A sample of paramagnetic salt contains `2xx10^(24)` atomic dipoles, each of moment `1.5xx10^(-23)JT^-1`. The sample is placed under a homogeneous magnetic field of `0.64T` and cooled to a temperature of `4.2K`. The degree of magnetic saturation archieved is equal to `15%`. What is the total dipole moment of the sample for a mangetic field of `0.98T` and a temperature of `2.8K`. (Assume Curie's law).

Options

A. `8 xx 10^(3) J//T`
B. ` 10.34 J//T`
C. ` 8 xx 10 ^(-3) J//T`
D. ` 25 J//T`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the total dipole moment of the sample.
The total dipole moment \( M_{\text{total}} \) can be calculated using the formula:

\[ M_{\text{total}} = N \cdot \mu \]

where:
- \( N = 2 \times 10^{24} \) (number of atomic dipoles),
- \( \mu = 1.5 \times 10^{-23} \, \text{J/T} \) (dipole moment of each atomic dipole).

Substituting the values:

\[ M_{\text{total}} = (2 \times 10^{24}) \cdot (1.5 \times 10^{-23}) = 30 \, \text{J/T} \]

### Step 2: Calculate the effective magnetic moment at the first condition.
Given that the degree of magnetic saturation achieved is \( 15\% \), we can find the effective magnetic moment \( M_{\text{effective}} \) at the first condition (magnetic field \( B_1 = 0.64 \, \text{T} \) and temperature \( T_1 = 4.2 \, \text{K} \)):

\[ M_{\text{effective}} = 0.15 \cdot M_{\text{total}} = 0.15 \cdot 30 = 4.5 \, \text{J/T} \]

### Step 3: Apply Curie’s Law to find the constant \( C \).
According to Curie’s Law:

\[ M_{\text{effective}} = C \cdot \frac{B_1}{T_1} \]

We can rearrange this to find \( C \):

\[ C = M_{\text{effective}} \cdot \frac{T_1}{B_1} \]

Substituting the values:

\[ C = 4.5 \cdot \frac{4.2}{0.64} = 29.53 \, \text{J/T}^2 \cdot \text{K} \]

### Step 4: Calculate the effective magnetic moment at the second condition.
Now we need to find the effective magnetic moment \( M_{\text{effective}}' \) for the new conditions (magnetic field \( B_2 = 0.98 \, \text{T} \) and temperature \( T_2 = 2.8 \, \text{K} \)):

\[ M_{\text{effective}}' = C \cdot \frac{B_2}{T_2} \]

Substituting the values:

\[ M_{\text{effective}}' = 29.53 \cdot \frac{0.98}{2.8} = 10.34 \, \text{J/T} \]

### Final Answer:
The total dipole moment of the sample for a magnetic field of \( 0.98 \, \text{T} \) and a temperature of \( 2.8 \, \text{K} \) is \( 10.34 \, \text{J/T} \).

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