The pressure applied from all direction on a cube is P. How much its temperature should be raised to maintain the original volume ? The volume elasticity of the cube is `beta` and the coefficient of volume expansion is `alpha`

Options

A. `(p)/(alphabeta)`
B. `(palpha)/(beta)`
C. `(pbeta)/(alpha)`
D. `(alphabeta)/(p)`

Answer

To solve the problem, we need to find out how much the temperature of a cube should be raised to maintain its original volume when a pressure \( P \) is applied from all directions. We will use the concepts of volume elasticity and coefficient of volume expansion.

### Step-by-Step Solution:

1. **Understanding Volumetric Stress**:
The volumetric stress \( P \) is defined as the pressure applied from all directions. It can be expressed in terms of volume elasticity \( \beta \) and the change in volume \( \Delta V \):
\[ P = \beta \left( \frac{\Delta V}{V} \right) \]
where \( V \) is the original volume of the cube.

2. **Expressing Change in Volume**:
Rearranging the equation for \( \Delta V \):
\[ \Delta V = \frac{P V}{\beta} \]
Since the pressure causes a decrease in volume, we have \( \Delta V < 0 \).

3. **Volume Expansion Due to Temperature Change**:
The change in volume due to a change in temperature \( \Delta T \) is given by:
\[ \Delta V = V \cdot \alpha \cdot \Delta T \]
where \( \alpha \) is the coefficient of volume expansion.

4. **Setting the Changes Equal**:
To maintain the original volume, the increase in volume due to the temperature change must equal the decrease in volume due to pressure:
\[ V \cdot \alpha \cdot \Delta T = -\Delta V \]
Substituting \( \Delta V \) from step 2:
\[ V \cdot \alpha \cdot \Delta T = -\left( \frac{P V}{\beta} \right) \]

5. **Solving for Change in Temperature**:
Canceling \( V \) from both sides (assuming \( V \neq 0 \)):
\[ \alpha \cdot \Delta T = -\frac{P}{\beta} \]
Rearranging gives:
\[ \Delta T = -\frac{P}{\alpha \beta} \]
Since we are interested in the temperature increase to counteract the decrease in volume, we take the absolute value:
\[ \Delta T = \frac{P}{\alpha \beta} \]

### Final Answer:
The temperature should be raised by:
\[ \Delta T = \frac{P}{\alpha \beta} \]