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The original PDF calls ${ϵ}_r$ a relative magnetic permeability while also specifying a nonmagnetic dielectric. These descriptions are inconsistent if read literally. The physically consistent interpretation of the notation is relative permittivity ${ϵ}_r=2$ and relative magnetic permeability ${\mu }_r=1$.

Let $a=3\mathrm{mm}$ and let $b=6\mathrm{mm}$ be the inner radius of the outer conductor. For line charge density $\lambda$, Gauss's law gives $E_r=\lambda /(2\pi ϵr)$, henceThe magnetic field in the dielectric is $H_{\phi }=I/(2\pi r)$. Its energy per unit length givesUsing $b/a=2$,These are the lossless coaxial line parameters, assuming negligible field penetration into the conductors. If the quoted radii instead referred to an outer diameter or if ${\mu }_r=2$ were intended, the data would need clarification; the stated dielectric and nonmagnetic assumptions support the interpretation above.

In a homogeneous Ohmic conductor, $\mathbf{J}=\sigma \mathbf{E}$. Combining this with Maxwell's equations givesFor a good conductor, $\sigma \gg \omega ϵ$, so conduction dominates displacement current. With time dependence $e^{-i\omega t}$ and depth $z>0$, take $E\propto e^{ikz}$. ThenThe field is consequently proportional to $e^{-z/\delta }{e}^{iz/\delta }$: it both attenuates and changes phase with depth. The skin depth is the distance for amplitude to fall by $e^{-1}$; power density falls as $e^{-2z/\delta }$, so its attenuation length is $\delta /2$.

Physically, changing fields induce currents whose magnetic field opposes penetration. Equivalently, after neglecting displacement current, magnetic diffusion obeys ${\partial }_t\mathbf{B}=(\mu \sigma {)}^{-1}{\nabla }^2\mathbf{B}$. Faster changes have less time to diffuse into the metal. Increasing frequency, conductivity or magnetic permeability reduces $\delta$, provided the relevant material parameters remain applicable.

For a thick conductor the tangential surface electrical impedance is $Z_s=(1-i)/(\sigma \delta )$ with the stated time convention and consistent field orientation. Its real part $1/(\sigma \delta )$ represents dissipative resistance per square. Current crowding near the surface raises the AC resistance of a wire when $\delta$ is small compared with its radius; thin strands and laminated magnetic cores reduce suitable eddy-current losses. When $\sigma /\gg \omega ϵ$, the full complex propagation constant must replace the good-conductor approximation. In a magnetic metal the magnetic permeability can itself be nonlinear and frequency dependent.

In the small-body approximation, write ${\chi }_{\mathrm{eff}}=\chi /(1+N_d\chi )$, with $\chi =\mu -1$. The induced magnetic dipole moment is $\mathbf{m}=VM\hat{\mathbf{z}}=V{\chi }_{\mathrm{eff}}{H}_0\hat{\mathbf{z}}$. Its effective interaction potential energy at fixed source current isThe factor $1/2$ accounts for inducing the moment: differentiating $-m(z)B_0(z)$ as if the moment were permanent would double the force. Equivalently, the instantaneous dipole force is $F_z=md{B}_0/dz$. Thus the force on a small linearly magnetizable body isFrom part (b), $d{H}_0/dz=-3a^{2}Iz/[2(a^2+z^2{)}^{5/2}]$. HenceFor the ideal slender rod, ${\chi }_{\mathrm{eff}}\simeq \mu -1$. If the source's printed magnetisation is imposed instead, its corresponding conditional answer isFor $\mu >1$ and $z>0$, both expressions point toward the loop, where the applied magnetic field is stronger; for $z<0$ the direction reverses toward the loop plane. A material with diamagnetism, ${\chi }_{\mathrm{eff}}<0$, reverses this attraction. The force is in newtons, since a magnetic dipole moment times a field gradient has units $\mathrm{A}{\mathrm{m}}^2\mathrm{T}{\mathrm{m}}^{-1}$. Both formulas assume linear response, negligible saturation and approximately uniform applied field across the sample.

