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Akira Tonomura used electron holography to test the Aharonov-Bohm effect with magnetic flux confined away from the electron paths. A coated magnetic toroid let electron waves pass through its hole and outside it while a superconducting shield excluded the magnetic field from those paths. The measured relative phase still depended on the enclosed flux. The experiment isolates a quantum phase effect that cannot be inferred from the local classical magnetic force along either path.^[ref]

A physicist who developed electron holography and investigated the Aharonov-Bohm effect.^[ref]

Hiroji Yamada coauthored an Aharonov-Bohm effect experiment using electron holography. Electron paths passing through and outside a coated toroid acquired a relative phase shift even though superconductivity confined the magnetic flux away from the electron paths.^[ref]

For a uniformly permeable toroid with inner radius $a$, outer radius $b$ and axial height $h$, an ideal poloidal winding gives $H_{\varphi }=NI/(2\pi r)$. Integrating the magnetic flux density across its section gives $\Phi =\mu NIh\ln (b/a)/(2\pi )$ and hence the displayed self-inductance. It retains radial field variation when $b-a$ is not small. For two fully linked windings, replace $N^2$ by $N_1{N}_2$ to obtain their mutual inductance. Discrete turns, leakage and nonlinear core response limit the ideal formula.

For a slender winding around a toroid with $N$ turns, core area $A$, mean magnetic path length $\ell$ and approximately constant magnetic permeability $\mu$, Ampère's law gives $H\ell =NI$. The linked magnetic flux is $N\Phi =N\mu HA$, so $L=N\Phi /I=\mu N^{2}A/\ell$. This approximation neglects flux leakage and variation across the core and assumes operation below magnetic saturation. With a nonlinear magnetic core, an incremental inductance instead depends on differential permeability at the operating point.

Let a slender toroid have core cross-section $A$, mean length $\ell$ and large relative magnetic permeability $\mu$. Cut a narrow gap of perpendicular width $t$ whose faces make angle $\theta$ with the local core direction. Continuity of normal magnetic flux density gives $B_{g,n}=B\sin \theta$; continuity of tangential magnetic field intensity gives $H_{g,t}=B\cos \theta /(\mu {\mu }_0)$. The gap field is nearly normal to the faces for large $\mu$. Its effective flux area is $A/\sin \theta$, so its leading magnetic reluctance is $t\sin \theta /({\mu }_{0}A)$. Adding core reluctance gives the displayed self-inductance. At $45^{\circ }$, $L\simeq {\mu }_0{N}^{2}A/(\ell /\mu +t/\sqrt{2})$ and $H_g\simeq NI/(t+\sqrt{2}\ell /\mu )$. This retains $\mu t/\ell$ while neglecting fringing and corrections of relative order $t/\ell$ or $1/\mu$.

Junji Endo coauthored an Aharonov-Bohm effect experiment using electron holography. Electron paths passing through and outside a coated toroid acquired a relative phase shift even though superconductivity confined the magnetic flux away from the electron paths.^[ref]

Nobuyuki Osakabe coauthored an Aharonov-Bohm effect experiment using electron holography. Electron paths passing through and outside a coated toroid acquired a relative phase shift even though superconductivity confined the magnetic flux away from the electron paths.^[ref]

For fixed, magnetically linear geometry, self-inductance relates a coil's own magnetic flux linkage to its electric current, ${\Lambda }_i=L_i{I}_i$. mutual inductance relates the linkage in one coil to the current in the other, ${\Lambda }_{21}=M{I}_1$. These coefficients enter the induced electromotive forces through ${\mathcal{E}}_i=-d{\Lambda }_i/dt$; reversing a chosen winding orientation reverses the sign assigned to $M$.

In the ideal confined-field toroid, a circular magnetostatic Ampère law contour encloses $N_{1}I$, soWrite $w=b-a$ for the height of the square cross-section. The inductance of a rectangular-section toroid follows by integrating the magnetic flux density over that cross-section; it cannot be replaced by its value at an arbitrary mean radius:Consequently, for winding normals chosen consistently,The equality means complete shared magnetic flux linkage; with leakage one instead has $|M|<\sqrt{L_1{L}_2}$.

For the crack, use the radial-path approximation for a cracked toroid, with an air length $d$ on each path and core length $2\pi r-d$. Continuity of the normal magnetic flux density at a crack face gives $H_g={\mu }_r{H}_c$. HenceIntegrating the radial paths yieldsHere $q=1.5915\mathrm{cm}$ and $M_d/M_0=0.47079$: about a 53% reduction. The small ratio $d/a$ does not justify expanding in ${\mu }_{r}d/a$, which is about ten. The crack's low magnetic permeability gives appreciable magnetic reluctance. This is an estimate within a confined-flux model; actual redistribution near the slit is outside its accuracy. Narrow-gap circuit models assume small gap width relative to transverse dimensions and approximately directed flux.^[ref]

For the intact ring, ideal poloidal winding currents and the no-leakage boundary condition admit $\mathbf{H}=N_{1}I\hat{\mathbf{\varphi }}/(2\pi r)$ with zero radial and vertical components. It is curl-free inside the current-free core, divergence-free for uniform magnetic permeability, tangent to the core boundary, and has the required circulation; uniqueness of the Neumann problem removes an additional homogeneous scalar-potential field. This explains zero non-azimuthal components in the stipulated intact-ring idealization. A literal slit breaks that rotational symmetry, so zero transverse components everywhere is not exact for the cracked ring: magnetic fringing and flux redistribution occur. In fact, extending the approximate $B(r)\propto 1/(r+q)$ unchanged around a current-free core would give $[\nabla \times \mathbf{H}{]}_z\ne 0$. Thus that radial-path calculation and a claim of exact purely azimuthal field in the cracked geometry cannot both be exact. The numerical answer above retains the intended circuit approximation while making its limitation explicit.

