Intensity of magnetization is `4` (units) at temperature `6K` and `B = 0.4 T`. what is the intensity of magnetization at temperature `24K` in `B = 0.3T`

Options

A. 0.75
B. 0.25
C. 0.5
D. 1

Answer

To solve the problem, we will use the relationship between the intensity of magnetization (M), magnetic field (B), and temperature (T). The intensity of magnetization is directly proportional to the magnetic field and inversely proportional to the temperature.

### Step-by-Step Solution:

1. **Understanding the relationship**:
The intensity of magnetization (M) can be expressed as:
\[ M \propto \frac{B}{T} \]
This implies:
\[ M = k \frac{B}{T} \]
where \( k \) is a constant.

2. **Setting up the equations for the two cases**:
For the first case (1):
- \( M_1 = 4 \) units
- \( B_1 = 0.4 \) T
- \( T_1 = 6 \) K

For the second case (2):
- \( M_2 \) is what we need to find
- \( B_2 = 0.3 \) T
- \( T_2 = 24 \) K

3. **Writing the equations**:
From the relationship:
\[ M_1 = k \frac{B_1}{T_1} \quad \text{(1)} \]
\[ M_2 = k \frac{B_2}{T_2} \quad \text{(2)} \]

4. **Taking the ratio of the two equations**:
Dividing equation (2) by equation (1):
\[ \frac{M_2}{M_1} = \frac{B_2/T_2}{B_1/T_1} \]

5. **Substituting the known values**:
\[ \frac{M_2}{4} = \frac{0.3/24}{0.4/6} \]

6. **Calculating the right-hand side**:
Simplifying the right-hand side:
\[ \frac{M_2}{4} = \frac{0.3 \cdot 6}{0.4 \cdot 24} \]
\[ = \frac{1.8}{9.6} = \frac{3}{16} \]

7. **Finding \( M_2 \)**:
Now, we can find \( M_2 \):
\[ M_2 = 4 \cdot \frac{3}{16} \]
\[ = \frac{12}{16} = 0.75 \text{ units} \]

### Final Answer:
The intensity of magnetization at temperature 24 K and magnetic field 0.3 T is \( 0.75 \) units.

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