For a fixed magnetic dipole moment in an externally imposed uniform magnetic field, the torque is $\mathbf{\tau }=\mathbf{m}\times \mathbf{B}$. Its component in the direction of increasing $\theta$ is $-mB\sin \theta$. Since ${\tau }_{\theta }=-dU/d\theta$, integration giveswhere the arbitrary additive constant has been chosen so that $U=0$ at $\theta =\pi /2$.

The aligned orientation $\theta =0$ has the minimum $U=-mB$ and is stable against small rotations; the anti-aligned orientation $\theta =\pi$ has the maximum $U=+mB$ and is unstable. The graph repeats every $2\pi$. Rotating quasistatically from alignment to anti-alignment requires external work $\Delta U=2mB$.

Because the magnetic field obeys $\nabla \cdot \mathbf{B}=0$, it can be represented locally as the curl of a magnetic vector potential:This automatically enforces the absence of magnetic monopoles. It is also a global representation throughout ordinary all-space problems without monopole singularities. It is not unique: the gauge transformationleaves $\mathbf{B}$ unchanged because the curl of a gradient vanishes. In a time-dependent problem,makes the same transformation leave both physical fields unchanged and automatically satisfies Faraday's law.

In vacuum magnetostatics, choose the Coulomb gauge $\nabla \cdot \mathbf{A}=0$. Substitution into Ampère's law and the identity $\nabla \times (\nabla \times \mathbf{A})=\nabla (\nabla \cdot \mathbf{A})-{\nabla }^2\mathbf{A}$ givesfor a localized steady electric current density and a potential vanishing at infinity. Taking the curl recovers the Biot-Savart law. The magnetic vector potential converts a constrained vector field problem into componentwise Poisson equations once a gauge is chosen.

For a uniform magnetic field, $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{r}$ is a convenient choice. For a localized magnetic dipole moment $\mathbf{m}$, the leading far-field potential isAnother useful relation is Stokes theorem, which relates a closed-path integral to the magnetic flux through a spanning surface:The units of $\mathbf{A}$ are $\mathrm{T}\mathrm{m}$, equivalently $\mathrm{Wb}{\mathrm{m}}^{-1}$.

An electric dipole consists of opposite charges separated by a small displacement, with electric dipole moment $\mathbf{p}=q\mathbf{d}$ directed from negative to positive. At distances large compared with its size,The net charge term vanishes; the field decays faster than a monopole field. In an imposed slowly varying electric field,for a permanent fixed dipole. A uniform field produces a torque but no net force. For a linear induced dipole $\mathbf{p}=\alpha \mathbf{E}$, the interaction energy is instead $-\alpha E^2/2$, because building the dipole also costs energy.

A small planar current loop has magnetic dipole moment $\mathbf{\mu }=IA\hat{\mathbf{n}}$, with the normal fixed by the right-hand rule. Outside the source,Magnetic field lines form closed loops; a current-loop dipole does not require magnetic monopoles. Dipole moments describe the leading directional field of a neutral localized source, and their torques explain alignment in external fields.

In vacuum, Maxwell's equations in SI units areThe displacement current term makes the last equation consistent with the charge continuity equation. In a source-free region, take the curl of Faraday's law and use $\nabla \times (\nabla \times \mathbf{E})=\nabla (\nabla \cdot \mathbf{E})-{\nabla }^2\mathbf{E}$. This givesA plane electromagnetic wave therefore propagates at $c=1/\sqrt{{\mu }_0{ϵ}_0}$. The divergence equations make both fields transverse; the curl equations giveThe electric fields and magnetic fields are in phase and perpendicular to one another and to the propagation direction. Their energy densities are equal:For electric-field peak amplitude $E_a$, the time-averaged intensity is $⟨S⟩={ϵ}_{0}c{E}_a^2/2$. Mutually coupled changing electric fields and magnetic fields transport energy through vacuum, even where no charges or conduction currents are present.

