advanced35 min studyLesson 278 in Physics

Magnetic permeability and resonance

Distinguish static magnetization, linear dynamic permeability and a passive resonant magnetic response.

Before you begin

On this page

The question for today

What you’ll understand

Separate a differential small-signal permeability from a hysteresis curve.
Analyze a scalar magnetic resonance with consistent damping signs.
Connect negative real permeability to dispersion rather than gain.

Magnetic permeability is often introduced as a constant multiplying magnetic intensity. That description can fail near a resonance or inside a hysteresis loop. A useful dynamic permeability describes a small oscillatory perturbation about a specified material state, frequency band and bias. This lesson analyzes a passive scalar resonance as an illustrative model, then identifies the additional information needed for a real magnetic medium or an engineered resonant loop array.

Specify what the linear response describes

Magnetization is magnetic dipole moment per volume, and magnetic induction includes both the vacuum intensity contribution and the material magnetization. In a weak isotropic linear response, magnetization is proportional to magnetic intensity. The proportionality may depend on frequency and can be complex because magnetic degrees of freedom retain memory and dissipate energy. A quoted static permeability therefore does not automatically predict the response to a rapidly varying field.

A ferromagnet can have multiple stable domain configurations, saturation and hysteresis. Its ratio of total induction to total intensity along a large excursion differs from a differential small-signal permeability around a chosen bias. Strong bias can also select an axis and make the alternating response tensorial, with circularly polarized eigenmodes and gyrotropic coupling. The scalar model below deliberately describes only an isolated effective mode in a regime where those complications can be neglected; it is not a universal equation for ferromagnetic resonance.

\mathbf B=\mu_0(\mathbf H+\mathbf M),\qquad \mathbf M(\omega)=\chi_m(\omega)\mathbf H(\omega),\qquad \mu_r=1+\chi_m

B is in T; H and M in A/m. χm and μr are dimensionless. The scalar relation assumes a local isotropic small-signal response about a specified material state.↔ Scroll wide equations horizontally.

Build a passive magnetic pole

A damped magnetic mode can be parameterized with a Lorentz denominator. The driving magnetic intensity excites a mode whose restoring and inertial behavior sets a natural frequency, while damping removes the mode energy. For an engineered loop, current circulating around its area provides a magnetic dipole and circuit resistance supplies dissipation. Different microscopic mechanisms can have different numerators and high-frequency backgrounds, so the chosen simple pole is a local spectral model with an explicitly declared positive oscillator strength.

Using $e^{-i\omega t}$ puts a negative imaginary damping term in that denominator. The imaginary permeability is then nonnegative at positive frequency for this passive scalar model. Cycle-averaged magnetic loss is proportional to that imaginary part and the squared peak magnetic intensity. This expression should not be double counted with a separate explicit current-loss model for the same resonator. When effective parameter retrieval involves spatial dispersion or electric-magnetic coupling, testing passivity requires the full response rather than independently imposing scalar signs on questionable fitted parameters.

\mu_r(\omega)=1+\frac{F_m}{\omega_m^2-\omega^2-i\gamma_m\omega},\qquad \langle p_m\rangle=\frac{\omega\mu_0}{2}\operatorname{Im}\mu_r\,|H|^2

Fm is positive and in s⁻², while ωm and γm are positive angular rates in s⁻¹. The loss formula gives W/m³ for a local scalar constitutive response represented by μr and peak H.↔ Scroll wide equations horizontally.

Interpret a negative real response

Just above a magnetic resonance, the real resonant contribution is negative. If its magnitude exceeds the background, the real permeability can become negative over a finite interval. The imaginary contribution remains positive, so the material can absorb energy throughout that interval. Negative real permeability is therefore not equivalent to negative resistance, gain or negative total stored energy. Its interpretation must include the associated frequency dependence and the material dynamics that produced it.

