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Step-by-Step Solution
Step 1: Understand the Given Information
We are given a paramagnetic sample with a net magnetization of 6 A/m when placed in an external magnetic field of 0.4 T at a temperature of 4 K. We want to find its magnetization when placed in a 0.3 T field at 24 K.
Step 2: Recall Curie’s Law
For a paramagnetic material, Curie’s law states that:
$$\chi = \frac{C}{T},$$
where
$\chi$ is the magnetic susceptibility,
$C$ is the Curie constant, and
$T$ is the absolute temperature.
The net magnetization $M$ is related to $\chi$ and $B_{\text{ext}}$ by:
$$M = \chi \, B_{\text{ext}}.$$
Step 3: Determine the Curie Constant from Initial Conditions
From the first scenario:
Magnetization, $M_1 = 6 \text{ A/m}$
External field, $B_1 = 0.4 \text{ T}$
Temperature, $T_1 = 4 \text{ K}$
Using Curie’s law in the form $M = \frac{C\, B_{\text{ext}}}{T}$, we have:
$$6 = \frac{C \times 0.4}{4}.$$
Solving for $C$:
$$C = \frac{6 \times 4}{0.4} = \frac{24}{0.4} = 60.$$
Step 4: Apply Curie’s Law to the Second Scenario
For the new scenario:
External field, $B_2 = 0.3 \text{ T}$
Temperature, $T_2 = 24 \text{ K}$
Using $M = \frac{C \, B_{\text{ext}}}{T}$ again:
$$M_2 = \frac{60 \times 0.3}{24}.$$
Simplify:
$$M_2 = \frac{60 \times 3}{240} \;=\; \frac{180}{240} \;=\; 0.75 \,\text{A/m}.$$
Step 5: State the Final Answer
The magnetization of the sample in the second scenario is
$$\boxed{0.75 \,\text{A/m}}.$
