The value of atmosphere pressure is `10^(6)` dynes/`cm^(2)`. Find the value in S.I. using dimensional analysis?

Explanation

To convert the atmospheric pressure from CGS (centimeter-gram-second) to SI (International System of Units), we follow these steps:

### Step 1: Understand the given value
The atmospheric pressure is given as \(10^6\) dynes/cm².

### Step 2: Identify the units
- In the CGS system, pressure is measured in dynes per square centimeter (dyn/cm²).
- In the SI system, pressure is measured in Newtons per square meter (N/m²).

### Step 3: Convert dynes to Newtons
1 Newton (N) is defined as \(10^5\) dynes. Therefore, we can convert dynes to Newtons using the following relationship:
\[ 1 \text{ dyne} = 10^{-5} \text{ N} \]

### Step 4: Convert cm² to m²
1 square meter (m²) is equal to \(10^4\) square centimeters (cm²). Therefore, we can convert cm² to m² using the following relationship:
\[ 1 \text{ cm}^2 = 10^{-4} \text{ m}^2 \]

### Step 5: Set up the conversion
Now, we can express the conversion from dynes/cm² to N/m²:
\[ \text{Pressure in N/m²} = \text{Pressure in dyn/cm²} \times \left(\frac{10^{-5} \text{ N}}{1 \text{ dyne}}\right) \times \left(\frac{1 \text{ cm}^2}{10^{-4} \text{ m}^2}\right) \]

### Step 6: Substitute the values
Substituting the given pressure value:
\[ \text{Pressure in N/m²} = 10^6 \text{ dyn/cm}^2 \times 10^{-5} \text{ N/dyne} \times 10^4 \text{ cm}^2/\text{m}^2 \]

### Step 7: Simplify the expression
Now, we simplify the expression:
\[ \text{Pressure in N/m²} = 10^6 \times 10^{-5} \times 10^4 \text{ N/m}^2 \]
\[ = 10^{6 - 5 + 4} \text{ N/m}^2 \]
\[ = 10^{5} \text{ N/m}^2 \]

### Final Answer
Thus, the value of atmospheric pressure in SI units is:
\[ \text{Pressure} = 10^5 \text{ N/m}^2 \]

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