A sample of the paramagnetic salt to which the magnetization...

A sample of the paramagnetic salt to which the magnetization curve of Fig. $32-14$ applies is immersed in a uniform magnetic field of $2.0 \mathrm{~T}$. At what temperature will the degree of magnetic saturation of the sample be (a) $50 \%$ and (b) $90 \% ?$

Key Concepts

Curie’s Law

Curie’s Law states that the magnetization of a paramagnetic material is directly proportional to the applied magnetic field and inversely proportional to temperature. It highlights the role of thermal energy in disrupting the alignment of magnetic moments. Although Curie’s Law applies best in the linear regime at higher temperatures and lower fields, it lays the groundwork for understanding how temperature influences the degree of magnetic saturation.

Boltzmann Distribution

The Boltzmann distribution describes the statistical distribution of particles among different energy states at thermal equilibrium. For magnetic dipoles in an external field, it determines the fraction of dipoles with their magnetic moments aligned versus anti-aligned with the field. This distribution underlies the temperature dependence of magnetization, as the difference in occupation of energy states, affected by the magnetic energy relative to thermal energy, controls the alignment fraction.

Magnetic Saturation

Magnetic saturation occurs when nearly all the magnetic dipoles in a material are aligned with the applied magnetic field. At saturation, increasing the magnetic field further does not lead to a significant increase in magnetization because the maximum alignment possible has been achieved. In the context of paramagnets, achieving a particular percentage of saturation involves balancing magnetic energy against thermal agitation.

Paramagnetism

Paramagnetism is the property of materials that have unpaired electrons, causing them to possess permanent magnetic moments. When exposed to an external magnetic field, these magnetic moments tend to align with the field, which produces a net magnetization. The degree of alignment depends on the competition between the aligning effect of the magnetic field and the randomizing effect of thermal energy.

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