Magnetism and Matter

We use Curie's law to understand how the magnetism of a paramagnetic material changes with temperature and magnetic field.
According to Curie's law, the magnetisation (M) is directly proportional to the magnetic field (B0) and inversely proportional to the temperature (T) :
M∝‌

B0

T

or, M=C⋅‌

B0

T

where C (Curie's constant) stays the same for the same substance.
Curie's constant can be written as: C=‌

MT

B0

where:
T= temperature in Kelvin
B0= external magnetic field
M= magnetisation (total dipole moment)
Since C (Curie's constant) is the same for all measurements of the same sample, we can relate two cases as:
‌

M1T1

B01

=‌

M2T2

B02

So, M2=M1⋅‌

T1

T2

⋅‌

B02

B01

Now, let's fill in the given values:
Number of dipoles =2×1024
Dipole moment per dipole =15×10−23JT−1
Step 1: Find Initial Total Dipole Moment
Total dipole moment possible (if 100% aligned): 2×1024×15×10−23=3×102JT−1 But, at first, only 20% saturation is achieved: M1=0.2×3×102=60JT−1
Step 2: Use Curie's Law for New Conditions
Magnetic field changes from B01=0.6T to B02=0.9T and temperature from T1=4.2K to T2=2.8K.
Now, use the formula: M2=M1×‌

T1

T2

×‌

B02

B01

Substitute the numbers in: M2=60×‌

4.2

2.8

×‌

0.9

0.6

Calculate step by step:
‌‌

4.2

2.8

=1.5
‌‌

0.9

0.6

=1.5
So, M2=60×1.5×1.5=135JT−1