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 need to find the intensity of magnetization (M2) at a new temperature (T2 = 24 K) and magnetic field (B2 = 0.3 T), given the initial conditions (M1 = 4 units, T1 = 6 K, B1 = 0.4 T).

### Step-by-Step Solution:

1. **Understand the Relationship**:
The intensity of magnetization (M) is directly proportional to the magnetic field (B) and inversely proportional to the temperature (T). This can be expressed mathematically as:
\[ M \propto \frac{B}{T} \]
Therefore, we can write:
\[ M = k \frac{B}{T} \]
where \( k \) is a constant.

2. **Set Up the Ratios**:
For two different states, we can write:
\[ \frac{M_2}{M_1} = \frac{B_2/T_2}{B_1/T_1} \]

3. **Substitute Known Values**:
We know:
- \( M_1 = 4 \) units
- \( T_1 = 6 \) K
- \( B_1 = 0.4 \) T
- \( T_2 = 24 \) K
- \( B_2 = 0.3 \) T

Substituting these values into the ratio gives:
\[ \frac{M_2}{4} = \frac{0.3/24}{0.4/6} \]

4. **Simplify the Right Side**:
Calculate the right side:
\[ \frac{M_2}{4} = \frac{0.3 \times 6}{0.4 \times 24} \]
Simplifying the numerator and denominator:
\[ = \frac{1.8}{9.6} = \frac{3}{16} \]

5. **Calculate M2**:
Now, we can find \( M_2 \):
\[ M_2 = 4 \times \frac{3}{16} = \frac{12}{16} = 0.75 \text{ units} \]

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

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