In magnetostatics Ampère's circuital law is $\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\mathrm{enc}}$. Its general vacuum version adds ${\mu }_0{ϵ}_{0}d{\Phi }_E/dt$ on the right. Faraday's law is $\mathcal{E}=\oint \mathbf{E}\cdot d\mathbf{l}=-d{\Phi }_B/dt$ for a fixed loop. A circular Amperian path around a long straight wire giveswith direction given by the right-hand rule. The force on a wire element is $d\mathbf{F}=Id\mathbf{l}\times \mathbf{B}$; its magnitude is $IBdl\sin \chi$ and its direction follows this cross product.

Choose the straight-wire current rightward and the square below it, as printed. Its positive magnetic field there is into the page. With an into-page loop normal, the magnetic flux linkage isHere $\mathcal{M}$ denotes mutual inductance, not a mass. Choose positive coil current to produce an out-of-page field. Then its induced current in the resistance-dominated approximation isThis reproduces the specified sign with the declared orientation. It neglects coil self-inductance, requiring $\omega L_{\mathrm{coil}}\ll R$, as implicit in the requested expression.

The nearest side contains N parallel current segments and is at a uniform distance $s$, so its force magnitude isThe two sides perpendicular to the wire have opposite currents and equal magnetic fields at corresponding distances; their forces cancel pairwise. The far side opposes the near-side force and has the smaller field. ThusWhen $I$ and $i$ have the same sign under the chosen conventions, the near-side coil current opposes the straight-wire current and the total force is away from the wire; otherwise it is toward it. In particular the signed force alternates with $\sin (\omega t)\cos (\omega t)$. The coil is fixed, so these forces do no mechanical work on it. The mean Joule heating is

The Lorentz force determines the trajectory:Thus a magnetic field changes momentum direction but does no work; an electric field can change the particle's kinetic energy. In a uniform electric field with negligible magnetic field and nonrelativistic speed, $\mathbf{a}=q\mathbf{E}/m$ is constant. A transverse initial velocity then gives a parabola.

In a uniform magnetic field, the parallel velocity is unchanged and the perpendicular motion is circular. Nonrelativistically,the combined trajectory is a helix, with circulation reversed when the charge sign reverses. Relativistically $r_L=p_{\perp }/(|q|B)$ and the gyrofrequency is $|q|B/(\gamma m)$.

For perpendicular uniform fields in the magnetic-dominated case, the drift velocity is ${\mathbf{v}}_d=\mathbf{E}\times \mathbf{B}/B^2$, which obeys $\mathbf{E}+{\mathbf{v}}_d\times \mathbf{B}=0$. A particle at this velocity is undeflected: the crossed-field velocity selector selects speed $E/B$ in the appropriate direction, independent of mass and charge. Radiation reaction and collisions are neglected in these ideal trajectories.

In vacuum the electric and magnetic field energy densities areFor example, charging a capacitor requires $U={\int }_0^Q(q/C)dq=Q^2/(2C)=C{V}^2/2$. A parallel-plate capacitor with $C={ϵ}_{0}A/d$ and $E=V/d$ then gives $U={ϵ}_0{E}^{2}Ad/2$, identifying the electric field energy density. Similarly an inductor stores $L{I}^2/2$; applying this to a long solenoid yields $B^2/(2{\mu }_0)$ times its interior volume.

Electromagnetic energy is transported by the Poynting vector $\mathbf{S}=\mathbf{E}\times \mathbf{B}/{\mu }_0$. Poynting's theorem,expresses local energy conservation and transfer to matter. In a vacuum plane wave $E=cB$, so the two energy densities are equal and the energy flux has magnitude $c(u_E+u_B)$. For nondispersive linear media the corresponding reversible expressions are $\mathbf{E}\cdot \mathbf{D}/2$ and $\mathbf{B}\cdot \mathbf{H}/2$; they must not be applied unchanged to arbitrary dispersive or hysteretic materials.

