`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 regarding the susceptibility of a paramagnetic material at different temperatures, we can follow these steps:

### Step 1: Understand the relationship between susceptibility and temperature
According to Curie's law of paramagnetism, the magnetic susceptibility (χ) of a paramagnetic material is inversely proportional to its absolute temperature (T). This can be expressed mathematically as:
\[ \chi \propto \frac{1}{T} \]

### Step 2: Express the relationship for two different temperatures
Let χ₁ be the susceptibility at temperature T₁ and χ₂ be the susceptibility at temperature T₂. From the relationship established in Step 1, we can write:
\[ \chi_1 = \frac{C}{T_1} \]
\[ \chi_2 = \frac{C}{T_2} \]
where C is a constant for the material.

### Step 3: Set up the ratio of susceptibilities
From the expressions for χ₁ and χ₂, we can set up the following ratio:
\[ \frac{\chi_1}{\chi_2} = \frac{T_2}{T_1} \]

### Step 4: Rearranging the ratio
Rearranging the above equation gives us:
\[ \chi_1 T_1 = \chi_2 T_2 \]

### Step 5: Conclusion
Thus, we have derived the relationship:
\[ \chi_1 T_1 = \chi_2 T_2 \]
This means that the product of susceptibility and temperature remains constant for a paramagnetic material.

### Final Answer
The correct relationship is:
\[ \chi_1 T_1 = \chi_2 T_2 \]