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At a room temperature of $27\phantom{\rule{0.2em}{0ex}}\text{\xb0C},$ the thermal energy per atom is
which is about 220 times greater than ${U}_{B}.$ Clearly, energy exchanges in thermal collisions can seriously interfere with the alignment of the magnetic dipoles. As a result, only a small fraction of the dipoles is aligned at any instant.
The four sketches of [link] furnish a simple model of this alignment process. In part (a), before the field of the solenoid (not shown) containing the paramagnetic sample is applied, the magnetic dipoles are randomly oriented and there is no net magnetic dipole moment associated with the material. With the introduction of the field, a partial alignment of the dipoles takes place, as depicted in part (b). The component of the net magnetic dipole moment that is perpendicular to the field vanishes. We may then represent the sample by part (c), which shows a collection of magnetic dipoles completely aligned with the field. By treating these dipoles as current loops, we can picture the dipole alignment as equivalent to a current around the surface of the material, as in part (d). This fictitious surface current produces its own magnetic field, which enhances the field of the solenoid.
We can express the total magnetic field $\overrightarrow{B}$ in the material as
where ${\overrightarrow{B}}_{0}$ is the field due to the current ${I}_{0}$ in the solenoid and ${\overrightarrow{B}}_{m}$ is the field due to the surface current ${I}_{m}$ around the sample. Now ${\overrightarrow{B}}_{m}$ is usually proportional to ${\overrightarrow{B}}_{0},$ a fact we express by
where $\chi $ is a dimensionless quantity called the magnetic susceptibility . Values of $\chi $ for some paramagnetic materials are given in [link] . Since the alignment of magnetic dipoles is so weak, $\chi $ is very small for paramagnetic materials. By combining [link] and [link] , we obtain:
For a sample within an infinite solenoid, this becomes
This expression tells us that the insertion of a paramagnetic material into a solenoid increases the field by a factor of $(1+\chi ).$ However, since $\chi $ is so small, the field isn’t enhanced very much.
The quantity
is called the magnetic permeability of a material. In terms of $\mu ,$ [link] can be written as
for the filled solenoid.
Paramagnetic Materials | $\chi $ | Diamagnetic Materials | $\chi $ |
---|---|---|---|
Aluminum | $2.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ | Bismuth | $\mathrm{-1.7}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ |
Calcium | $1.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ | Carbon (diamond) | $\mathrm{-2.2}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ |
Chromium | $3.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}$ | Copper | $\mathrm{-9.7}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}$ |
Magnesium | $1.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ | Lead | $\mathrm{-1.8}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ |
Oxygen gas (1 atm) | $1.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}$ | Mercury | $\mathrm{-2.8}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ |
Oxygen liquid (90 K) | $3.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}$ | Hydrogen gas (1 atm) | $\mathrm{-2.2}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-9}}$ |
Tungsten | $6.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ | Nitrogen gas (1 atm) | $\mathrm{-6.7}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-9}}$ |
Air (1 atm) | $3.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}$ | Water | $\mathrm{-9.1}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}$ |
A magnetic field always induces a magnetic dipole in an atom. This induced dipole points opposite to the applied field, so its magnetic field is also directed opposite to the applied field. In paramagnetic and ferromagnetic materials, the induced magnetic dipole is masked by much stronger permanent magnetic dipoles of the atoms. However, in diamagnetic materials, whose atoms have no permanent magnetic dipole moments, the effect of the induced dipole is observable.
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