In the Lorentz law, $q$ is signed electric charge, $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic fields at the bead, and the velocity in $q\mathbf{V}\times \mathbf{B}$ is its full laboratory velocity. Here $\mathbf{V}=v\hat{\mathbf{r}}+w\hat{\mathbf{\theta }}$. Since $\hat{r}\times \hat{z}=-\hat{\theta }$ and $\hat{\theta }\times \hat{z}=\hat{r}$,The tangential component integrated around an instantaneous circle of nearly fixed radius givesThis is the stated tangential circulation work expression in the $w\gg v$ limit. It is tangential circulation work. The tangential circulation work versus particle work distinction matters because the bead actually follows an expanding spiral. The radial displacement may be small, but its force is proportional to $w$, so its power is of the same order as the tangential power. In factThe cancellation expresses the fact that magnetic forces do no work. Ignoring radial work while retaining the first-order tangential effect would incorrectly conclude that a magnetic field supplies net particle energy.

The electric field giving the same tangential force would be $\boxed{{\mathbf{E}}_{\mathrm{eq}}=-vB\hat{\theta }}$. This is a force-equivalent field for the moving guide, not evidence that a real circulating electric field has appeared in the laboratory merely because the loop expands in a static magnetic field. An electromotive force is the tangential driving-force circulation per unit charge, with units of energy per charge. For a material circuit moving with local velocity ${\mathbf{v}}_{\mathrm{loop}}$,Orient the ring counterclockwise with positive normal $+\hat{z}$. Its flux is $\Phi =B\pi R^2$, so $\dot{\Phi }=2\pi BRv$. The tangential circulation therefore givesIt is the motional term that matters here. For a fixed loop, the circulation of the actual electric field instead follows the partial time derivative of the magnetic field. The distinction between moving-wire magnetic forces and induction in a stationary loop is essential to the flux rule.^[ref]

For the magnetic power balance on an expanding circular guide, the guide must supply an additional inward reaction $-qBw\hat{r}$ opposing the radial magnetic force. With radial guide speed $v$, this contribution to its power on the bead is $P_{\mathrm{guide}}^{(B)}=-qBvw$, equal to the tangential magnetic power. The radial magnetic power is positive by the same amount, so the total magnetic power still vanishes; the mechanical exchange is associated with the moving mechanical constraint.

If “the ring's work” is meant as its full work on a finite-mass bead, the source statement needs a further qualification. For a frictionless bead of mass $m_b$ with constant imposed radial speed, its radial acceleration is $-w^2/R$, so the complete radial guide reaction and power areThe first term is present even when $B=0$ and describes the familiar change of tangential kinetic energy on an expanding guide. The requested $-qBvw$ is exactly the additional magnetic-field-dependent term, not that full reaction power. No bead mass or extra agency fixing $w$ is supplied, so it would be unjustified to assign a numerical total mechanical power or silently discard its centripetal contribution.

A time-independent conservative force defines a conservative vector field on a domain when the work between two points is independent of path, equivalently when every closed-path work integral vanishes. Define a potential energy $U$ by $\boxed{\mathbf{F}=-\nabla U}$ and $U(B)-U(A)=-{\int }_A^B\mathbf{F}\cdot d\mathbf{r}$. The additive constant is arbitrary; only differences affect work. A vanishing curl is necessary, and is sufficient on a simply connected domain under the usual smoothness assumptions. Local zero curl by itself need not guarantee a single-valued global potential on a domain with holes.

For a test mass outside a point or spherical mass, $U=-GMm/r$ with zero at infinity reproduces the inward gravitational force. Near a surface with approximately uniform downward gravity, $U=mgz$ when $z$ increases upward. A linear spring has $U=k{x}^2/2$. In electrostatics the electric potential is energy per unit charge, so $U=qV$ and $\mathbf{E}=-\nabla V$; potential and potential energy have different units and roles.

For motion under a time-independent conservative force, $d(K+U)/dt=0$. nonconservative forces change the mechanical energy by their work, and an explicitly time-dependent potential can exchange energy with the external agency changing it. A time-varying magnetic field can give a circulating electric field, so not every electric field is electrostatic or conservative. Conversely, zero magnetic work on a charge does not turn a velocity-dependent magnetic force into an ordinary position-potential force.

