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Ignoring magnetization, electric polarization charge transport gives bound current ${\mathbf{J}}_{\mathrm{b}}={\partial }_t\mathbf{P}$. The assumed weak dielectric contrast therefore gives ${\mathbf{J}}_{\mathrm{b}}\simeq -i\omega {ϵ}_0({ϵ}_r-1){\mathbf{E}}_0{e}^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}$ inside the sphere. This is the first Born approximation: the incident field supplies the electric polarization to leading order in ${ϵ}_r-1$.

For $\mathbf{x}=r\mathbf{n}$, the far-distance expansions are $|\mathbf{x}-{\mathbf{x}}^{\prime }|=r-\mathbf{n}\cdot {\mathbf{x}}^{\prime }+O(a^2/r)$ and $|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^{-1}\simeq r^{-1}$. Substitute the retarded time into the phase in the Lorenz gauge vector potential. The phase becomes $kr-\omega t+(\mathbf{k}-k\mathbf{n})\cdot {\mathbf{x}}^{\prime }$, so, using ${\mu }_0{ϵ}_0=c^{-2}$,The assumptions $r\gg a$ and $k{a}^2/r\ll 2\pi$ control amplitude and phase errors respectively. In the radiation zone the supplied electric and magnetic fields satisfy ${\mathbf{B}}_{\mathrm{scatt}}=c^{-1}\mathbf{n}\times {\mathbf{E}}_{\mathrm{scatt}}$. Complex phasors therefore give the time-averaged radial Poynting flux $|{\mathbf{E}}_{\mathrm{scatt}}{|}^2/(2{\mu }_{0}c)$. The incident flux is $|{\mathbf{E}}_0{|}^2/(2{\mu }_{0}c)$, soOrient spherical coordinates along $\mathbf{q}$ to evaluate the form factor:ConsequentlyAt $q=0$ the bracket is interpreted by its limit $a^3/3$, not a singularity. The transverse factor accounts for polarization of an electromagnetic wave; for complex polarization of an electromagnetic wave its squared norm means the Hermitian norm.

For a time-independent current, the Maxwell equations reduce to magnetostatics:Write $\mathbf{B}=\nabla \times \mathbf{A}$ using a vector potential and choose the Coulomb gauge $\nabla \cdot \mathbf{A}=0$. The vector identity for the curl of a curl gives the Poisson equationThe free-space Green function therefore givesTaking the curl with respect to $\mathbf{r}$, moving it under the integral, and using $\nabla (1/|\mathbf{r}-{\mathbf{r}}^{\prime }|)=-(\mathbf{r}-{\mathbf{r}}^{\prime })/|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3$ gives the Biot-Savart law

The magnetization $M$ is magnetic dipole moment per unit volume. It produces bound volume and surface current densityStarting from the vector potential of magnetic dipoles and integrating by parts giveswhich has precisely the form generated by those bound currents.

Write the cylinder radius as $R$ and use cylindrical radius $\rho$. Ampère's law for the free axial current gives, inside the material,At $\rho =R$ this is tangential to the cylinder. Since $n=\hat{r}$,andFor $\mu >{\mu }_0$, the bound surface current therefore runs opposite to the free wire current.

Using the Levi-Civita symbol and the epsilon-delta identity,ThereforeOne particularly simple vector potential for $u$ isDirect differentiation gives $\nabla \cdot A=0$ and $\nabla \times A=(y^2,z^2,x^2)=u$.

Parametrize the curved cone byThe outward oriented surface element isSubstitution and integration over $0\le z\le h$, $0\le \phi <2\pi$ give

Because $\nabla \cdot u=0$, the divergence theorem predicts that the two outward fluxes sum to zero. On the top disk $S_2$, $d\mathbf{S}=(0,0,1)dxdy$ and $u_z=x^2$, so, with $R=h\tan \alpha$,as predicted.

The outward orientation on $S_2$ induces the stated anticlockwise orientation on $C$, whereas the outward orientation on $S_1$ induces the reverse orientation. Since $u=\nabla \times A$, Stokes theorem predictsOn $C$, put $(x,y,z)=(R\cos \phi ,R\sin \phi ,h)$. Thenverifying both identities.

