= Solution
The <method of images> replaces induced charges or currents on a boundary by fictitious sources outside the physical region, chosen to satisfy that boundary's conditions. For example, a point charge $q$ a height $d$ above a grounded conducting plane has an image $-q$ at the reflected point. The sum has zero potential on the plane, and the electrostatic uniqueness theorem identifies it with the physical solution in the upper half-space. Images do not describe a second physical charge below the metal.
For the superconducting plane, use the different <boundary condition> \b[$B_z=0$ at its surface]: <magnetic flux> cannot enter the ideal bulk. Choose the real ring moment $\mathbf m=m\widehat z$, where $m=I\pi R^2$, at $z=h$. Its image has moment $-m\widehat z$ at $z=-h$. This <superconducting plane image dipole> makes the normal fields cancel at $z=0$, while their tangential fields add. The image is a computational continuation; the physical field inside the plane is zero. The original sketch shows the dipole-scale field, appropriate to $h\gg R$; the fine field close to the ring wire is outside that approximation.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2016/paper-1-superconductor-field.png]
{title=Original dipole-scale field above a flux-excluding plane, with the opposite image moment shown schematically}
In the vacuum region the <magnetic scalar potential> is
$$
\phi_m(\rho,z)=\frac m{4\pi}\left[\frac{z-h}{(\rho^2+(z-h)^2)^{3/2}}-\frac{z+h}{(\rho^2+(z+h)^2)^{3/2}}\right],\qquad \mathbf B=-\mu_0\nabla\phi_m.
$$
At the surface $z=0$, it reduces to $-mh/[2\pi(\rho^2+h^2)^{3/2}]$. Its tangential derivative therefore gives
$$
\boxed{\mathbf B(a,0)=-\frac{3\mu_0mha}{2\pi(a^2+h^2)^{5/2}}\widehat{\boldsymbol\rho}=-\frac{3\mu_0IR^2ha}{2(a^2+h^2)^{5/2}}\widehat{\boldsymbol\rho}.}
$$
Both $B_z$ and $B_\phi$ vanish there; the sign is inward radially for the chosen positive moment. The expression is the leading dipole result, not an exact finite-ring formula at arbitrary $R/h$.
For the <force>, calculate the external image field on the axis near the real dipole:
$$
B_{{\rm im},z}(z)=-\frac{\mu_0m}{2\pi(z+h)^3}.
$$
Differentiate this field with the image location held fixed, then evaluate at $z=h$. The <dipole levitation force above a superconducting plane> is
$$
F_z=m\left.\frac{\partial B_{{\rm im},z}}{\partial z}\right|_{z=h}=\frac{3\mu_0m^2}{32\pi h^4}=\boxed{\frac{3\pi\mu_0I^2R^4}{32h^4}},
$$
directed upward. Differentiating a substituted separation $2h$ while moving both dipoles in the <force> derivative would incorrectly double the result.
To levitate ring <mass> $M$, balance this <force> against $Mg$:
$$
\boxed{I_{\rm lev}=\sqrt{\frac{32Mgh^4}{3\pi\mu_0R^4}}.}
$$
A light, thin wire and the smallest practical gap reduce current. At fixed gap and wire cross-section $s$, $M=2\pi R s\rho_w$ gives $I_{\rm lev}\propto h^2\sqrt{s\rho_w}/R^{3/2}$, favouring a larger ring until the dipole-limit constraint on $R/h$ matters. There is no unique least current without geometric and material constraints. As a geometrically consistent order-of-magnitude example, take $R=1\,\mathrm{mm}$, $h=10\,\mathrm{mm}$, wire radius $50\,\mathrm{\mu m}$ and an assumed metallic-scale <density> $\rho_w=6.0\times10^3\,\mathrm{kg\,m^{-3}}$. Then $M=2.96\times10^{-7}\,\mathrm{kg}$ and \b[$I_{\rm lev}\simeq2.8\times10^2\,\mathrm A$]. The large value reflects the weak $h^{-4}$ <force> in the stipulated far-field geometry. Uniformly shrinking this geometrical example reduces current as size to the <power> $3/2$, but increases required <electric current density>; arbitrarily thin wire is therefore not a free optimization.
This balance estimate is not a verified superconducting design. The actual wire must carry that current without quenching, its self-field counts toward the limit, the plane must remain in its assumed flux-excluding state, and leads or supports add load. Transport critical-current measurements specify <temperature>, applied field and the <voltage> criterion rather than a material-independent limiting current.<Standard Reference Materials: Critical current measurements on an NbTi superconducting wire standard reference material by A. F. Clark and others>[NBS SP 260-91, Section 7 and Table 3, field/temperature-dependent critical-current specification]{ref}
Back to article page