In SI units, with microscopic charge density ${\rho }_e$ and current density $\mathbf{J}$, Maxwell's equations can be writtenThe first relates electric flux to charge. The second states that the net magnetic flux through a closed surface is zero. Faraday's law relates electric circulation to changing magnetic flux. The Ampère-Maxwell equation includes both conduction current and displacement current; the latter is essential to time-dependent fields and local charge conservation.

For fixed contours and spanning surfaces, the integral curl equations are $\oint \mathbf{E}\cdot d\mathbf{l}=-d{\Phi }_B/dt$ and $\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\mathrm{enc}}+{\mu }_0{\epsilon }_{0}d{\Phi }_E/dt$. For a moving material contour the first must include the motional force term; its total emf is $\oint (\mathbf{E}+{\mathbf{v}}_{\mathrm{loop}}\times \mathbf{B})\cdot d\mathbf{l}$. This prevents confusing a changing circuit area with a partial time derivative of a static magnetic field.

Taking the divergence of the Ampère-Maxwell equation and using electric Gauss's law gives ${\partial }_t{\rho }_e+\nabla \cdot \mathbf{J}=0$. In a source-free vacuum, combining the two curl equations gives ${\nabla }^2\mathbf{E}-{\mu }_0{\epsilon }_0{\partial }_t^2\mathbf{E}=0$, and similarly for $\mathbf{B}$, predicting electromagnetic waves of speed $c=1/\sqrt{{\mu }_0{\epsilon }_0}$. The Lorentz force law then specifies how these fields act on a charge. Macroscopic matter descriptions may instead use $\mathbf{D}$ and $\mathbf{H}$ with free sources and constitutive relations; those should not be silently mixed with the microscopic-source equations above.

For a steady thin-wire electric current, the Biot-Savart law iswhere $\mathbf{R}$ points from the source element to the field point and $d\mathbf{\ell }$ follows positive electric current. On a circular loop's axis the transverse contributions cancel by symmetry. The axial numerator is $a^{2}d\phi$ and the distance is $\sqrt{a^2+z^2}$, givingIts graph is positive and even, with maximum ${\mu }_{0}I/(2a)$ at the centre and a $|z{|}^{-3}$ far-field decay. Reversing the electric current reverses its sign.

For two equally oriented $N$-turn coils centred at $\pm D$, superposition givesConsequentlyThis is an even differentiable function, so every odd derivative at zero vanishes, including the first and third. To check the second derivative explicitly, let $f(s)=(a^2+s^2{)}^{-3/2}$. ThenThus $2D=a$ makes the quadratic variation vanish, which is the Helmholtz coil condition. The central value and leading axial variation are

Near the centre, the coils' magnetic field lines are nearly straight and parallel to the axis. On the axis rotational symmetry makes $B_x=B_y=0$ exactly; in the central plane the radial components from the two coils cancel. More quantitatively, in the current-free central volume, axisymmetry and $\nabla \cdot \mathbf{B}=0$ implyThe suppressed axial derivatives therefore also suppress transverse magnetic fields near the centre. The smallness applies locally compared with the coil radius; it is not a claim of an exactly uniform field throughout the apparatus.

Electromagnetic induction is the production of an electromotive force by changing magnetic flux or by moving a conductor through a magnetic field. With the loop and surface orientations related by the right-hand rule, Faraday's law of induction isHere $\mathbf{v}$ is the local wire velocity. For a stationary loop, the induced electric field can have a nonzero closed-path integral; its differential form is $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$. For a straight rod moving perpendicularly to itself and to a uniform magnetic field, the motional electromotive force has magnitude $B\ell v$.

The minus sign expresses Lenz's law: the induced electric current produces a magnetic field that opposes the change in linked magnetic flux. Pulling a conducting loop through a magnetic field therefore requires work against an electromagnetic resisting force; that work supplies the resulting heat or stored energy.

For an inductor with constant self-inductance, $\mathcal{E}=-LdI/dt$ and its stored magnetic energy is $L{I}^2/2$. A changing electric current in one coil can induce an electromotive force in another through mutual inductance, the operating principle of a electrical transformer. These examples connect the flux law, the direction of the response and conservation of energy.

