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 1: Understand the formula for Bulk Modulus
The bulk modulus \( B \) is defined as:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \]
where:
- \( \Delta P \) is the change in pressure,
- \( \Delta V \) is the change in volume,
- \( V \) is the original volume.

### Step 2: Convert the given values
1. The pressure applied is \( 100 \) atmospheres. We need to convert this into dyne/cm².
- \( 1 \text{ atmosphere} = 1.013 \times 10^6 \text{ dyne/cm}^2 \)
- Therefore, \( 100 \text{ atmospheres} = 100 \times 1.013 \times 10^6 \text{ dyne/cm}^2 = 1.013 \times 10^8 \text{ dyne/cm}^2 \).

2. The volume reduction is \( 0.01\% \).
- This can be expressed as a fraction:
\[ \frac{\Delta V}{V} = \frac{0.01}{100} = 0.0001 \]

### Step 3: Substitute values into the Bulk Modulus formula
Now we can substitute the values into the bulk modulus formula:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} = -\frac{1.013 \times 10^8 \text{ dyne/cm}^2}{0.0001} \]

### Step 4: Calculate the Bulk Modulus
Calculating the above expression gives:
\[ B = -\frac{1.013 \times 10^8}{0.0001} = -1.013 \times 10^{12} \text{ dyne/cm}^2 \]

Since bulk modulus is a positive quantity, we can express it as:
\[ B = 1.013 \times 10^{12} \text{ dyne/cm}^2 \]

### Final Answer
The bulk modulus of the material of the rubber is approximately:
\[ B \approx 1.013 \times 10^{12} \text{ dyne/cm}^2 \]

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