$. The minus sign expresses diamagnetism; it does not make this dipole's backward radiation cancel the electric contribution.^[ref]

A magnetic skin layer is the region penetrated by an oscillating magnetic field in a conducting material. With constant magnetic permeability $\mu$ and electrical conductivity $\sigma$, define the magnetic diffusivityHere $\eta$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$; it is not the electrical resistivity $1/\sigma$. Negligible material motion reduces the resistive induction equation to magnetic diffusion.

Take the conductor to occupy $y>0$ and use time dependence $e^{i\omega t}$ with $\omega >0$. For a tangential magnetic field, the one-dimensional equation isThe solution decaying into the conductor hasThus the skin depth $\delta$ is both the amplitude decay length and the distance for one radian of phase lag. Raising frequency by a factor of four halves $\delta$. ^[ref]

If the boundary amplitude varies slowly as $C(x)$, the solenoidal magnetic-field constraint additionally requires, to leading order,Indeed ${\partial }_x{\hat{B}}_x+{\partial }_y{\hat{B}}_y=0$. This normal component is smaller by the ratio of $\delta$ to the along-boundary variation length; a predominantly tangential field need not be exactly tangential.

A demonstration described by Hermann A. Haus and James R. Melcher drives a winding on a magnetic core passing through a slit in an aluminium block. A magnetometer probe samples the field through a measuring slit. Comparing 60 and 240 Hz tests the predicted factor-of-two change in penetration depth. ^[ref]

antenna quantities. radiation resistance represents emitted power by $P_{\mathrm{rad}}=I_{\mathrm{rms}}^2{R}_{\mathrm{rad}}$ at the feed. It is not Joule heating in the antenna; a separate loss resistance accounts for that. antenna gain is the directional far-field intensity relative to an isotropic radiator supplied with the same input power:It includes radiation efficiency and equals antenna directivity only for a lossless antenna. antenna effective area is defined by the power available to a matched load, $P_{\mathrm{av}}=A_{\mathrm{eff}}{S}_{\mathrm{inc}}$, for incidence from a specified direction and matched polarization. It need not equal a literal geometric collecting area.

Gain–area relation. To derive the antenna gain-effective-area relation, take a passive reciprocal antenna. Its electromagnetic reciprocity makes the receive pattern proportional to the transmit pattern; write $A_{\mathrm{eff}}(\Omega )=KG(\Omega )$. The normalization follows from thermal equilibrium. In the Rayleigh-Jeans law limit, blackbody radiation radiance per unit frequency per unit solid angle is $2k_{B}T/{\lambda }^2$ for two polarizations. A single antenna polarization couples to one half, so the available incident noise power isThe open-circuit Johnson voltage spectrum of the antenna resistance is $4k_{B}T{R}_{\mathrm{rad}}d\nu$. A matched load receives half the open-circuit voltage, giving available power $k_{B}Td\nu$. In a lossless antenna at thermal equilibrium these powers agree; hence $\int A_{\mathrm{eff}}d\Omega ={\lambda }^2$. Since $\int Gd\Omega =4\pi$,The same efficiency factor multiplies receiving area and gain if reciprocal dissipative losses are included consistently. Impedance mismatch or polarization mismatch reduces delivered power further. The factor of two for thermal polarization modes and the factor of four between open-circuit voltage variance and matched-load power are separate effects.

Normalize the radar beam. Interpret the azimuth as signed about the beam centre at $\varphi =0$, so both sides of the narrow lobe are included. Put $u=\theta -\pi /2$. To leading order in the small angular width, $\sin \theta \simeq 1$ and the angular integrals can be extended over the real line:The Gaussian antenna beam solid angle gives the peak on the positive $x$-axis consequently hasHere $N$ is power per solid angle, equal to $r^2{S}_r$ in the far field. The leading neglected solid-angle correction is relative order ${\alpha }^2$.

Return using the supplied electric-dipole response. The electric-dipole radar cross section of a small conducting sphere determines this return. At target range $D$, the incident intensity is $S_t=C/D^2$. Write its root mean square electric field as $E_t$, with $S_t={ϵ}_{0}c{E}_t^2$. The induced electric dipole is $p_{\mathrm{rms}}=4\pi {ϵ}_0{a}^3{E}_t$. A dipole's angular radiation pattern, normalized by $\int {\sin }^2\chi d\Omega =8\pi /3$, gives $d{P}_{\mathrm{sc}}/d\Omega =(3P_{\mathrm{sc}}/8\pi ){\sin }^2\chi$. The backward direction is perpendicular to the induced dipole, so the supplied total dipole power givesThe received intensity is this angular power divided by $D^2$. Using the monostatic radar equation, or directly multiplying by the receive area, givesThe receiver threshold is a voltage across the matched load, so the detectable power is $V_m^2/R$, without another matching factor:This is the electric-dipole result following from the response data provided. It assumes $ka\ll 1$, far-field propagation on each leg, alignment at the beam maximum and no attenuation or additional receiver losses.

