= Field-cooled Meissner experiment
{c}
The <Field-cooled Meissner experiment> starts above the <superconducting transition temperature> with a weak, steady applied <magnetic field> already penetrating a solid sample. Cooling while keeping the applied <magnetic field> fixed tests whether the interior <magnetic flux> changes. This is the decisive contrast with cooling in zero <magnetic field> and switching the <magnetic field> on afterwards. The diagram shows the two histories for the same final applied <magnetic field>.
\Image[/superconductivity-field-history.png]
{title=Field cooling distinguishes equilibrium Meissner screening from perfect-conductor flux memory}
{description=Analytic idealizations, not measurements: a tangential <magnetic field> beside a planar half-space. The <perfect conductor> preserves its initial interior <magnetic field>; equilibrium <London equations> instead give exponential screening in both histories. $\lambda_L$ is the <London penetration depth> for <superconductivity> and merely a common plotting length for the <perfect conductor>.}
Meissner and Ochsenfeld reported magnetic expulsion in 1933, showing that the transition involves more than vanishing <electrical resistance>. A modern motivating version appears in MIT's lead-sample experiment: surround the sample with a <solenoid> supplying a steady <magnetic field> and a separate pickup coil connected to a recording instrument. Apply the <magnetic field> while the lead is still normal, then cool through its transition without changing the <solenoid> <electric current>. Expulsion produces a transient pickup-coil <electric potential difference>; warming through the transition produces the opposite sign as <magnetic flux> returns. MIT also checks the signal direction by switching off the applied <magnetic field> while the sample is normal. https://hoffman.physics.harvard.edu/materials/SCintro.php[Hoffman laboratory, magnetic-expulsion discovery]{ref} https://ocw.mit.edu/courses/8-13-14-experimental-physics-i-ii-junior-lab-fall-2016-spring-2017/ec45c3030c7ec2ba0f0537ab84d09c17_MIT8_13-14F16-S17exp39.pdf#page=10[MIT Junior Lab, §5.2.2: Meissner effect in lead]{ref}
<Faraday's law of induction> explains the readout. For a stationary $N$-turn pickup coil, with the same linked <magnetic flux> $\Phi$ per turn and a fixed orientation,
$$
V_{\rm pick}(t)=-N\frac{d\Phi}{dt},\qquad
\int_{t_1}^{t_2}V_{\rm pick}(t)\,dt=-N[\Phi(t_2)-\Phi(t_1)].
$$
Thus a transient measures a change in linked <magnetic flux>, not directly the static <magnetic field> at every point in the sample. For stipulated $N=100$ and $\Delta\Phi=-2\times10^{-8}\,\mathrm{Wb}$ per turn, the signed pulse area is $+2\times10^{-6}\,\mathrm{V\,s}$. The apparatus geometry determines how a change in interior <magnetic field> contributes to that linked <magnetic flux>. Keeping the applied <magnetic field> fixed, repeating on warming, and checking a normal-state control make the expulsion interpretation testable; a stationary pickup coil has no sustained signal after the <magnetic flux> has stopped changing.
Why does the preparation history matter? In the stationary, ideal <perfect conductor> model, the interior <electric field> is zero. The differential form of <Faraday's law of induction> then gives
$$
\frac{\partial\mathbf B}{\partial t}=-\nabla\times\mathbf E=0.
$$
If the interior <magnetic field> was $\mathbf B_0$ when perfect conduction began, it remains $\mathbf B_0$; if it was zero, it stays zero when an external <magnetic field> is subsequently applied. Both a <perfect conductor> prepared in zero <magnetic field> and a sample exhibiting the <Meissner effect> can therefore exclude a newly applied weak <magnetic field>. Only the field-cooled comparison tests expulsion of an already present <magnetic field>. This argument concerns a stationary sample and the ideal zero-<electric field> model, not a moving conductor.
The equilibrium <London equations> supply the additional constraint. In a homogeneous region without penetrating flux tubes, they imply $\nabla^2\mathbf B=\mathbf B/\lambda_L^2$. For a half-space $z\geq0$ with a <magnetic field> parallel to its flat boundary, fixed surface value $B_s$, and a bounded solution deep inside, this gives
$$
\mathbf B(z)=B_s e^{-z/\lambda_L}\,\widehat{\mathbf x}.
$$
At $z=5\lambda_L$ only $e^{-5}\simeq0.00674$ of the surface <magnetic field> remains. The screening <electric currents> near the surface produce this reduction. Unlike the frozen interior <magnetic field> of a <perfect conductor>, this equilibrium profile does not retain an arbitrary initial $\mathbf B_0$. The extra constitutive relation, and not zero <electrical resistance> by itself, selects the screened state. Expulsion occurs during the transition, when transient <electric fields> can be present; the steady-state zero-<electric field> argument does not describe the process of establishing <superconductivity>. The diagram gives equilibrium end states, not transition kinetics. https://doi.org/10.1098/rspa.1935.0048[F. and H. London (1935), magnetic screening beyond the acceleration equation]{ref}
These ideal comparisons concern a solid sample in equilibrium and a sufficiently weak <magnetic field>. Stronger <magnetic fields> can destroy <superconductivity> or permit <magnetic flux> to penetrate parts of a sample, so complete bulk expulsion is not a claim about every superconducting state. https://www.ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf#page=6[Terry P. Orlando, Applied Superconductivity, Lecture 1 (2005), mixed-state magnetic penetration]{ref} A hole in a superconducting loop can retain <magnetic flux> even while the material surrounding it screens its own interior; see <flux conservation in a superconducting loop>. The hole is not superconducting material, so its retained <magnetic flux> does not contradict the <Meissner effect>. https://www.feynmanlectures.caltech.edu/III_21.html#Ch21-S7[Feynman, Leighton and Sands, III.21.7: field cooling a superconducting ring]{ref}
Back to article page
