Assume that we have a very tiny spherical or disk-like superconducting material that is subjected to an externally uniform magnetic field. I want to know if the net force that accelerates the superconductor is great or negligible. Is it true if, for instance, the thickness of the disk tends to zero, the net force approaches zero? (the disk is normal to the external magnetic field) Does the size of the superconductor affect the net force? In other words, assume that we repeat the experiment for disk-like superconductors of different sizes (radii) of $1 m$, $0.1 m$, and $0.001 m$ with negligible thicknesses. Do the forces remain the same or vary?
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asked Jan 14 at 11:41
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2$\begingroup$ Somebody correct me here, but there is no net force on a magnetic material (para-, dia-, ferromagnet or superconductor) in a homogenous field. There would be a net torque depending on the symmetry of the magnetic body and its orientation in the field. A force would only occur in a non-uniform field and it would be proportional to the field gradient, if I am not mistaken. So, no, the answer is that there is no difference in your scenario because the force is always zero, but that's a pathological case. It seems to me that you should modify the question using a gradient field. $\endgroup$– FlatterMannCommented Jan 14 at 14:02
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2 Answers
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Superconductivty is shielding the total of a volume by surface currents such that the magnetic field of the surface currents exactly cancels an external or internal field.
On any superconducting body, toplogically being a 2-sphere, surface currents must have zeros, at least two, producing dipole fields, that canot be constant inside.
That's the famous theorem, that is impossible to comb a coconut (hedgehog in other languages)
For effects of superconductivity one needs a current loop, a torus. The torus has a basis of two linear independent current fields, one along its surface as a wire loop, shielding the inner volume, and one through the donought, producing an innner field, with zero external field.
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For pure superconductors (simplification), the magnetic field in the volume is zero $\vec{B}=\vec{0}$, the field outside the S must be tangent to its surface, as we know that for an ordinary conductor in a magnetic field, the force per surface calculated from Maxwell stress tensor : $$\sigma_{ij}=\frac{1}{\mu_{0}}(B_{i}B_{j}-\frac{B^{2}}{2\mu_{0}}\delta_{i}^{j})$$ as $\vec{n}.\vec{B}=0$, near the surface $$\sigma_{ij}=-\frac{B^{2}}{2\mu_{0}}$$ the force per surface is $$\vec{F}_{sur}=-\frac{B^{2}}{2\mu_{0}}\vec{n}$$ i.e. the surface is subjected to compressive pressure.
The thickness must be greater than the penetration distance of the field in the S (about $10^{-5}$ cm}).
