Animated Solution for Physics - Magnetic Effects of Current: A paramagnetic sample shows a net magnetisation of $6\text{ A/m}$, when it is placed in an external magnetic field of $0.4\text{ T}$ at a temperature of $4\text{ K}$. When the sample is placed in an external magnetic field of $0.3\text{ T}$ at a temperature of $24\text{ K}$, then the magnetisation will be

The behavior of paramagnetic materials in an external magnetic field is a fascinating interplay between order and chaos. On one hand, the external magnetic field tries to align the atomic dipoles, creating a net magnetisation. On the other hand, thermal agitation (temperature) acts as a disruptive force, causing the dipoles to jiggle around and lose their alignment.

This tug-of-war is elegantly described by Curie's Law, which states that the magnetic susceptibility ${\chi }_m$ of a paramagnetic material is inversely proportional to its absolute temperature $T$. Mathematically, we write this as:

But what exactly is susceptibility? It is defined as the ratio of the induced magnetisation $I$ to the magnetising field intensity $H$.

Furthermore, the magnetising field $H$ is directly related to the external magnetic field $B_0$ by the permeability of free space ${\mu }_0$:

By substituting these definitions back into Curie's Law, we can uncover a direct relationship between the variables we actually care about: magnetisation, external field, and temperature.

Since ${\mu }_0$ is just a constant, we can absorb it into the proportionality, leaving us with a beautifully simple master equation:

This tells us exactly what we intuitively guessed: magnetisation increases if you crank up the magnetic field, but decreases if you heat up the material!

Because we are dealing with the same paramagnetic sample under two different sets of conditions, we can set up a ratio to eliminate any proportionality constants.

Notice how the temperatures are flipped ($T_2/T_1$) because of the inverse relationship. This is a common place where students make silly mistakes, so always double-check your indices!

Now, we simply plug in the values provided in the problem. For the first state, we have $I_1=6\text{ A/m}$, $(B_0{)}_1=0.4\text{ T}$, and $T_1=4\text{ K}$. For the second state, $(B_0{)}_2=0.3\text{ T}$ and $T_2=24\text{ K}$.

Simplifying the fractions on the right side:

Finally, solving for $I_2$:

And there we have it! The new magnetisation is $0.75\text{ A/m}$. By understanding the physical principles behind the formula, the math becomes a breeze.