Q.A paramagnetic sample shows a net magnetisation of $8\ \text{Am}^{-1}$ when placed in an external magnetic field of $0.6\ \text{T}$ at a temperature of $4\ \text{K}$. When the same sample is placed in an external magnetic field of $0.2\ \text{T}$ at a temperature of $16\ \text{K}$, the magnetisation will be

Concept: Magnetic Poles — The magnetisation of a paramagnetic material follows Curie’s law: $M \propto B/T$ (for fixed sample and moderate fields).

Reasoning:

1. Curie’s law states $M = C \frac{B}{T}$, where $C$ is the Curie constant for the sample.

2. From the first condition:

   $8 = C \cdot \frac{0.6}{4} \quad\Rightarrow\quad C = \frac{8 \times 4}{0.6} = \frac{32}{0.6} = \frac{160}{3}\ \text{A m}^{-1}\text{T}^{-1}\text{K}$.

3. For the second condition: …

The key to this problem is recognising that paramagnetic materials obey Curie’s law under the conditions given. Curie’s law states that the magnetisation $M$ of a paramagnetic sample is directly proportional to the applied magnetic field $B$ and inversely proportional to the absolute temperature $T$:

Why does this make sense physically? In a paramagnet, each atom has a tiny magnetic moment. Without an external field, thermal agitation keeps these moments randomly oriented — no net magnetisation. When you apply a field, it tries to align the moments, but temperature works against it, jumbling them up. So a stronger field gives more alignment (higher $M$), while a higher temperature gives less alignment (lower $M$). The ratio $B/T$ captures this competition neatly.

Now, the problem gives us two different situations for the same sample. Since the material and its properties (like the number of atoms per volume) don’t change, the constant of proportionality in Curie’s law stays the same. That means we can write:

or more simply:

Let’s work through the numbers step by step.

1. List what we know.

   First case: $M_1 = 8\ \text{A m}^{-1}$, $B_1 = 0.6\ \text{T}$, $T_1 = 4\ \text{K}$.

   Second case: $B_2 = 0.2\ \text{T}$, $T_2 = 16\ \text{K}$, and $M_2$ is what we need.

2. Set up the proportion.

   From Curie’s law: $M \propto B/T$, so for the same sample:

This is valid because the proportionality constant cancels out.

3. Plug in the numbers.

Simplify each fraction inside:

$\frac{0.6}{4} = 0.15, \quad \frac{0.2}{16} = 0.0125$ …

Method: Solving Ratio Problems With Curie's Law

Use this technique whenever a paramagnetic sample's magnetisation is given under one set of $(B,T)$ conditions and you must find it under a different set for the same sample.

Steps

Step 1: Write Curie's law as a proportionality

where $C$ (the Curie constant) depends only on the sample itself, never on the applied field or temperature.

Step 2: Form a ratio to eliminate the unknown constant

Since $C$ is identical in both situations for the same sample,

You never need to compute $C$ itself — taking the ratio cancels it out.

Step 3: Substitute the known values and simplify

Plug in $B_1,T_1,M_1$ and $B_2,T_2$; reduce each $B/T$ fraction separately first, then take the ratio of the two fractions before solving. …