Enclosing free currents with zero-permeability media

Consider now that there is a free current density J_f inside the hole of a perfect diamagnetic material, so that ∇ ×H=Jf 6= 0. For the sake of simplicity, let us start by assuming a long (along thez−axis) hollow perfect diamagnetic material with a long straight wire of current I placed inside its hole, as sketched in Fig. 2.5.

Because the line integral of the magnetic field along any closed path enclosing the hole must fulfill H

H·dl = I, the magnetic field in the material volume, H^PD, and the magnetic field in the external region,H^EXT, must be finite and different from zero.

This leads to some general properties of µ → 0 materials. First, since H^PD is finite, the magnetic induction inside the material is zero; B^PD = µµ₀H^PD → 0 for µ → 0.

Second, because H^EXT 6= 0, the magnetic field reaches the external region, indicat-ing that perfect diamagnetic materials cannot shield the field created by a long wire.

Because B^PD →0 and the normal component of the magnetic induction must be con-tinuous, the magnetic field induction reaches the external region tangential to S^EXT (B_n^EXT

S^EXT = B_n^PD

S^EXT →0). Similarly, the magnetic field induction inside the hole reaches the material tangential toS^INT (B_n^INT

S^INT = B_n^PD

S^INT →0). These properties are illustrated in Fig. 2.6.

Figure 2.5: Sketch of the cross section of a long (alongz) PD material of external surface S^EXT enclosing a long straight wire of current I inside a hole of surface S^INT.

20 Shaping magnetic fields with extreme permeabilities

Figure 2.6: Numerical calculations of the magnetic induction field lines and (in colors) the modulus of the magnetic induction in arbitrary units when a long straight wire of current I is surrounded by different long (along z) perfect diamagnetic materials (µ= 10⁻⁵). In (a)-(d) the cross section of the external surface of the material is a star, while in (e)-(h) it is a rectangle. In (a) and (e) the wire is centered inside a centered hole with squared cross section. In (b) and (f) the wire position is the same as in (a) and (e) but it is located inside a cylindrical discentered hole. In (c) and (g), the wire is centered inside the cylindrical hole. In (d) and (f) the wire is centered inside a centered hole with rectangular cross section.

In materials with intermediate positive values of µ, the magnetic field distribution both inside the hole and in the external region depends on the hole surfaceS^INT, on the hole position, on the material external surfaceS^EXT, and on the wire position inside the hole. Here we demonstrate that this does not occur in the case of perfect diamagnetic materials withµ→0. For this discussion, we use the magnetic vector potentialA, which is defined as∇ ×A=Band for the chosen symmetry can be written asA=A(x, y)uz. In the volume of a hollow perfect diamagnetic material, sinceB^PD =∇ ×A^PD →0, the magnetic vector potential must be uniform, A^PD =a^PD, where a^PD is a constant.

Due to the continuity of the vector potential, the moduli of the magnetic vector potential along the internal and the external boundaries of the material, respectively, are

A^INT

S^INT= A^PD

S^INT =a^PD, (2.6)

A^EXT

S^EXT = A^PD

S^EXT =a^PD. (2.7)

2.2 Enclosing magnetic field sources 21 These equations set a Dirichlet boundary condition for A^INT and forA^EXT, respec-tively (see Appendix A2). The constant a^PD may depend on the chosen material con-figuration but this dependence, similar to the arbitrary choice of the reference magnetic vector potential, only contributes toA^INT and A^EXT as an additive constant that does not have an effect on the field distribution in these regions (B = ∇ ×A). Therefore, the conditions in Eqs. (2.6) and (2.7) uniquely determine both B^INT and B^EXT. On the one hand, the magnetic field induction inside the hole only depends on J_f and on the surface S^INT; it does not depend on the material outer surface (Fig. 2.6). On the other hand, the magnetic field induction in the external region only depends on the total current enclosed by the material and on the external surfaceS^EXT. For this reason, the field reaching the exterior of a perfect diamagnetic material does not depend on the wire position inside the hole, nor on the hole position or geometry (Fig. 2.6).

In this section, we have discussed the case of a long straight wire inside a long perfect diamagnetic material, but similar ideas apply to other geometries. We now briefly discuss the case of a circular loop carrying a current I embedded inside the hole of a perfect diamagnetic volume with axial symmetry, as sketched in Fig. 2.7.

As above, the propertyB^PD →0 uniquely determines the magnetic induction distri-bution both inside the material hole and in the external region by requiring the magnetic vector potential both at S^INT and S^EXT to be a constant. Consider first the geometry in Fig. 2.7a. Since the net free current enclosed by the material external surface S^EXT is zero, the material can shield the external region from the field created by the wire, as occurred for irrotational fields. This is shown in Fig. 2.8a. Consider now the toroidal geometry in Fig. 2.7b. In this case, one can think of closed paths in the external region that enclose a total free current different from zero. Thus, in order for the Maxwell equation ∇ ×H=J_f to be fulfilled, the field created by the current loop must exit the toroid. As illustrated in the examples in Figs. 2.8b-d, the external field distribution only depends on the currentI and on the geometry of the surfaceS^EXT.

Figure 2.7: Sketch of the cross-section of two hollow perfect diamagnetic materials with a circular loop carrying a currentI inside the hole. In (a) the axis of revolution crosses the material, while in (b) there is a space between the axis and the material.

22 Shaping magnetic fields with extreme permeabilities

Figure 2.8: Numerical calculations of the modulus of the magnetic induction in arbitrary units when a circular loop of current I is enclosed by different perfect diamagnetic materials with µ = 10⁻⁵. In (a) the wire is covered by a cylindrical material with a circular toroidal hole. In (b), (c) and (d) the wire is covered by a cylindrical shell with a toroidal hole of (b) circular and (c) and (d) squared cross-section. In (a), (b), and (c) the wire is centered inside the toroidal hole, while in (d) it is decentered.

The results presented in this section show that perfect diamagnetic materials can re-shape the magnetic field created by a wire in novel ways. Since the magnetic field exiting the material is always tangential to the material surface (B_n^EXT

S^EXT = B^PD_n

S^EXT →0), one can design the adequate material shape to meet specific needs. For example, a cylindrical shell enclosing a single circular current loop provides a uniform magnetic field distribution similar to that of a solenoid (Figs. 2.8b-d). This is reminiscent of the results obtained for epsilon-near-zero media embedding free currents, which have been used to derive highly directive emitters with tailored phases [52, 58].