Yes. For any sufficiently smooth scalar $\chi$, the gauge transformationleaves its curl unchanged because $\nabla \times \nabla \chi =0$. This freedom is much larger than adding a constant vector: for example, $\chi =xy$ adds the nonconstant vector $(y,x,0)$. Two vector potentials for the same field differ by a curl-free field; on a simply connected domain that difference is a gradient, while nontrivial topology may allow additional global freedom.

The integrand field is exactly the curl computed in part (a), so use its polynomial vector potential $\mathbf{G}=(z^3,x^3,y^3)/3$. The upper cap meets the plane $z=0$ in an ellipse. Since the parameters are only assumed nonzero, set $a_0=|a|$ and $b_0=|b|$ to use positive semiaxes. Its correctly oriented boundary isOn that boundary, $G_x=0$ and $dz=0$, so only $G_{y}dy=x^{3}dy/3$ contributes. The Stokes theorem therefore givesThe last integral follows by expanding ${\cos }^{4}t=(3+4\cos 2t+\cos 4t)/8$. ThusThe result is independent of $c$. A sign check uses the divergence theorem: closing the cap with its base disk adds outward normal $-\hat{\mathbf{z}}$ there. Since $\nabla \cdot \mathbf{F}=0$, the cap flux equals ${\int }_{\mathrm{disk}}{x}^{2}dxdy>0$, consistent with the boxed expression and its absolute values.

The modified upper surface has exactly the same ellipse at $z=0$ and the same boundary orientation. The field retains the global polynomial vector potential from part (a). By the Stokes theorem, both surface fluxes equal the same boundary line integral; the cap's height and interior shape do not enter. HenceThis is an instance of the fact that solenoidal flux depends on the oriented boundary, with the domain condition automatically satisfied by the smooth polynomial field on all of ${\mathbb{R}}^3$. When $a,b$ are chosen positive, the formula simplifies to $\pi a^{3}b/4$; the absolute-value form respects the original allowance of arbitrary nonzero signs.

An empty wave is a nonzero component of the wave function in Bohmian mechanics containing no actual particle configuration. Its persistence is a question about ontology and explanation, not an inconsistent prediction of duplicate detected particles. In de Broglie-Bohm theory, the wave function evolves by the Schrödinger equation while the actual configuration $Q$ evolves by the guidance equation. For spinless particles without a magnetic vector potential,The physical state includes both elements, so a component of $\Psi$ need not contain another set of particles.^[ref]

After a position-recording interaction, write the universal wave function as $\Psi (q,y)=a{\psi }_0(q){\Phi }_0(y)+b{\psi }_1(q){\Phi }_1(y)$, with apparatus configurations $y$ supported in disjoint regions. If the actual $Y$ lies in the first region, ${\Phi }_1(Y)$ and its relevant local derivatives vanish in the ideal disjoint-support model. Evaluating the guidance equation at $(Q,Y)$ therefore gives the velocity field of the occupied component alone. For realistic nearly disjoint packets this is an approximation justified by decoherence. The other component is empty in configuration space, even when the system packets overlap in ordinary SPACE. This is the effective wave function in Bohmian mechanics.

The empty component cannot simply be deleted from the fundamental theory. In a coherent interferometer, separated packets can later overlap and both contribute to the actual probability current. Their relative quantum phase can change the outgoing particle's path. An empty arm is therefore capable of future dynamical significance. When an environment records the alternatives in very many degrees of freedom, restoring their configuration-space overlap is overwhelmingly difficult; that explains effective collapse without a literal change to the universal wave function.

There are two serious objections. First, if every branch has the detailed functional organization of a detector and observer, what additional work do the particles do in making only the occupied branch a world with records? A many-worlds critic can question whether the particle ontology is doing explanatory work rather than merely labelling one branch. The Bohmian response is that ordinary objects and records are constituted by actual configurations, not by every formally similar pattern in $\Psi$. Assessing that response requires an account of material constitution; counting amplitudes alone settles neither side.