The Lorentz force determines the motion of a charged particle:At nonrelativistic speeds, $\mathbf{p}=m\mathbf{v}$. The electric field does work at rate $q\mathbf{E}\cdot \mathbf{v}$, while the magnetic part of the Lorentz force is perpendicular to velocity and does no work. A uniform electric field gives constant acceleration $q\mathbf{E}/m$; with a transverse initial velocity, the trajectory is a parabola.

In a uniform magnetic field, the parallel velocity is constant, while the perpendicular component undergoes circular motion. The angular cyclotron frequency, orbit radius and helical pitch areThe sign of $q$ determines the sense of rotation. For perpendicular uniform electric fields and magnetic fields, the guiding centre has the electric drift velocityIndeed, $\mathbf{E}+{\mathbf{v}}_D\times \mathbf{B}=0$, so the remaining motion is cyclotron rotation about that drifting centre, with possible motion along the magnetic field. The drift is independent of electric charge sign and mass. At relativistic energies, retain $\mathbf{p}=\Gamma m\mathbf{v}$ and the corresponding relativistic orbit radius rather than assuming constant nonrelativistic acceleration.

A charge $q$ moving nonrelativistically in prescribed fields obeys the Lorentz force lawThe magnetic part is perpendicular to velocity, so $d(m{v}^2/2)/dt=q\mathbf{E}\cdot \mathbf{v}$: the electric field can change kinetic energy, while the magnetic field alone changes direction without doing work.

In a uniform electric field with no magnetic field, acceleration is constant: $\mathbf{v}={\mathbf{v}}_0+(q\mathbf{E}/m)t$ and $\mathbf{r}={\mathbf{r}}_0+{\mathbf{v}}_{0}t+q\mathbf{E}{t}^2/(2m)$. The trajectory is a parabola when there is initial motion perpendicular to the field, and a line when motion is purely along it. Reversing the charge reverses the electric acceleration.

In a uniform magnetic field with no electric field, resolve velocity into parallel and perpendicular components. The parallel component is constant; the perpendicular component rotates at the cyclotron frequencyThe resulting path is a helix, or a circle if $v_{\parallel }=0$, with pitch $2\pi v_{\parallel }/{\omega }_c$. Charge sign determines its sense of rotation. These relations underlie magnetic bending and mass/charge analysis. At relativistic energies, use momentum $\mathbf{p}=\gamma m\mathbf{v}$; at fixed energy the cyclotron frequency becomes $|q|B/(\gamma m)$.

For uniform crossed fields $\mathbf{E}\perp \mathbf{B}$, the velocity ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/B^2$ satisfies $\mathbf{E}+{\mathbf{v}}_D\times \mathbf{B}=0$. The orbit is a gyromotion about a guiding center drifting withindependent of mass and charge sign. A velocity selector transmits particles with the appropriate direction and speed $E/B$ without deflection. If the electric field has a component along the magnetic field, that component accelerates the parallel motion instead. Nonuniform fields can add other drifts; the simple helix and uniform crossed-field expressions require the stated uniform-field assumptions.

For steady currents in vacuum, the Biot-Savart law gives the magnetic field by summing oriented current elements. With $\mathbf{R}=\mathbf{r}-{\mathbf{r}}^{\prime }$ from the source to the observation point,For a thin wire it becomes $\mathbf{B}=({\mu }_{0}I/4\pi )\int d{\mathbf{l}}^{\prime }\times \hat{\mathbf{R}}/R^2$. The cross product fixes the direction by the right-hand rule. Fields from different steady circuits add by superposition; the entire current path, including return wires where relevant, belongs in the calculation.

An infinite straight wire gives circular field lines of magnitude $\boxed{B={\mu }_{0}I/(2\pi s)}$ at perpendicular distance $s$. This follows by integrating the transverse component of each element along the wire. A circular loop of radius $a$ has cancelling transverse components on its axis, leavingMultiple coincident turns multiply the field by their number. Far from the loop this is the field of a magnetic dipole of moment $\mathbf{\mu }=I\pi a^2\hat{\mathbf{n}}$. Integrating over many loops gives the approximately uniform interior field $B\simeq {\mu }_{0}nI$ of a long solenoid, with $n$ turns per unit length. Finite coils have fringing fields, and appropriately spaced coil pairs can produce a more uniform measurement region.

