8.1 Model a solid superconducting sphere in a uniform magnetic field in axial symmetry; (a) plot the attenuation of magnetic        field along the z-axis outside of the sphere and (b) plot the tangential H-field equal to the supercurrent density along the        boundary contour of the sphere.

FIGURE E-8.1 Superconducting sphere in uniform field. Geometry: Axial symmetry.

8.2 Simulate a solid superconducting sphere near a current loop in axial symmetry; (a) compare contour plots of the magnetic        field along the z-axis and also between the current loop and the sphere when the sphere is in the superconducting and the        normal state and (b) calculate the force between the current loop and the sphere for several locations above the sphere.

FIGURE E-8.2 Superconducting sphere near a current loop. Geometry: Axial symmetry.

8.3 Model a current loop inside a hollow superconducting sphere in axial symmetry. Treat the superconducting shell as a single        boundary with (a) calculate the total magnetic field energy inside the sphere for several sphere radii and (b) plot        the induced supercurrent density (tangential H) on the inner surface of the sphere.

FIGURE E-8.3 Current loop inside a hollow superconducting sphere. Geometry: Axial symmetry.

8.4 Magnetic levitation: Model a permanent magnet above a superconducting plane in x-y symmetry; (a) compare the force              between the magnet and the superconducting plane when the coercive force vector is oriented normal and tangential to the        superconducting plane and (b) compare the supercurrent density (tangential H) at the surface of the superconductor for              each case.

FIGURE E-8.4 Permanent magnet above a superconducting plane. Geometry: x-y symmetry.

8.5 Simulate a line current; (a) outside a superconducting and (b) a permeable cylinder in x-y symmetry. Calculate the force               between a line current and the cylinder when the cylinder is superconducting (take μ_r = 10^-6) and compare to the force               between the permeable cylinder (take μ_r = 10⁴) with the same geometry.

FIGURE E-8.5 Line current outside a superconducting or permeable cylinder. Geometry: x-y symmetry.

8.6 Plot the z-component magnetic field along the axis of a superconducting disk of radius a in a uniform field modeled in r-z        coordinates. Compare to the analytical solution

      where z is the height above the disk and B₀is the flux density in absence of the disk.

FIGURE E-8.6 Superconducting disk. Geometry: Axial symmetry.

8.7 Model a superconducting tube in r-z coordinates. Treat the tube as a single boundary with zero vector potential. Plot the z      -component magnetic field along the axis of the superconducting tube and compare to the asymptotic value

      where z is the distance inside the tube of radius a and A is a constant.

FIGURE E-8.7 Superconducting tube. Geometry: Axial symmetry.

8.8 Model a superconducting ring in r-z coordinates with a specified inner and outer radius. Treat the ring as a single boundary.       Compare the z-component magnetic field along a radial contour inside the ring for both field-cooled and zero field-cooled       boundary conditions. Show that the total flux through the zero field-cooled ring is zero.

FIGURE E-8.8 Superconducting ring. Geometry: Axial symmetry.

8.9 Magnetic levitation: Model a permanent magnet above a superconducting plane in r-z symmetry; (a) calculate the force       between the magnet and the superconducting plane with the coercive force vector oriented normal to the superconducting       plane and (b) calculate the supercurrent density (tangential H) at the surface of the superconductor.

FIGURE E-8.9 Permanent magnet above a superconducting plane.

8.10 Model a superconducting can of radius a in an axial magnetic field. Make a contour plot of B_z along the axis of the cup.                 Show that the value of B_z inside the cup measured a distance z from the bottom is given by

       where A is a constant.

FIGURE E-8.10 Superconducting can. Geometry: Axial symmetry.

8.11 Model the dual layer superconducting |l-metal bowl in a uniform magnetic field in axial symmetry. Compare contour plots       of the tangential field along the inside surface of the bowl with the outer μ-metal layer to contour plots without the outer       μ-metal layer.

8.12 Model a Type-II hollow superconducting sphere in a uniform field. Plot the penetration of flux into the sphere for several                 external field values.

8.13 A superconducting oblate spheroid is positioned in a uniformusing separation of variables in oblate spheroidal                        coordinates. Apply the boundary condition of zero normal field on the surface of the oblate spheroid.

8.14 Repeat the calculation in 8.13 for a superconducting prolate spheroid in a uniform field.

8.15 Repeat the calculation in 8.13 for a field cooled superconducting toroid positioned in a uniform z-directed field.

8.16 Model a zero field cooled toroid in a uniform z-directed magnetic field. Solve the vector Laplacian for A_ø in axial symmetry          using separation of variables in toroidal coordinates. Apply the zero field-cooled condition A_ø = 0 on the boundary of the                 toroid. Find the radial position of the nodal points of vanishing B_z in the center of the ring.