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

The correct option is P/αß.

To find how much the temperature of the cube should be raised to maintain its original volume, we need to balance the volume contraction due to the applied pressure with the volume expansion due to the rise in temperature.

Step 1: Find the change in volume due to applied pressure
The volume elasticity (also known as Bulk Modulus), represented by the symbol ß, is defined as the ratio of bulk stress (pressure P) to bulk strain (fractional change in volume).

$ß=\frac{\text{Volume Stress}}{\text{Volume Strain}}=\frac{P}{-\frac{\Delta V_1}{V}}$

Here, V is the original volume, and $\Delta V_1$ is the change in volume due to pressure. Rearranging the formula gives the fractional change in volume due to pressure:

$\frac{\Delta V_1}{V}=-\frac{P}{ß}$

Step 2: Find the change in volume due to temperature rise
Let the temperature be raised by $\Delta T$. The fractional change in volume due to thermal expansion is given by:

$\frac{\Delta V_2}{V}=\alpha \Delta T$

where α is the coefficient of volume expansion.

Step 3: Apply the condition for maintaining original volume
To maintain the original volume, the net change in volume must be zero:

$\Delta V_1+\Delta V_2=0$

Dividing both sides by the original volume V:

$\frac{\Delta V_1}{V}+\frac{\Delta V_2}{V}=0$

Substituting the expressions from Step 1 and Step 2:

$-\frac{P}{ß}+\alpha \Delta T=0$

$\alpha \Delta T=\frac{P}{ß}$

Solving for the change in temperature $\Delta T$:

$\Delta T=\frac{P}{\alpha ß}$