When a pressure of 100 atmosphere is applied on a spherical ball, then its volume reduces to `0.01%`. The bulk modulus of the material of the rubber in dyne/`cm^2` is

Options

A. `10xx10^12`
B. `100xx10^12`
C. `1xx10^12`
D. `20xx10^12`

Explanation

To find the bulk modulus of the material of the rubber when a pressure of 100 atmospheres is applied and the volume reduces by 0.01%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Bulk Modulus (K)**:
The bulk modulus is defined as the ratio of the change in pressure (ΔP) to the relative change in volume (ΔV/V). Mathematically, it is expressed as:
\[ K = -\frac{\Delta P}{\frac{\Delta V}{V}} \]

2. **Convert the Given Pressure**:
The pressure applied is given as 100 atmospheres. We need to convert this to dyne/cm².
- 1 atmosphere = \(1.013 \times 10^6\) pascals.
- Therefore, 100 atmospheres = \(100 \times 1.013 \times 10^6 \, \text{Pa} = 1.013 \times 10^8 \, \text{Pa}\).
- Since \(1 \, \text{Pa} = 10 \, \text{dyne/cm}^2\), we convert the pressure:
\[ \Delta P = 1.013 \times 10^8 \, \text{Pa} \times 10 = 1.013 \times 10^9 \, \text{dyne/cm}^2 \]

3. **Calculate the Change in Volume**:
The volume reduces by 0.01%, which can be expressed as:
\[ \Delta V = 0.01\% \text{ of } V = \frac{0.01}{100} \times V = 0.0001V \]
Thus, the relative change in volume is:
\[ \frac{\Delta V}{V} = 0.0001 \]

4. **Substitute Values into the Bulk Modulus Formula**:
Now we can substitute the values into the bulk modulus formula:
\[ K = -\frac{\Delta P}{\frac{\Delta V}{V}} = -\frac{1.013 \times 10^9 \, \text{dyne/cm}^2}{0.0001} \]

5. **Calculate the Bulk Modulus**:
Performing the calculation gives:
\[ K = -\frac{1.013 \times 10^9}{0.0001} = -1.013 \times 10^{13} \, \text{dyne/cm}^2 \]
Since the bulk modulus is a positive quantity, we take the absolute value:
\[ K = 1.013 \times 10^{13} \, \text{dyne/cm}^2 \]

### Final Answer:
The bulk modulus of the material of the rubber is approximately \(1.013 \times 10^{13} \, \text{dyne/cm}^2\).