A medium with positive real permittivity and negative real permeability is usually evanescent in the ideal lossless local isotropic limit. Obtaining a propagating negative-index branch instead requires additional response conditions, which will be examined later. Real resonance experiments also involve demagnetizing fields, sample geometry, tensor eigenmodes and finite linewidths. The mechanical lab represents only damped free motion of a prepared scalar coordinate; it can illustrate linewidth intuition but does not calculate magnetization precession, hysteresis or an alternating permeability spectrum.

\operatorname{Re}(\mu_r-1)=\frac{F_m(\omega_m^2-\omega^2)}{(\omega_m^2-\omega^2)^2+\gamma_m^2\omega^2},\quad \operatorname{Im}\mu_r=\frac{F_m\gamma_m\omega}{(\omega_m^2-\omega^2)^2+\gamma_m^2\omega^2}

These expressions describe only the stated single-mode scalar model. Positive imaginary permeability is compatible with a negative real part. Magnetic mode frequency and damping may both depend on bias and geometry in a real sample.↔ Scroll wide equations horizontally.

Work it through

A passive negative magnetic response

At one frequency $\omega=2\Omega$ , let $\omega_m=\Omega$ , $\gamma_m=\Omega/2$ and $F_m=10\Omega^2$ . Find the relative permeability of the stated scalar model and decide whether it represents gain.

The real denominator is $\Omega^2-4\Omega^2=-3\Omega^2$ . Its imaginary term is $-i(\Omega/2)(2\Omega)=-i\Omega^2$ , so the resonant contribution is $10/(-3-i)$ .
Rationalizing gives $10(-3+i)/10=-3+i$ . Including the unit background yields $\mu_r=-2+i$ , with negative real and positive imaginary components.
The time-averaged magnetic loss is $\omega\mu_0|H|^2/2$ times the imaginary part, which is one. It is positive for a nonzero field, so the response absorbs energy despite its negative real permeability.

The result

The model gives μr = −2 + i and describes passive absorption, not amplification.

The living notebook

Explore the model

Interactive lab

Interactive lab · oscillator

Damped magnetic-mode analogy

Reset and set BOTH Damping ratio ζ = 0.10 and Natural frequency ω₀ = 2. Repeat a fresh run after Reset with ζ = 0.40 and ω₀ = 2. Predict the shorter ringdown in the second run, then Step or Play. The positive-stiffness oscillator starts with unit displacement and zero velocity; it has no magnetic bias, spin precession, hysteresis or driven resonance controls.

Displacement x(t)Current time

Displacement x(t): 1
Velocity v(t): 0
Regime: underdamped

Damping ratio ζ0.2Natural frequency ω₀1.4

t = 0 s

Unit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA broader dissipative resonance is connected to shorter memory, but magnetic spectra require their own response model.

Pause & reason

Check your understanding

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Read the checkpoint explanation

The positive imaginary component gives positive cycle-averaged magnetic dissipation. The real sign concerns phase response and must be interpreted with dispersion.

Your turn

Try it yourself

At exact model resonance let $F_m=3\Omega^2$ , $\omega_m=\Omega$ and $\gamma_m=0.25\Omega$ . Find μr and its imaginary part if the damping is doubled. Explain why a large static permeability need not equal either result.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

At ω = ωm the real detuning is zero; retain the unit permeability background.

Compare with the solution

Initially the resonant term is $3/(-0.25i)=12i$ , so $\mu_r=1+12i$ . Doubling damping changes it to $1+6i$ . These are finite-frequency small-signal responses. The static value of this same model is $\mu_r(0)=1+F_m/\omega_m^2=4$ , independent of the damping coefficient. A real hysteretic material adds dependence on bias and domain state, so an unspecified static ratio cannot replace the required dynamic constitutive law. All three values answer different measurement questions, even though they can belong to one simple material model.

Keep the idea

Take it with you

Dynamic permeability is a response about a specified state and can be tensorial.
Positive damping gives positive magnetic loss in the declared scalar convention.
A negative real permeability requires dispersive energy accounting and does not imply gain.

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