The law is useful for predicting fields of wires and coils where direct symmetry is insufficient, designing electromagnets and estimating stray fields. It is a magnetostatic law: steady charge flow satisfies charge continuity. An instantaneous application to rapidly changing currents neglects propagation and radiation; such problems require the time-dependent Maxwell's equations and retarded fields. A quasistatic approximation can be valid when circuit dimensions are small compared with the light-travel distance during the variation timescale.

Faraday's law of induction relates an induced electromotive force to the rate of change of magnetic flux. Choose a positive orientation around a loop and its associated surface normal by the right-hand rule. For a stationary conducting loop,Unlike an electrostatic field, the induced electric field generally has nonzero curl; the differential form is $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$. For a moving circuit, magnetic flux transport for a moving circuit includes the magnetic Lorentz force on the charges:where $\mathbf{u}$ is the local wire velocity. Thus changing the field or moving a conductor can produce induction.

The minus sign is Lenz's law: an induced electric current, when a closed conducting path permits it, produces a magnetic field opposing the change of linked flux. It need not oppose the original field itself. For example, increasing flux through a loop produces a current tending to reduce that increase; decreasing the same flux reverses the current. The opposition is consistent with conservation of energy: external mechanical or electrical work is needed to maintain a change against its inductive reaction.

For a coil of $N$ turns, define the magnetic flux linkage $\Lambda ={\sum }_{j=1}^N{\Phi }_j$, reducing to $N{\Phi }_B$ when every turn links the same flux. For fixed geometry and a linear magnetic response, self-inductance $L$ is defined by $\Lambda =LI$, and the coil's induced electromotive force isThe corresponding terminal voltage drop, with the passive sign convention, is $V_L=LdI/dt$. It opposes changes in electric current; it does not require the current itself to vanish.

Building the current from zero requires power $P=I{V}_L=LIdI/dt$. Integrating gives the energy stored by an ideal inductor:This energy is stored in the magnetic field and can be returned when the current falls. In a series RL circuit driven by a constant voltage $V_0$, Kirchhoff's voltage law gives $L\dot{I}+RI=V_0$. Starting with zero current, $I=(V_0/R)(1-e^{-tR/L})$, with time constant $L/R$. This makes the resistance to rapid current changes directly observable. Two coils also have mutual inductance: changing primary current induces secondary electromotive force $-Md{I}_1/dt$ with a polarity determined by the chosen winding orientations, the principle behind a transformer.

Choose a positive unit normal $\hat{\mathbf{n}}$ to the loop and choose positive circulation by the right-hand rule. Write the applied magnetic field as $\mathbf{B}(t)=B_0\cos (\omega t)\hat{\mathbf{b}}$, where $\hat{\mathbf{b}}\cdot \hat{\mathbf{n}}=\cos \theta$. Because the field is uniform, the external magnetic flux isFaraday's law of induction gives the induced electromotive force, and Ohm's law gives the current through the loop's electrical resistance:This is the induced current in a tilted resistive loop with the self-generated magnetic-flux contribution neglected as requested. The factor has units $\mathrm{Wb}{\mathrm{s}}^{-1}{\mathrm{\Omega }}^{-1}=\mathrm{A}$. A field parallel to the loop plane has $\cos \theta =0$ and induces no current in this uniform-field model.

For $\cos \theta >0$, the initial normal flux is positive. At $\omega t=\pi /2$, the external magnetic field is instantaneously zero but its normal component is decreasing, so $d{\Phi }_B/dt<0$ and $I>0$. The current is counterclockwise when viewed from the positive-normal side. Its own magnetic field points along $+\hat{\mathbf{n}}$ through the loop, preserving the initial normal flux in accordance with Lenz's law.

For $\cos \theta <0$, both circulation and the induced normal-field direction reverse. The current follows the rate of change of magnetic flux; the quarter-cycle current is nonzero even though the external field itself vanishes then.