Second, the universal wave function affects particles without the usual reciprocal influence of particles on it. This can look unlike a material field. A nomological interpretation of the wave function treats its role more like law than additional matter, although time-dependent effective subsystem waves make that position philosophically demanding.^[ref]

Neither objection licenses assigning ordinary particle energy separately to every empty component as though it contained a second classical apparatus. Such an assignment would require further assumptions. The definite mathematical theory reproduces standard equilibrium statistics while retaining empty components. The problem is the cost and intelligibility of that ontology, especially relative to alternatives, not an unexplained failure of its specified guidance equation.

A pure stationary state of a time-independent Hamiltonian operator is $\Psi (\mathbf{r},t)=\psi (\mathbf{r})e^{-iEt/\hslash }$. Its probability density and all expectations of time-independent observables are constant. This does not require zero probability current.

For real scalar potential energy and no vector potential, substitute $i\hslash \dot{\Psi }=-({\hslash }^2/2m){\nabla }^2\Psi +V\Psi$ and its complex conjugate into $\dot{\rho }={\Psi }^{\ast }\dot{\Psi }+{\dot{\Psi }}^{\ast }\Psi$. The potential terms cancel:Thus the probability current is $\mathbf{j}=(\hslash /m)\mathrm{Im}({\Psi }^{\ast }\nabla \Psi )$. In a stationary state $\nabla \cdot \mathbf{j}=0$: flow may circulate without changing the density.

Here only the azimuthal phase contributes an imaginary logarithmic derivative. Since $\nabla \varphi =\hat{\mathbf{\varphi }}/(r\sin \theta )$,There is no radial or polar flow; the azimuthal current is independent of $\varphi$, so its divergence vanishes. The apparent coordinate singularity at $\sin \theta =0$ cancels in the final expression.

The z-component of orbital angular momentum follows from the moment of the mass current:Indeed $A^2(24a_0^5)(4/3)(2\pi )=1$. This agrees with $L_z=-i\hslash {\partial }_{\varphi }$ acting on the wavefunction.

For the Schrödinger equation with a real scalar potential and no vector potential, the probability current isTogether with the probability density $\rho =|\Psi {|}^2$, it satisfies the continuity equation ${\partial }_t\rho +\nabla \cdot \mathbf{j}=0$. Substituting ${\partial }_x\Psi =ik\Psi$ givesA complex wavefunction requires both its real and imaginary parts in the sketch. Choosing the irrelevant constant phase so that $A$ is real, they are $A\cos kx$ and $A\sin kx$, while the probability density is horizontal. It is the spatial phase gradient of a Schrodinger wavefunction ${\partial }_x\arg \Psi =k$ that determines the sign of the probability current. In particular, $e^{ikx}$ and $e^{-ikx}$ have the same real part but opposite probability currents, so the slope of the real part cannot determine the direction.

A nonzero plane wave on the whole real line is not square integrable. The displayed density is understood with box normalization, generalized normalization, or as a local description of a broad wave packet; it does not define a normalized probability distribution on the whole line.

It can describe the motion per unit mass of a charged particle in the azimuthal magnetic field of a long straight current-carrying wire. Choose magnetic vector potentialThen $\mathbf{B}=\nabla \times \mathbf{A}=(mC/(qR))\hat{\mathbf{\varphi }}$ and $(T+q\mathbf{A}\cdot \mathbf{v})/m$ is the given Lagrangian. In terms of a straight-wire current $I$, $B_{\varphi }={\mu }_{0}I/(2\pi R)$, so $C=q{\mu }_{0}I/(2\pi m)$ with signs fixed by charge and current direction.

A static magnetic force does no work, consistent with the conserved kinetic energy. Its radial component is $-q\dot{z}{B}_{\varphi }$ and its axial component is $q\dot{R}{B}_{\varphi }$, reproducing the radial and axial couplings above. The reference length in the logarithm is a gauge convention, not a new physical wire radius in the ideal exterior-field model.