Perfect-conductor qualification. perfect-conductor sphere dipole interference arises because the sphere also excludes the oscillating magnetic field. Solving the magnetic boundary problem gives $\mathbf{m}=-2\pi a^3{\mathbf{H}}_t$ in addition to the electric dipole. The magnetic polarizability of a perfectly conducting sphere follows from its induced surface currents.^[ref] In the backward direction the magnetic contribution has half the electric scattering amplitude and the same sign, so the returned intensity is $9/4$ of the electric-only value. Equivalently, the monostatic radar cross section is ${\sigma }_{\mathrm{back}}=9\pi k^4{a}^6$, rather than $4\pi k^4{a}^6$. Including both leading dipoles therefore givesBoth results have been distinguished because the supplied auxiliary data describe only the electric dipole, while the literal perfect-conductor boundary conditions also generate the magnetic one. The backward interference is a leading-order effect, not a higher-order correction in $ka$.^[ref]

State a time convention before using complex response: take fields proportional to $e^{i(kz-\omega t)}$, with $k=k_r+i\kappa$ and $\kappa \ge 0$ for a passive wave entering the material. In a homogeneous Ohmic medium, Maxwell equations with $\mathbf{J}=\sigma \mathbf{E}$ giveThe ohmic-conductor complex wavenumber describes both phase advance and amplitude decay. Real constant conductivity is only a low-frequency approximation. In the Drude model, $m\dot{\mathbf{v}}+m\gamma \mathbf{v}=-e\mathbf{E}$ giveswith a fixed positive ionic background and no additional bound-electron contribution in this simple model. The plasma frequency marks the inertial electron-response scale. More elaborate band structure, interband transitions and nonlocal transport limit this scalar treatment.^[ref]

In the collisionless nonmagnetic limit, the dispersion relation is ${\omega }^2={\omega }_p^2+c^2{k}^2$. For $\omega >{\omega }_p$,A crest can move faster than $c$ without carrying a new signal faster than light. Below cutoff the ideal wave is evanescent, with penetration length $c/\sqrt{{\omega }_p^2-{\omega }^2}$; it does not propagate as an ordinary real-$k$ bulk wave. Evanescence in the lossless limit is not Joule dissipation. Nonzero collision rate gives a dissipative conductivity and smooths this ideal cutoff picture.

For a good conductor with approximately real $\sigma \gg \omega ϵ$, the spatial branch becomesA field amplitude falls as $e^{-z/\delta }$, while intensity falls as $e^{-2z/\delta }$. The phase velocity is $\omega \delta$ in this limit. Magnetic and electric amplitudes are not related by the vacuum ratio: $Z_s=E/H=(1-i)\sqrt{\mu \omega /(2\sigma )}$ with the chosen time convention. Its positive real part is the surface resistance. The local mean heating is $\frac{1}{2}\mathrm{Re}\sigma |E{|}^2$, consistent with electromagnetic energy flowing into the metal. skin depth is an amplitude length; interpreting it as an intensity length gives a factor-of-two error.

A transmission-line mode propagates chiefly through the dielectric between conductors, even though finite metal conductivity causes loss at its boundaries. With per-length parameters $R^{\prime },L^{\prime },G^{\prime },C^{\prime }$, the telegrapher's equations give, for the same $e^{-i\omega t}$ convention,Writing a forward wave as $e^{-\Gamma z}$, $\Gamma =\alpha -i\beta$, gives ${\Gamma }^2=(R^{\prime }-i\omega L^{\prime })(G^{\prime }-i\omega C^{\prime })$. In the low-loss regime $R^{\prime }\ll \omega L^{\prime }$ and $G^{\prime }\ll \omega C^{\prime }$, set $Z_0\simeq \sqrt{L^{\prime }/C^{\prime }}$ to obtainThis low-loss transmission-line attenuation separates series conductor loss from dielectric leakage loss. Average forward power falls as $e^{-2\alpha z}$. In the normal skin-effect regime, reduced conducting area makes the AC conductor resistance increase roughly as $\sqrt{\omega }$; dielectric loss can supply a separate frequency dependence. Reflection from a mismatched termination changes delivered power without being, by itself, dissipation in the line.^[ref]

To test loss, compare several lengths of otherwise identical line with matched terminations, calibrate connectors and incident/reflected power, and monitor temperature. Separate amplitude attenuation from phase delay and from standing-wave minima. Bulk penetration, line attenuation and dielectric propagation speed are related electromagnetic quantities but describe different experimental geometries.

An electrical transformer transfers energy between electromagnetic coils through electromagnetic induction. A changing shared magnetic flux produces an electromotive force in each coil. In the ideal model, both coils link the same flux $\Phi (t)$; for $N_1$ and $N_2$ turns, Faraday's law of induction gives induced voltages proportional to $N_{1}d\Phi /dt$ and $N_{2}d\Phi /dt$. Their relative signs depend on the winding orientations.^[ref]

For sinusoidal operation, the ideal root mean square voltage magnitudes satisfyThe ideal lossless model also conserves transferred power. With negligible excitation electric current, its electric current magnitudes obey $I_1/I_2=N_2/N_1$. For a load electrical impedance $Z_L$, combining the voltage and electric current ratios gives the input electrical impedanceThus an electrical transformer changes the voltage-current relationship seen by a source as well as the voltage.^[ref]

For an ideal 10:1 turns ratio, a 230 V sinusoidal input produces 23 V at the secondary. A 23 ohm resistive load draws 1 A, corresponding to 0.1 A at the primary: both sides transfer 23 W. These are ideal-model calculations. Winding electrical resistance, magnetic hysteresis, eddy currents and incomplete shared flux introduce losses or departures from those ratios in a real electrical transformer. A constant flux produces no sustained induced secondary voltage.