It is known from Curie's law that magnetic moment (m) of a paramagnetic sample is directly proporational to the ratio of the external magnetic field (B) to the absolute temperature (T). A sample of paramagnetic salt contains `2.0 xx 10^(24)` atomic dipoles each of dipole moment `1.5xx 10^(-23) JT^(-1)`.
Sample is placed under a homogeneous magnetic field of `0.84 T` and cooled to a temperature of 4.2. The drgree of magnetic saturation achevied is equal to `15%` . You may assume that `85%` dipoles are randomly oriented and do not contributes to the magnetisation . The total diploe moment of the sample , for a magnetic field of `0.98T` and a temperature of `2.8 K`, is

Options

A. `4.5 J//T`
B. `30 J//T`
C. `7.9 J//T`
D. `3 J//T`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript while ensuring clarity and understanding.

### Step-by-Step Solution:

1. **Calculate the Total Dipole Moment (m1) at the Initial Conditions:**
- Given:
- Number of atomic dipoles, \( N = 2.0 \times 10^{24} \)
- Dipole moment of each dipole, \( p = 1.5 \times 10^{-23} \, \text{J/T} \)
- Degree of magnetic saturation, \( S = 15\% = 0.15 \)

- The total dipole moment \( m_1 \) can be calculated as:
\[ m_1 = N \cdot p \cdot S \]
\[ m_1 = (2.0 \times 10^{24}) \cdot (1.5 \times 10^{-23}) \cdot 0.15 \]
\[ m_1 = 2.0 \times 1.5 \times 0.15 \times 10^{24 - 23} = 0.45 \times 10^{1} = 4.5 \, \text{J/T} \]

2. **Use Curie’s Law to Relate the Magnetic Moments:**
- According to Curie's law, the ratio of the magnetic moments at two different conditions can be expressed as:
\[ \frac{m_1}{m_2} = \frac{B_1}{B_2} \cdot \frac{T_2}{T_1} \]
- Where:
- \( B_1 = 0.84 \, \text{T} \) (initial magnetic field)
- \( B_2 = 0.98 \, \text{T} \) (final magnetic field)
- \( T_1 = 4.2 \, \text{K} \) (initial temperature)
- \( T_2 = 2.8 \, \text{K} \) (final temperature)

3. **Rearranging to Find \( m_2 \):**
- Rearranging the equation gives:
\[ m_2 = m_1 \cdot \frac{B_2}{B_1} \cdot \frac{T_1}{T_2} \]
- Substituting the known values:
\[ m_2 = 4.5 \cdot \frac{0.98}{0.84} \cdot \frac{4.2}{2.8} \]

4. **Calculating \( m_2 \):**
- First, calculate the ratios:
\[ \frac{0.98}{0.84} \approx 1.1667 \]
\[ \frac{4.2}{2.8} = 1.5 \]
- Now substitute these ratios back into the equation:
\[ m_2 = 4.5 \cdot 1.1667 \cdot 1.5 \]
\[ m_2 \approx 4.5 \cdot 1.75 \approx 7.875 \, \text{J/T} \]

5. **Final Result:**
- The total dipole moment of the sample at the new conditions is approximately:
\[ m_2 \approx 7.9 \, \text{J/T} \]