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)`

Explanation

To solve the problem, we will use Curie's law for paramagnetic materials, which states that the magnetization \( M \) is proportional to the magnetic field \( B \) and inversely proportional to the temperature \( T \).

### Step-by-step Solution:

1. **Identify the Given Values:**
- First magnetization, \( M_1 = 8 \, \text{A/m} \)
- First magnetic field, \( B_1 = 0.6 \, \text{T} \)
- First temperature, \( T_1 = 4 \, \text{K} \)
- Second magnetic field, \( B_2 = 0.2 \, \text{T} \)
- Second temperature, \( T_2 = 16 \, \text{K} \)

2. **Use Curie's Law:**
According to Curie's law, the relationship can be expressed as:
\[ M \propto \frac{B}{T} \]
This implies:
\[ \frac{M_1}{M_2} = \frac{B_1}{B_2} \cdot \frac{T_2}{T_1} \]

3. **Rearranging the Formula:**
We can rearrange the formula to find \( M_2 \):
\[ M_2 = M_1 \cdot \frac{B_2}{B_1} \cdot \frac{T_1}{T_2} \]

4. **Substituting the Values:**
Now, substituting the known values into the equation:
\[ M_2 = 8 \cdot \frac{0.2}{0.6} \cdot \frac{4}{16} \]

5. **Calculating Step-by-Step:**
- First calculate \( \frac{0.2}{0.6} = \frac{1}{3} \)
- Next, calculate \( \frac{4}{16} = \frac{1}{4} \)
- Now substitute these values back into the equation:
\[ M_2 = 8 \cdot \frac{1}{3} \cdot \frac{1}{4} \]
- This simplifies to:
\[ M_2 = 8 \cdot \frac{1}{12} = \frac{8}{12} = \frac{2}{3} \, \text{A/m} \]

6. **Final Result:**
The magnetization \( M_2 \) when the sample is placed in an external magnetic field of \( 0.2 \, \text{T} \) at a temperature of \( 16 \, \text{K} \) is:
\[ M_2 = \frac{2}{3} \, \text{A/m} \]

### Conclusion:
Thus, the answer is \( \frac{2}{3} \, \text{A/m} \).

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