A volume of \(200 \mathrm{~cm}^{3}\) of a paramagnetic substance is maintained at constant temperature. The magnetic field is increased quasi-statically and isothermally from 0 to \(10^{6} \mathrm{~A} / \mathrm{m}\). Assume the Curie law to hold and the Curie constant per unit volume to be \(1.885 \mathrm{~K} / \mathrm{m}^{3}\) (a) How much work would have to be done if no material were present? (b) How much work is done to change the total magnetization of the material when the temperature is \(300 \mathrm{~K}\) and when it is \(1 \mathrm{~K}\) ? (c) How much work is done to change the total magnetization by the generator supplying the current?

Curie law helps us understand how the magnetization \(M\) of a paramagnetic substance changes with temperature \(T\). This law states that the magnetization of a substance is directly proportional to the applied magnetic field \(H\) and inversely proportional to the temperature. That is: \[ M = \frac{C \cdot H}{T} \] Here, \(C\) is the Curie constant, which depends on the material's properties.
In simpler terms, when you increase the magnetic field or decrease the temperature, the magnetization increases. If you increase the temperature, the magnetization decreases. This relationship helps us predict how a material will react in different conditions.

A paramagnetic substance is a type of material that is weakly attracted to magnetic fields. This happens because it has some unpaired electrons that align with the magnetic field. Unlike ferromagnetic materials (like iron), paramagnetic substances do not retain the magnetization when the external field is removed.
Key features of paramagnetic substances include:

• Weak, temporary attraction to magnetic fields
• Do not retain magnetization after the external field is removed
• Properties heavily influenced by temperature

The Curie law is particularly useful in describing paramagnetic substances and allows us to calculate how magnetized they become when subjected to external magnetic fields and varying temperatures.

To calculate the magnetization \(M\) of a paramagnetic substance, you use Curie’s law as stated before: \[ M = \frac{C \cdot H}{T} \] Let's apply this to a practical example:
Suppose we have a paramagnetic substance with a Curie constant \(C\) of \(1.885 K/m^3\), and we increase the magnetic field \(H\) to \(10^6 A/m\). If the temperature \(T\) is \(300 K\), the magnetization is:
\[ M = \frac{1.885 \cdot 10^6}{300} = 6283.33 A/m \]
Next, if we lower the temperature to \(1 K\), the magnetization becomes much larger:
\[ M = \frac{1.885 \cdot 10^6}{1} = 1885 A/m \]
Magnetization increases significantly as the temperature drops.

Magnetic energy describes the amount of work needed to magnetize a material. In the context of our problem, we must distinguish between the work done in a vacuum and the work done when a paramagnetic material is present.
The magnetic energy per unit volume for the paramagnetic material is given by: \[ U = -\frac{1}{2} \cdot M \cdot H \]
For a temperature of \(300 K\) and magnetization of \(6283.33 A/m\), the energy per unit volume becomes:
\[ U = -\frac{1}{2} \cdot 6283.33 \cdot 10^6 = -3.14 \cdot 10^9 J/m^3 \]
We then multiply by the volume to get the total work done:
\[ W = -3.14 \cdot 10^9 J/m^3 \cdot 200 \times 10^{-6} m^3 = -628 J \]
Similarly, for \(1 K\), we get a much larger energy: \[ W = -1885 \cdot 10^6 A/m \cdot 200 \times 10^{-6} m^3 = -377,000 J \]