According to Debye`s law the specific heat at extremely low temperatures value with temperature T as

Options

A. T
B. `T^2`
C. `T^3`
D. `T^4`

Answer

To solve the question regarding Debye's law and the specific heat at extremely low temperatures, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Debye's Law**: Debye's law states that at extremely low temperatures, the specific heat of solids behaves differently than at higher temperatures. It is important to recognize that this law is applicable to the vibrational modes of atoms in a solid.

2. **Specific Heat and Temperature Relationship**: According to Debye's law, the specific heat (C) at constant volume (V) of a solid is proportional to the cube of the absolute temperature (T) when the temperature is very low. This can be mathematically expressed as:
\[ C \propto T^3 \]

3. **Mathematical Expression**: From the proportionality, we can write:
\[ C = k \cdot T^3 \]
where \( k \) is a constant that depends on the material properties of the solid.

4. **Conclusion**: Therefore, the specific heat at extremely low temperatures varies with temperature \( T \) as:
\[ C \propto T^3 \]

5. **Final Answer**: The specific heat at extremely low temperatures according to Debye's law is given by:
\[ C \sim T^3 \]