A localized magnetic field H in an electrically conducting...

A localized magnetic field $H$ in an electrically conducting medium of permeability $\mu$ and conductivity $\sigma$ will diffuse through the medium in the $x$ -direction at a rate given by $$ \frac{\partial H}{\partial t}=\frac{1}{\mu \sigma} \frac{\partial^{2} H}{\partial x^{2}} $$ Show that the time of decay of the field is given approximately by $L^{2} \mu \sigma$, where $L$ is the extent of the medium, and show that for a copper sphere of radius $1 \mathrm{~m}$ this time is less than $100 \mathrm{~s}$. $$ \begin{aligned} &\mu(\text { copper })=1 \cdot 26 \times 10^{-6} \mathrm{Hm}^{-1} \\ &\sigma(\text { copper })=5 \cdot 8 \times 10^{7} \mathrm{~S} \mathrm{~m}^{-1} \end{aligned} $$ (If the earth's core were molten iron its field would freely decay in approximately $15 \times 10^{3}$ years. In the sun the local field would take $10^{10}$ years to decay. When $\sigma$ is very high the local field will change only by being carried away by the movement of the medium-such a field is said to be 'frozen' into the medium - the field lines are stretched and exert a restoring force against the motion.)

A localized magnetic field $H$ in an electrically conducting medium of permeability $\mu$ and conductivity $\sigma$ will diffuse through the medium in the $x$ -direction at a rate given by
$$
\frac{\partial H}{\partial t}=\frac{1}{\mu \sigma} \frac{\partial^{2} H}{\partial x^{2}}
$$
Show that the time of decay of the field is given approximately by $L^{2} \mu \sigma$, where $L$ is the extent of the medium, and show that for a copper sphere of radius $1 \mathrm{~m}$ this time is less than $100 \mathrm{~s}$.
$$
\begin{aligned}
&\mu(\text { copper })=1 \cdot 26 \times 10^{-6} \mathrm{Hm}^{-1} \\
&\sigma(\text { copper })=5 \cdot 8 \times 10^{7} \mathrm{~S} \mathrm{~m}^{-1}
\end{aligned}
$$
(If the earth's core were molten iron its field would freely decay in approximately $15 \times 10^{3}$ years. In the sun the local field would take $10^{10}$ years to decay. When $\sigma$ is very high the local field will change only by being carried away by the movement of the medium-such a field is said to be 'frozen' into the medium - the field lines are stretched and exert a restoring force against the motion.)

Key Concepts

Frozen-In Field Concept

The frozen-in field concept arises in the context of highly conductive media. When ? is very large, the magnetic field changes primarily due to the movement of the medium rather than diffusion. In such cases, the magnetic field lines are effectively 'frozen' into the conductive fluid, meaning they move along with it. This concept is crucial in the study of magnetohydrodynamics and helps explain the magnetic behavior in astrophysical and geophysical environments.

Electrical Conductivity and Magnetic Permeability

Electrical conductivity (?) and magnetic permeability (?) are key intrinsic properties of a material that determine its response to electromagnetic fields. The conductivity influences how easily induced currents, which affect the magnetic field, can flow in the medium. Permeability governs the medium’s ability to support the formation and diffusion of magnetic fields. Together, they determine the magnetic diffusivity, which in turn controls the diffusion and decay rate of the magnetic field.

Magnetic Diffusion

Magnetic diffusion describes the process by which magnetic fields spread out and weaken over time in a conducting medium. This process is governed by a diffusion equation analogous to the heat equation, where the rate of change of the magnetic field is proportional to its curvature (second spatial derivative), modulated by the magnetic diffusivity.

Diffusion Time Scale

The diffusion time scale provides an estimate for how long it takes a magnetic field to decay in a medium. By considering the balance between the time derivative and the spatial derivative in the magnetic diffusion equation, one finds that the decay time is on the order of L² multiplied by the product of magnetic permeability (?) and electrical conductivity (?). This scaling captures how the characteristic length of the medium influences the rate at which the magnetic field dissipates.

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