Use a complex pair order parameter $\Psi =|\Psi |e^{i\theta }$ with signed charge $q$, $|q|=2e$. A local isotropic Ginzburg-Landau theory of superconductivity has free energyFor $\alpha <0$, $\beta >0$, the uniform amplitude is $|{\Psi }_0{|}^2=-\alpha /\beta$. Variation gives the static field equation and supercurrentThe vector potential obeys Maxwell's equations, including $\nabla \times \mathbf{B}={\mu }_0({\mathbf{j}}_s+{\mathbf{j}}_n)+{\mu }_0{ϵ}_0{\partial }_t\mathbf{E}$ and $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$, with $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$. The static approximation drops the displacement current. A common dissipative near-transition dynamical extension is the time-dependent Ginzburg-Landau equationIt is a phenomenological relaxation law in its applicable regime, not a universal microscopic Schrödinger equation for every superconductor. Its covariant time derivative and the covariant spatial derivative ensure consistent phase and electromagnetic gauge transformations.

The amplitude and screening lengths areFor $\kappa >1/\sqrt{2}$ the ideal model is type II. An Abrikosov vortex has a $2\pi n$ phase winding, a core with suppressed $|\Psi |$ of size $\xi$, and circulating screening currents over $\lambda$. Single-valuedness implies fluxoid and flux quantization in a superconductor:The sign of $n$ depends on the chosen circulation and charge convention. Far from the core, where the contour current term is negligible, each singly wound vortex carries one flux quantum in magnitude.

The equilibrium lower critical magnetic field marks favorable entry of the first vortex; for $\kappa \gg 1$, ${\mu }_0{H}_{c1}\simeq [{\Phi }_0/(4\pi {\lambda }^2)](\ln \kappa +C)$ with an order-one core correction $C$. The upper critical magnetic field follows by linearizing the order-parameter equation in the field:Between these fields lies the superconducting mixed state, often with an Abrikosov vortex lattice. The ideal equilibrium entry and exit fields need not yield large irreversible loops. Flux pinning and surface barriers impede vortex motion; field sweeps then retain history-dependent flux and produce magnetic hysteresis and dissipative losses.

A superconducting weak link, such as a tunnel junction or narrow constriction, couples the phases of two superconductors. With consistent choices of phase and voltage orientation, the Josephson effect givesThe Gauge-invariant Josephson phase includes the line integral of $\mathbf{A}$ across the link. A zero voltage can sustain a phase-dependent steady current; a constant nonzero voltage makes that contribution oscillate. Microwave irradiation can lock this oscillation to an applied frequency, giving voltage steps at $V_n=nhf/(2e)$, an experimental test of coherent pair transport rather than an ordinary resistive current law.

A DC SQUID places two weak links in a loop. For identical junction critical currents $I_0$ and negligible self-inductance, flux quantization fixes their relative phase difference to $2\pi {\Phi }_{\mathrm{ext}}/{\Phi }_0$. Adding the two sinusoidal currents and maximizing over their common phase givesThe flux-periodic response provides sensitive magnetic measurements. Junction asymmetry or appreciable loop inductance changes the ideal contrast and can introduce additional hysteresis; the phase-coherent interference principle remains. The Jaklevic Josephson interference experiment is a motivating experimental realization of this link between a superconducting loop, weak links and magnetic flux.

Take the stated static vector potential with zero scalar potential. In the laboratory,Let $S^{\prime }$ move along $z$ at the electron's reference velocity $v$. The inverse transformation is $z=\gamma (z^{\prime }+v{t}^{\prime })$. The Lorentz transformation of electromagnetic fields giveswith all other components zero. The sign follows from $\mathbf{v}\times \mathbf{B}=v{B}_x{\mathbf{e}}_y$.

