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Magnetism and Matter — Physics STD 12 Science — Question

Bihar BoardEnglish MediumSTD 12 SciencePhysicsMagnetism and Matter5 Marks
Question
Define magnetic permeability and magnetic susceptibility. Prove that $\mu_{r} = (1+\chi)$.
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Answer

Magnetic Permeability : It is the ability of a material to permit the passage of magnetic field lines through it. Hence it is the degree to which the magnetic field can penetrate the material.
For example, when a piece of iron is placed in a magnetising field, lines of force become much more concentrated in iron (where they become lines of induction) than in air. Thus iron is said to have greater permeability than air. The permeability of a medium may be defined as ratio of the lines of induction per unit normal area to the number of lines of force of the magnetising field per unit normal area. The former measures the magnetic induction B in the medium and the latter measures the strength of the magnetising field H. Hence permeability µ is defined as the ratio of the magnetising induction (B) in the specimen to the magnetising field (H). Thus $\mu=\frac{B}{H}$.
The ratio of magnetic permeability of the material $\mu$ and magnetic permeability of free space $\left(\mu_0\right)$ is called relative permeability of the substance denoted by $\mu_r$ i.e. $\mu_r=\mu / \mu_0$.
where $\mu_0 \times 4 \pi \times 10^{-7} Wb / Am$
Magnetic Susceptibility : For a large number of magnetic materials the intensity of magnetisation I is proportional to magnetising field strength, H. i.e.
$I \propto H$ or $I=\chi . H$
Where $\chi$ is a proportionality constant and is known as magnetic susceptibility of magnetic material.
Now $I =\chi H \Rightarrow \chi= I / H$
Hence the magnetisation per unit field strength is called the magnetic susceptibility. Since I and H have the same dimensions the quantity $\chi$ is dimensionless i.e. it has no unit.
The magnetic susceptibility of a specimen of a magnetic material measures the ease with which the specimen can be magnetised. Since I is equal to the magnetic moment per unit volume, hence the susceptibility as defined above is also called volume susceptibility of the material.
Derivation of $\mu _r=(1+\chi)$ : When a magnetic material is placed in a magnetising field of magnetic intensity $\overrightarrow{ H }$, the material gets magnetised. In this state the total magnetic induction $\overrightarrow{ B }$ is the sum of the magnetic induction $\overrightarrow{ B }_0$ in vaccum produced by the magnetic intensity $\overrightarrow{ H }$ and magnetic induction $\overrightarrow{ B }_m$ due to magnetisation of the material i.e. intensity of magnetisation.
Hence $\overrightarrow{ B }=\overrightarrow{ B }_0+\overrightarrow{ B }_m$
but $\overrightarrow{ B }=\mu \overrightarrow{ H }, B _0=\mu_0 \overrightarrow{ H }$ and $\overrightarrow{ B }_m=\mu_0 \overrightarrow{ I }$
$\mu H =\mu_0( H + I )$
$\Rightarrow \quad \mu H =\mu_0 H \left(1+\frac{ I }{ H }\right)$
$\Rightarrow \quad \frac{\mu}{\mu_0}=\left(1+\frac{ I }{ H }\right)$. But $\mu / \mu_0=\mu_r$ and $I / H =\chi$
$\therefore \quad \mu_r=(1+\chi)$

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