Question

The susceptibility at 300 K is \[1.2 \times {10^{ - 5}}.\] what will be its susceptibility at 200 K?

Answer 1

Hint: In order to solve this question, we need to first rewrite the given information and then use the proper formula to find the susceptibility at 200 K. Here, susceptibility of a magnetic material is defined as the ratio of the magnetization vector (M), and the applied magnetic field. The magnetic susceptibility gives the measure of how much the material gets magnetized upon application of applied magnetic field.

Formula Used:
The magnetic susceptibility of a material at temperature T is given as:
\[{X_m} = \dfrac{C}{T}\]

Complete step by step solution:
Curie law states that the magnetic susceptibility is inversely proportional to the temperature of the material. This can be expressed mathematically as:
\[{X_m} = \dfrac{C}{T}\]
C= Curie temperature
T= Temperature in Helium
\[{\text{So, }}X{\text{ }}\propto {\text{ }}\dfrac{1}{T}\]
It’s given that,
\[{X_1} = 1.2 \times {10^{ - 5}}\]
\[{T_1} = 300\]
\[{T_2} = 200\]
\[{X_2} = ?\]
We know that X is inversely proportional to T.
So, we can write it as
\[\dfrac{{{X_1}}}{{{X_2}}} = \dfrac{{{T_2}}}{{{T_1}}}\]
Substituting the given values
\[ \Rightarrow \dfrac{{1.2 \times {{10}^{ - 5}}}}{{{X_2}}} = \dfrac{{200}}{{300}}\]
Further solving we get,
\[ \Rightarrow {X_2} = 1.2 \times {10^{ - 5}} \times \dfrac{{300}}{{200}}\]
\[ \Rightarrow {X_2} = 1.8 \times {10^{ - 5}}\]
Hence, the susceptibility at 200 K will be \[1.8 \times {10^{ - 5}}\]

Note:
It should be remembered that one cannot substitute the temperature in degrees Celsius here. The magnetic susceptibility is inversely proportional to the absolute temperature of the material in Kelvins. Generally, at a temperature where the magnetic property of material changes from ferromagnetic to paramagnetic. Knowing this, one can know the trend as to whether the susceptible increases or decreases with temperature.