SaM - Magnetic Permeability • Magnetic Susceptibility • Magnetic Field

Remeber:

In a vacum:$\vec{B}={\mu }_0\vec{H}$In a material:$\vec{B}=\mu \vec{H}$Or:$\vec{B}={\mu }_0(\vec{H}+\vec{M})$Where:

• $\mu$ is the magnetic permeability, defined as $\mu ={\mu }_r\cdot {\mu }_0$.
  • ${\mu }_r=1+{\mathcal{X}}_M$, where ${\mathcal{X}}_M$ is the magnetic susceptibility, (“suscettività magnetica”) defined similar to the electric susceptibility.
  • $\mu$ is measured in $\left[\frac{\text{T}\cdot \text{m}}{\text{A}}\right]$
  • ${\mu }_0=4\pi \times 10^{-7}\frac{\text{T}\cdot \text{m}}{\text{A}}$
  • For non-magnetic materials such as: air, plastic, glass, wood, copper: ${\mu }_r\approx 1$.
  • For magnetic materials such as: iron, cobalt, nickel: ${\mu }_r\in [10^2÷10^5]$.
    ~ For example pure iron at $99.8\%$ has a ${\mu }_r=2\cdot 10^3$, while if it is pure at $99.95\%$ it has a ${\mu }_r=2\cdot 10^5$.IMPORTANTE
• $\vec{B}$ : magnetic field density $\left(\left[\text{T}\right]:\text{Tesla}\right)$
  Represents the total magneti field.
• $\vec{H}$ : magnetic field intensity $\left(\left[\frac{\text{A}}{\text{m}}\right]\right)$
  Represents the magnetic field produced solely by the presence of current (free current) and not influenced by the magnetic properties of materials.
• $M$ : magnetization $\left(M={\mathcal{X}}_M\cdot H\right)$

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Memory Card

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• $\mu$ is the magnetic permeability

• ${\mathcal{X}}_M$ is the magnetic susceptibility.
  ⇒ Derived from $\vec{M}$ the magnetization of the material.

• Note the similarities with the “Electrical Domain”:

  • $\mathcal{X}$ was the eletric susceptibility.
  • $\epsilon$ was the eletric permittivity.
  • $\vec{E}$: electric field.
  • $\vec{D}$: eletric displacement field.

• What do $\vec{B}$ and $\vec{H}$ represent for magnetism, can you help me understand and visualize them? (ChatGPT 3.5)

  • Sure! In magnetism, $\vec{B}$ and $\vec{H}$ are both vector fields that describe different aspects of the magnetic field.
    1. $\vec{B}$ (magnetic flux density or magnetic induction): This represents the total magnetic field, including the effects of both the magnetic material (if present) and free space. It measures the force experienced by a moving charged particle due to its velocity in a magnetic field. $\vec{B}$ is directly related to the strength and direction of the magnetic force exerted on a moving charge or a current-carrying wire.
    2. $\vec{H}$ (magnetic field intensity or magnetic field strength): This represents the magnetic field produced solely by the presence of current (free current) and not influenced by the magnetic properties of materials. In other words, $\vec{H}$ characterizes the magnetic field in the absence of magnetic materials. It’s essentially the magnetic field that would exist in space if there were no magnetic materials around.
  • To visualize them, imagine a scenario where you have a bar magnet placed on a table:
    • $\vec{B}$ would represent the total magnetic field produced by the magnet, including both the magnetic effects of the magnet itself and the surrounding space.
    • $\vec{H}$ would represent the magnetic field produced solely by the currents (if any) within the magnet, without considering any magnetic materials around it.
  • In summary, $\vec{B}$ accounts for all magnetic effects, while $\vec{H}$ isolates the contribution of free currents to the magnetic field.

• What do $\vec{B}$ and $\vec{H}$ represent for magnetism, can you help me understand and visualize them? (ChatGPT 4o)

  • Certainly! In magnetism, the vectors $\vec{B}$ and $\vec{H}$ represent different, yet closely related concepts within a magnetic field. Let’s break them down:
  • Magnetic Flux Density ($\vec{B}$):
    • Symbol: $\vec{B}$
    • Also Known As: Magnetic field, Magnetic induction
    • Unit: Tesla (T) or Weber per square meter (Wb/m²)
    • Definition and Properties:
      1. Physical Meaning: $\vec{B}$ describes the density and direction of the magnetic flux. It represents how much magnetic field is passing through a given area.
      2. Equation in Vacuum: In free space, $\vec{B}$ is related to the magnetic field strength $\vec{H}$ by the permeability of free space (${\mu }_0$):$\vec{B}={\mu }_0\vec{H}$
      3. Materials Influence: In materials, the relationship is influenced by the material’s permeability ($\mu$):$\vec{B}=\mu \vec{H}$ where $\mu ={\mu }_0{\mu }_r$, and ${\mu }_r$ is the relative permeability of the material.
      4. Visualization: The vector $\vec{B}$ can be visualized as the number of magnetic field lines passing through a given area. The denser the lines, the stronger the magnetic field at that point.
  • Magnetic Field Strength ($\vec{H}$):
    • Symbol: $\vec{H}$
    • Unit: Ampere per meter (A/m)
    • Definition and Properties:
      1. Physical Meaning: $\vec{H}$ represents the magnetizing force. It describes the amount of magnetizing effect produced by an electric current or a magnetic material.
      2. Equation in Vacuum: In free space, $\vec{H}$ is directly proportional to the current that produces it, following Ampère’s Law:$\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}$Where $\vec{J}$ is the current density and $\vec{D}$ is the electric displacement field.
      3. Materials Influence: In materials, $\vec{H}$ helps to determine the magnetization $\vec{M}$:$\vec{B}={\mu }_0(\vec{H}+\vec{M})$
      4. Visualization: The vector $\vec{H}$ can be visualized as the force that creates the magnetic field. It is often represented by lines around a current-carrying conductor, showing how the field is generated by the current.
  • Relationship and Visualization:
    • Interaction: Both $\vec{B}$ and $\vec{H}$ are essential to describe the magnetic properties of materials and fields. While $\vec{B}$ focuses on the magnetic flux through an area, $\vec{H}$ focuses on the magnetizing force creating the field.
    • Visualizing Together: In a diagram, $\vec{B}$ can be represented by dense lines indicating flux through an area, while $\vec{H}$ can be shown as arrows indicating the direction and strength of the magnetizing force. In a simple solenoid, for instance:
    • $\vec{H}$ is represented by arrows along the axis of the solenoid, showing the magnetizing force produced by the current.
    • $\vec{B}$ is represented by loops of flux lines around the solenoid, indicating the density and direction of the magnetic field.
  • Understanding $\vec{B}$ and $\vec{H}$ helps in analyzing and designing magnetic systems, from electromagnets to transformers, and is fundamental in fields like electrical engineering and physics.