Animated Solution for Physics - Magnetic Effects of Current: A paramagnetic material has $10^{28}{\text{ atoms/m}}^3$. Its magnetic susceptibility at temperature $350\text{ K}$ is $2.8\times 10^{-4}$. Its susceptibility at $300\text{ K}$ is

Paramagnetic materials are fascinating. Inside them, tiny atomic magnetic dipoles act like microscopic compass needles. When placed in an external magnetic field, these needles want to align perfectly with the field.

However, there is a chaotic villain in this story: temperature. Thermal energy causes the atoms to vibrate and jiggle, disrupting this perfect alignment. This tug-of-war between the magnetic field trying to align the dipoles and thermal agitation trying to randomize them is the essence of paramagnetism.

The Mathematical Bridge

The French physicist Pierre Curie discovered a beautifully simple relationship to describe this behavior. He found that the magnetic susceptibility $\chi$, which measures how easily a material gets magnetized, is inversely proportional to its absolute temperature $T$.

This is known as Curie's Law:

This means if you heat the material, its susceptibility drops. If you cool it down, it becomes easier to magnetize.

In our specific problem, we are given a paramagnetic material with a known susceptibility at a specific temperature: $T_1=350\text{ K}$ ${\chi }_1=2.8\times 10^{-4}$

We are asked to find the new susceptibility ${\chi }_2$ when the temperature is lowered to $T_2=300\text{ K}$.

Notice that the problem also mentions the atomic density ($10^{28}{\text{ atoms/m}}^3$). This is a classic JEE distractor! The susceptibility $\chi$ is a macroscopic property that already accounts for the number of atoms per unit volume. We don't need this number to find the new susceptibility.

Since $\chi$ and $T$ are inversely proportional, we can set up a simple ratio:

Let's substitute our known values into this master equation:

Now, it's just a matter of careful algebra. First, let's simplify the temperature ratio:

Now, isolate ${\chi }_2$:

Let's do the arithmetic:

As expected, since we cooled the material from $350\text{ K}$ to $300\text{ K}$, the thermal agitation decreased, and the magnetic susceptibility increased from $2.8\times 10^{-4}$ to $3.267\times 10^{-4}$. The physics perfectly matches the math!