Relationship of Relative Permeability, Initial Magnetic Susceptibility and Magnetic Field Strength

Ever wondered what makes some materials, like iron, so great at concentrating magnetic fields, while others, like plastic, just don't care? It all comes down to three key terms: relative permeability (${\mu }_r$), initial magnetic susceptibility (${\chi }_i$), and magnetic field strength ($H$).

Relative Permeability (μr​) and Initial Magnetic Susceptibility (χi​)

Imagine you're at a party. The magnetic field strength ($H$) is like the host trying to get everyone to dance. The host has a certain amount of energy and is trying to influence the atmosphere.

The material you put in this field (like an iron bar) is made up of tiny magnetic "party guests." These guests' reaction to the host's influence is what we measure.

Relative Permeability (${\mu }_r$): This tells you how good the material is at "helping" the host's influence. A material with a high μr​ is like a group of guests who are super enthusiastic about dancing and get everyone else on the floor. It's a measure of how easily a material can become a magnetic field conductor. For a vacuum, ${\mu }_r=1$, because there are no guests to help.

Relative permeability is a dimensionless quantity that measures how much a material can support the formation of a magnetic field within itself. 

It is the ratio of the material's permeability ($\mu$) to the permeability of free space (${\mu }_0$). For a vacuum, ${\mu }_r$ is exactly 1.

${\mu }_r=1+{\chi }_i$

The two quantities are not independent; they are just different ways of expressing the same material property.

Initial Magnetic Susceptibility (${\chi }_i$): This is another way of looking at the same thing. Susceptibility measures how much the guests themselves respond and get into the dancing mood. It's the material's internal reaction to the host's influence. The higher the ${\chi }_i$, the more magnetized the material becomes. For a vacuum, ${\chi }_i=0$, as there's no material to magnetize.

It is a dimensionless quantity that measures how easily a material becomes magnetized in response to an applied magnetic field. It is a measure of the material's magnetization ($M$) in relation to the magnetizing force ($H$). For a vacuum, ${\chi }_i$ is 0.

Magnetic Field Strength (H) and Magnetization (M)

Now, let's look at how different materials respond to the host "magnetic field strength (H)"

For Diamagnetic and Paramagnetic Materials:

Think of these as party guests who are either shy (diamagnetic, χi​ is slightly negative) or a little bit interested (paramagnetic, χi​ is slightly positive).

Their response is directly proportional to the host's influence. If the host doubles their effort (H), the guests' enthusiasm (M) doubles as well. The equation is a simple straight line: 

$M={\chi }_{i}H$

The magnetizing force ($H$), also called magnetic field strength, is a measure of the external magnetic field created by currents. It is a vector field and has units of amperes per meter (A/m). When this external field is applied to a material, the material becomes magnetized, creating its own internal magnetic field. This internal response is measured by the magnetization ($M$), also in A/m.

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