Negative permeability in magnetostatics and its experimental demonstration

The control of magnetic fields, essential for our science and technology, is currently achieved by magnetic materials with positive permeability, including ferromagnetic, paramagnetic, and diamagnetic types. Here we introduce materials with negative static permeability as a new paradigm for manipulating magnetic fields. As a first step, we extend the solutions of Maxwell magnetostatic equations to include negative-permeability values. The understanding of these new solutions allow us to devise a negative-permeability material as a suitably tailored set of currents arranged in space, overcoming the fact that passive materials with negative permeability do no exist in magnetostatics. We confirm the theory by experimentally creating a spherical shell that emulates a negative-permeability material in a uniform magnetic field. Our results open new possibilities for creating and manipulating magnetic fields, which can be useful for practical applications.

Figure 1

Normalized magnetization (in red), magnetic field (in green), magnetic induction (in blue), and energy density (in black) as a function of the permeability $\mu$ for (a) a sphere, (b) an infinite cylinder in perpendicular field, (c) an infinite thin strip, and (d) an infinite slab.

Figure 2

Magnetic induction field lines and normalized energy density $E/\left({\mu }_0{H}_0^2\right)$ (in colors) for the magnetic response of cylinders of (a) $\mu ={10}^4$, (b) $\mu ={10}^{-4}$, (c) $\mu =-1/2$, and (d) $\mu =-2$ to a vertically applied magnetic field $H_0$.

Figure 3

Normalized averaged magnetization $M^{*}$ as a function of the permeability $\mu$ for a spherical shell (blue dashed line) and a cylindrical shell (black solid line) for $f=1/2$.

Figure 4

Relations of nondistortion (red lines) and divergent fields (blue lines) between the permeabilities (a) ${\mu }_{\theta }$ and ${\mu }_r$ for a spherical shell and (b) ${\mu }_{\theta }$ and ${\mu }_{\rho }$ for a cylindrical one. $R_2/R_1=2$ for both cases. The regions filled in orange correspond to paramagnetic shells, while the white regions correspond to diamagnetic shells.

Figure 5

Magnetic induction field lines and normalized energy density $E/\left({\mu }_0{H}_0^2\right)$ in color scale for two cylindrical shells with radii ratio $R_2/R_1$ = 2 and magnetic permeabilities (a) ${\mu }_{\rho }=100$ and ${\mu }_{\theta }=0.01$ and (b) ${\mu }_{\rho }=-1/2$ and ${\mu }_{\theta }=-2$.

Figure 6

Magnetic induction field lines and normalized energy density $E/\left({\mu }_0{H}_0^2\right)$ in color scale for two cylindrical shells with radii ratio $R_2/R_1=2$. Both have ${\mu }_{\rho }=1$ and their corresponding ${\mu }_{\theta }$ is obtained from Eq. (10) for (a) $n=1$ and (b) $n=2$.

Figure 7

(a) Finite-element simulation of the $z$ component of B, normalized to $B_0$, when a field $B_0$ is applied in the $z$ direction to a spherical shell with $\mu =-0.5$ and radii ratio $R_2/R_1=2$. (b) Same for the discretized 6+6 current loops. (c) Three-dimensional sketch of the experimental negative-$\mu$ material, consisting of two sets of 6 circular current loops placed on a 3D-printed plastic former. (d) Picture of the actual experimental negative-$\mu$ material.

Figure 8

(a) Picture of the experimental setup with the spherical negative-$\mu$ material in the middle of two Helmholtz coils that create a uniform field in the $z$ direction; the tip of the Hall probe is shown on the left of the sphere. (b) Scheme of the feedback loop circuit.

Figure 9

(a) Experimental measurements (black squares), finite-element calculation for the discretized spherical shell with $\mu =-0.5$ (red line), and analytic results for the ideal material (blue line) for the $z$ component of B along the $x$ axis. (b) Same as (a) for the $z$ component of B along the $z$ axis. The shell region is painted in gray.