`x_1 and x_2` are susceptibility of a Paramagnetic material at temperatures `T_1, K and T_2K` respectively, then

Options

A. `x_1T_1 = x_2T_2`
B. `x_1T_2 = x_2T_1`
C. `x_1T_2 = x_2T_1`
D. `x_1sqrt(T_1) = x_2sqrt(T_1)`

Answer

To solve the problem, we need to establish the relationship between the magnetic susceptibility \( x_1 \) and \( x_2 \) of a paramagnetic material at temperatures \( T_1 \) and \( T_2 \) respectively. We will use the Curie Law, which states that the magnetic susceptibility \( x \) of a paramagnetic material is inversely proportional to its absolute temperature \( T \).

### Step-by-Step Solution:

1. **Understand Curie Law**: According to Curie Law, the magnetic susceptibility \( x \) is inversely proportional to the absolute temperature \( T \). Mathematically, this can be expressed as:
\[ x \propto \frac{1}{T} \]
This implies that:
\[ x \cdot T = \text{constant} \]

2. **Set Up the Relationship**: For two different temperatures \( T_1 \) and \( T_2 \), with corresponding susceptibilities \( x_1 \) and \( x_2 \), we can write:
\[ x_1 \cdot T_1 = k \quad \text{(1)} \]
\[ x_2 \cdot T_2 = k \quad \text{(2)} \]
where \( k \) is a constant.

3. **Equate the Constants**: Since both equations equal the same constant \( k \), we can set them equal to each other:
\[ x_1 \cdot T_1 = x_2 \cdot T_2 \]

4. **Rearranging the Equation**: From the above equation, we can express the relationship between the susceptibilities and temperatures:
\[ \frac{x_1}{x_2} = \frac{T_2}{T_1} \]

5. **Final Relation**: This gives us the desired relationship between the susceptibilities and temperatures:
\[ x_1 T_1 = x_2 T_2 \]

### Conclusion:
Thus, the relationship between the susceptibilities \( x_1 \) and \( x_2 \) at temperatures \( T_1 \) and \( T_2 \) is given by:
\[ x_1 T_1 = x_2 T_2 \]