A paramagnetic sample shows a net magnetisation of `8Am^-1` when placed in an external magnetic field of `0*6T` at a temperature of `4K`. When the same sample is placed in an external magnetic field of `0*2T` at a temperature of `16K`, the magnetisation will be

Options

A. `(32)/(3)Am^(-1)`
B. `(2)/(3)Am^(-1)`
C. `6Am^(-1)`
D. `2.4Am^(-1)`

Answer

To solve the problem, we will use Curie's law, which states that the intensity of magnetization (I) is directly proportional to the magnetic field (B) and inversely proportional to the temperature (T).

### Step-by-Step Solution:

**Step 1: Write down Curie's Law**
Curie's Law can be expressed as:
\[ I \propto \frac{B}{T} \]
This means that the magnetization (I) is directly proportional to the magnetic field (B) and inversely proportional to the temperature (T).

**Step 2: Set up the ratio for the two conditions**
For two different states, we can write:
\[ \frac{I_1}{I_2} = \frac{B_1}{B_2} \cdot \frac{T_2}{T_1} \]
Where:
- \( I_1 \) = initial magnetization = 8 A/m
- \( B_1 \) = initial magnetic field = 0.6 T
- \( T_1 \) = initial temperature = 4 K
- \( B_2 \) = final magnetic field = 0.2 T
- \( T_2 \) = final temperature = 16 K
- \( I_2 \) = final magnetization (unknown)

**Step 3: Substitute known values into the equation**
Substituting the known values into the equation gives:
\[ \frac{8}{I_2} = \frac{0.6}{0.2} \cdot \frac{16}{4} \]

**Step 4: Simplify the right side**
Calculating the right side:
\[ \frac{0.6}{0.2} = 3 \]
\[ \frac{16}{4} = 4 \]
Thus:
\[ \frac{8}{I_2} = 3 \cdot 4 = 12 \]

**Step 5: Solve for \( I_2 \)**
Now, we can rearrange to find \( I_2 \):
\[ I_2 = \frac{8}{12} = \frac{2}{3} \, \text{A/m} \]

### Final Answer:
The magnetization when the sample is placed in an external magnetic field of 0.2 T at a temperature of 16 K will be:
\[ I_2 = \frac{2}{3} \, \text{A/m} \]