# Magnetic Permeability And Susceptibility

## Magnetic Permeability

Permeability is an ability to permeate which means to pass into every part of space. Magnetic permeability is the measure of ‘how permeable’ a material is for passage of magnetic field lines through i.e. a measure of its ability to permit the passage of magnetic field lines through it.

Thus, the degree to which the magnetic field lines can penetrate or permeate a given medium is called its magnetic permeability.

It is experimentally found that the magnetic induction is directly proportional to the magnetizing field vector i.e. $B∝H \text{ (For para and di only)}$ $B=μH$

Where, $μ$ is a constant called magnetic permeability whose value depends upon the nature of the material. Thus, the magnetic permeability is mathematically defined as the ratio of magnetic induction and magnetizing field vector.

Furthermore, if $H=1$, $μ=B$. Thus, magnetic permeability can also be defined as the magnetic induction produced by a magnetizing field of unit strength.

A matter having more value of $μ$ means that more magnetic field lines will pass through the matter.

For para and ferro, $B>H$ so $μ>1$. Thus, more magnetic field lines will pass through parra and ferro-magnetic substances.

For di, $B<H$ so $μ<1$. Thus, less magnetic field lines will pass through diamagnetic substance.

## Magnetic Susceptibility

Susceptibility means sensitiveness. Magnetic susceptibility is a measure of ‘how susceptible’ a material is to get magnetized i.e. how easily and strongly a substance can be magnetized by magnetic field.

Thus, the ease with which a specimen of a magnetic material can be magnetized is called its magnetic susceptibility. It is found that soft ion can be easily magnetized compared to steel. Therefore, susceptibility of soft iron is more as compared to steel.

It is experimentally found that the intensity of magnetization is directly proportional to the magnetizing field vector i.e. $I ∝ H \text{ (For para and di only)}$ $I=χH$ $χ=\frac{I}{H}$

Where, $χ$ is a constant called magnetic susceptibility of a matter whose value depends upon the nature of the substance, the strength of the magnetizing field and the nature of the surrounding medium of substance.

Mathematically, it is defined as the ratio of intensity of magnetization and magnetizing field vector. Since $I$ and $H$ have same dimensions, $χ$ has no dimensions and units.

Furthermore, if $H=1$, $χ=I$. Thus, magnetic susceptibility can also be defined as the intensity of magnetization produced in it by magnetizing field of unit strength.

## Relation between Permeability and Susceptibility

Consider a current $i$ carrying solenoid. Then, the magnetic field due to a current carrying solenoid is given by, $B_0=μ_0ni \text{ ____(1)}$ [Magnetic Field at a Point on the Axis of a Solenoid]

Where, $μ_0$ is the permeability of free space. Here, $B_0$ refers to the field due to the current in the wire only. It is the field that would be present in the absence of ferromagnetic material.

If the solenoid contains a piece of ferromagnetic substance (like iron) inside it, then the field will be greatly increased. This happens because the magnetic domains in the iron become aligned in the external field. So, the additional magnetic field due to iron itself is given by, $B_m=μ_0nI_m\text{ ____(2)}$

Where, $I_m$ is the surface current that can produce surface density of magnetization. The total field $B$ is the sum of the magnetic field due to current and due to iron i.e. $B=B_0+B_m$ $B=μ_0ni+μ_0nI_m$ $B=μ_0(ni+nI_m)\text{ ____(3)}$

The total field inside the solenoid is also given by, $B=μni \text{ ____(4)}$

Where, $μ=μ_0μ_r$ is the permeability of the medium. From $(3)$ and $(4)$, $μni=μ_0(ni+nI_m)$ $μH= μ_0(H+I)$ $μH=μ_0(H+χH) \text{ [I=χH]}$ $\therefore μ=μ_0(1+χ)$

This gives the relation between permeability and susceptibility.

Also, $μ=μ_0μ_r$. Thus, $μ_0μ_r=μ_0(1+χ)$ $\therefore μ_r=1+χ$

This gives the relation between relative permeability and susceptibility.

We have, $\mu_r=\frac{\mu}{\mu_0}$

Thus, the ratio of permeability of medium to the permeability of free space is called relative permeability. It has no unit and dimension.

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