Although an observer fixed in $S^{\prime }$ sees an oscillating field, this is not an exact source-free propagating electromagnetic wave. The phase covector has ${\omega }^{\prime }=-\gamma v{k}_u$, $k_z^{\prime }=\gamma k_u$, so its squared norm isIt is the boost of a static spatial grating, not a null vacuum wave. Equivalently, $E^{\prime }/B^{\prime }=v<c$ and $B^{\prime 2}-E^{\prime 2}/c^2=B^2$ is nonzero away from nodes; a vacuum plane wave would have vanishing electromagnetic field invariants. In the ultrarelativistic limit it can be a useful nearly wave-like approximation along the electron's path, but the exact distinction remains.

A charged electron cannot actually maintain the given constant velocity through this nonzero transverse field without an external constraint: the Lorentz force deflects it. The calculation therefore uses the prescribed straight reference trajectory, a weak-deflection approximation, or an instantaneously comoving inertial frame. A globally fixed electron rest frame is not available for its exact deflected motion.

Work in temporal gauge, with zero scalar potential, and use the charge sign exactly as in the given covariant spatial derivative. A spatially uniform vector potential acts on a Fourier mode asThere are no spatial derivatives of $\mathbf{A}$. Substitution into the field equation and multiplication by $c^2$ therefore givesThe canonical wavevector $\mathbf{k}$ is conserved by spatial translation symmetry; the kinetic momentum is $\hslash \mathbf{k}-e\mathbf{A}(t)$. The absence of a term involving $\dot{\mathbf{A}}$ in this mode equation follows from temporal gauge and the spatial minimal-coupling operator. The electric field still affects evolution through the changing oscillator frequency.

Let $B$ be a continuously differentiable solenoidal vector field on an open star-shaped set containing the origin. Thenis a vector potential for $B$. For $D={\int }_0^{1}tB(tx)dt$, differentiation gives $\nabla \cdot D=0$ and $\nabla \times (D\times x)=2D+(x\cdot \nabla )D={\int }_0^1{\partial }_t(t^{2}B(tx))dt=B(x)$. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.

Suppose two oriented surfaces have the same oriented boundary and $S_1\cup (-S_2)$ bounds a volume inside a domain where the vector field $\mathbf{F}$ is smooth and solenoidal. The divergence theorem gives zero flux through their closed union, proving the displayed equality. If a global vector potential $\mathbf{F}=\nabla \times \mathbf{G}$ exists on the relevant surfaces, the Stokes theorem proves the equality directly from their common boundary. The domain condition matters: a singularity inside the joining volume can obstruct the divergence argument.

Where a nonzero wavefunction can be written as $\psi =\sqrt{\rho }{e}^{i\theta }$, and there is no vector potential, substituting into the Schrodinger probability current gives $\mathbf{j}=\rho \hslash \nabla \theta /m$. The direction of probability flow is determined by the phase gradient rather than the slope of the real part. A plane wave and its complex conjugate have the same real part and density but opposite currents. In a vector potential the gauge-invariant kinetic momentum replaces $\hslash \nabla \theta$.

The radial vector field $B=e_r/r^2$ on punctured three-dimensional Euclidean space has flux $4\pi$ through every centered sphere. It has no vector potential smooth on an entire enclosing sphere: the Stokes theorem on a closed surface would force zero flux. Two overlapping patches suffice. In spherical coordinates take $A_{+}=(1-\cos \theta )e_{\varphi }/(r\sin \theta )$, regular away from the negative $z$ axis, and $A_{-}=-(1+\cos \theta )e_{\varphi }/(r\sin \theta )$, regular away from the positive $z$ axis. Their curls both equal $B$.

The Cartesian forms $A_{+}=(-y,x,0)/[r(r+z)]$ and $A_{-}=(y,-x,0)/[r(r-z)]$ show that the first is smooth at the north pole and the second at the south pole. On the overlap their difference is $A_{+}-A_{-}=2\nabla \varphi$, where the azimuth $\varphi$ is defined locally. Its multivaluedness around the axis explains the circulation difference $4\pi$. Splitting the sphere along a latitude, applying the Stokes theorem with $A_{+}$ on the north and $A_{-}$ on the south, and using their opposite boundary orientations recovers the total flux. A coordinate singularity in the tangential spherical basis does not by itself make a vanishing smooth vector field singular at a pole.