In magnetostatics, Ampère's circuital law in vacuum isIt is especially useful when symmetry makes the tangential field constant on an Amperian loop. It gives $B={\mu }_{0}I/(2\pi r)$ for an infinite straight wire, $B\simeq {\mu }_{0}nI$ inside a long solenoid, and $B\simeq \mu NI/(2\pi r)$ in an ideal toroid. The long-solenoid and toroidal expressions neglect fringing.

For time-dependent fields, displacement current is essential:In matter, the corresponding macroscopic form is $\oint \mathbf{H}\cdot d\mathbf{\ell }=I_{\mathrm{free}}+d\int \mathbf{D}\cdot d\mathbf{A}/dt$. The enclosed electric current alone is sufficient only in the magnetostatic limit.

Inductance and interface conditions. For a linear coil, self-inductance is the ratio $L=\Lambda /I$ of its own magnetic flux linkage $\Lambda =N\Phi$ to current. Its induced electromotive force is $-LdI/dt$ for fixed $L$. Across an interface with unit normal $\mathbf{n}$ from medium 1 to 2,These electromagnetic boundary conditions follow from Gauss's law for magnetism and Ampère's law. Here ${\mathbf{K}}_f$ is free surface current. At the iron–air faces there is no free surface current, so normal magnetic flux density and tangential magnetic field intensity are continuous. In each isotropic linear medium, $\mathbf{B}={\mu }_0{\mu }_r\mathbf{H}$.

Complete core. A slender toroid has approximately uniform $H$ along mean path length $\ell$. Ampère's law gives $H\ell =NI$, so $B=\mu {\mu }_{0}NI/\ell$. The flux per turn is $\Phi =BA$, yielding the inductance of a toroidal winding,This uses constant relative magnetic permeability $\mu$ and neglects leakage, fringing and saturation.

Why the gap lowers inductance. The gap replaces an easy magnetic path in iron with air. Maintaining the same flux now requires a much larger magnetic field intensity in part of the path and therefore more ampere-turns. Equivalently, magnetic reluctance increases, so the flux linkage for a fixed current and hence self-inductance decrease. A narrow gap can matter greatly because its permeability is smaller by the factor $\mu$.

Oblique-gap calculation. In the original diagram $t$ is the perpendicular separation of the cut faces. Let $\mathbf{s}$ point along the core, and let $\mathbf{n},\mathbf{q}$ be normal and tangential to the gap faces. With the face at angle $\theta$ to the core direction, $\mathbf{s}=\sin \theta \mathbf{n}+\cos \theta \mathbf{q}$. Write the core flux density as $B\mathbf{s}$. The boundary conditions givesoThe gap area normal to its nearly normal field is $A/\sin \theta$, not $A$. Its leading reluctance is consequently $t\sin \theta /({\mu }_{0}A)$. Adding the core reluctance gives the inductance of a toroid with an oblique narrow air gap,Thus at $\theta =45^{\circ }$,The approximation retains the possibly order-one ratio $\mu t/\ell$ while neglecting $t/\ell$, fringing and the small tangential gap contribution. More precisely, a local uniform-field path along the core, with missing iron length $t/\sin \theta$ subtracted, gives denominator $\ell /\mu +t\sin \theta (1-1/\mu )$. Its correction is beyond the stipulated large-$\mu$, narrow-gap accuracy; simply neglecting $\mu t$ would not be justified.

Field in the slot. The component expression above givesFor the $45^{\circ }$ gap and $\mu \gg 1$,Its units are $\mathrm{A}{\mathrm{m}}^{-1}$. The field is almost perpendicular to the cut faces, rather than following the iron direction straight across the cut. Its small angle away from the face normal satisfies $\tan \alpha =1/\mu$ at $45^{\circ }$; the sense reverses if the current reverses.

Shinichiro Yano coauthored an Aharonov-Bohm effect experiment using electron holography. Electron paths passing through and outside a coated toroid acquired a relative phase shift even though superconductivity confined the magnetic flux away from the electron paths.^[ref]

Takeshi Kawasaki coauthored an Aharonov-Bohm effect experiment using electron holography. Electron paths passing through and outside a coated toroid acquired a relative phase shift even though superconductivity confined the magnetic flux away from the electron paths.^[ref]

A toroid is a ring-shaped surface of revolution obtained by rotating a closed planar profile about an axis in its plane that stays outside the profile. A circular profile gives a torus; a rectangular profile gives a ring with rectangular cross-section. The enclosed solid supplies the core geometry in a toroidal winding, while the mathematical surface is its boundary. The distinction matters when computing a surface area versus a volume.

Tsuyoshi Matsuda coauthored an Aharonov-Bohm effect experiment using electron holography. Electron paths passing through and outside a coated toroid acquired a relative phase shift even though superconductivity confined the magnetic flux away from the electron paths.^[ref]