Linear and Nonlinear Media
We have defined the magnetization \(\bfM\) and the auxiliary field \(\bfH\). In practical engineering and physics, the crucial question is: How does a specific material respond when I apply a magnetic field?
Does the material boost the field? Does it fight against it? Or does it do something more complex, like remembering its past state? This distinction divides materials into Linear and Nonlinear media.
1. Magnetic Susceptibility and Permeability
For an enormous class of materials (diamagnets and paramagnets), the magnetization \(\bfM\) is directly proportional to the applied auxiliary field \(\bfH\). These are called Linear Media.
We quantify this relationship with a dimensionless constant \(\chi_m\), called the Magnetic Susceptibility:
This linear relationship allows us to rewrite the total magnetic field \(\bfB\) in terms of \(\bfH\) alone. Since \(\bfB = \mu_0(\bfH + \bfM)\), we get:
Here, \(\mu\) is the Permeability of the material. It encompasses the total magnetic response of the medium.
Electric: \(\bfP = \epsilon_0 \chi_e \bfE\) and \(\bfD = \epsilon \bfE\). In
dielectrics, \(\chi_e\) is always positive (material aligns with field).
Magnetic: \(\bfM = \chi_m \bfH\) and \(\bfB = \mu \bfH\). Here, \(\chi_m\) can be
positive (Paramagnetic) OR negative (Diamagnetic).
This simple proportionality makes solving boundary value problems much easier, as the complex microscopic currents are hidden inside the constant \(\mu\).
2. Temperature Dependence
The magnetic behavior of materials is strongly temperature-dependent. Each material type responds differently as thermal energy increases:
Diamagnets
\(\chi_m < 0\) (small, ~\(10^{-5}\))
Temperature Dependence: Essentially none. Diamagnetism arises from induced orbital currents, which are essentially independent of temperature. The effect persists even at absolute zero.
Paramagnets
\(\chi_m > 0\) (small, ~\(10^{-3}\) to \(10^{-5}\))
Curie's Law: \(\chi_m = C/T\). Susceptibility is inversely proportional to temperature. At high T, thermal agitation randomizes the atomic dipoles, weakening the alignment with B.
Ferromagnets
\(\chi_m \gg 1\) (can be \(10^3\) to \(10^6\))
Curie Temperature (Tc): Above this critical temperature, the spontaneous alignment is destroyed. The ferromagnet becomes paramagnetic. (Iron: 770°C, Nickel: 358°C)
3. Ferromagnetism: Domains & Magnetization
Some materials, most notably Iron, Nickel, and Cobalt, are dramatically different. They are Nonlinear and can become permanent magnets.
Why Isn't Every Piece of Iron a Magnet?
In a ferromagnet, atomic dipoles spontaneously align with one another due to quantum-mechanical "exchange coupling"—even without an external field. If all the dipoles in a piece of iron were aligned, it would produce a strong magnetic field \(\bfB\).
But in practice, the alignment occurs only in small patches called domains. Within each domain, billions of dipoles point in the same direction. However, different domains point in different directions. Their magnetic fields largely cancel, so an ordinary piece of iron appears unmagnetized.
Zoom in to reveal the aligned atomic dipoles responsible for the magnet's field.
How to Magnetize a Material
To turn an ordinary piece of iron into a permanent magnet, you must align the domains. This is done by applying a strong external magnetic field \(\bfH\).
When \(\bfH\) is applied:
- Domains aligned with the field grow at the expense of unfavorably aligned domains. Domain walls shift.
- As \(\bfH\) increases further, domains rotate to align more closely with the field.
- At very high fields, the material reaches saturation—nearly all dipoles point the same way, and the material produces a strong net \(\bfB\).
Crucially, when you turn off the external field (\(\bfH \to 0\)), the domain walls don't fully return to their original positions. Some alignment remains. This residual magnetization is called Remanence.
The material now produces its own \(\bfB\) field even without an external source—it has become a permanent magnet.
Hysteresis: The Memory Effect
This irreversibility is called hysteresis. If you plot \(\bfB\) vs \(\bfH\), you don't get a straight line. Instead, the path going "up" (increasing \(\bfH\)) differs from the path going "down" (decreasing \(\bfH\)). The curve forms a loop.