A copper bar of length `L` and area of cross section `A` is placed in a chamber at atmospheric pressure. If the chamber is evacuated, the percentage change in its volume will be (compressibility of copper is `8 xx 10^(-12) m^(2)//N` and `1 atm = 10^(5) N//m^(2))`

Options

A. `8xx10^(-7)`
B. `8xx10^(-5)`
C. `1.25xx10^(-4)`
D. `1.25xx10^(-5)`

Explanation

To find the percentage change in the volume of a copper bar when the chamber it is placed in is evacuated, we can follow these steps:

### Step 1: Understand the Relationship Between Compressibility and Bulk Modulus
Compressibility (\( \beta \)) is defined as the reciprocal of bulk modulus (\( K \)):
\[ \beta = \frac{1}{K} \]
From this, we can express the change in volume (\( \Delta V \)) in terms of the change in pressure (\( \Delta P \)):
\[ \Delta V = -\beta V \Delta P \]

### Step 2: Calculate the Change in Pressure
When the chamber is evacuated, the pressure changes from atmospheric pressure (\( P_0 \)) to zero. Therefore, the change in pressure (\( \Delta P \)) is:
\[ \Delta P = P_0 = 1 \text{ atm} = 10^5 \text{ N/m}^2 \]

### Step 3: Calculate the Initial Volume
The initial volume (\( V \)) of the copper bar can be calculated using its length (\( L \)) and area of cross-section (\( A \)):
\[ V = A \cdot L \]

### Step 4: Substitute Values into the Volume Change Formula
Using the compressibility of copper (\( \beta = 8 \times 10^{-12} \text{ m}^2/\text{N} \)), we can substitute the values into the formula for volume change:
\[ \Delta V = -\beta V \Delta P = -\left(8 \times 10^{-12} \text{ m}^2/\text{N}\right) \cdot (A \cdot L) \cdot (10^5 \text{ N/m}^2) \]

### Step 5: Calculate the Percentage Change in Volume
The percentage change in volume (\( \frac{\Delta V}{V} \times 100 \)) can be calculated as follows:
\[ \frac{\Delta V}{V} = -\beta \Delta P \]
Substituting the values:
\[ \frac{\Delta V}{V} = -\left(8 \times 10^{-12}\right) \cdot (10^5) = -8 \times 10^{-7} \]
To convert this to a percentage:
\[ \text{Percentage change} = \left(-8 \times 10^{-7}\right) \times 100 = -8 \times 10^{-5} \% \]

### Final Result
The percentage change in the volume of the copper bar when the chamber is evacuated is:
\[ \text{Percentage change} = 8 \times 10^{-5} \% \]