If the velocity of light denoted by 'c', acceleration due to gravity 'g' and atmospheric pressure 'p' are taken as fundamental units, then the dimensional formula of length will be :

Options

A. `c^(2)//g`
B. `c//g`
C. `g//c`
D. `c//g`

Explanation

To find the dimensional formula of length when the velocity of light (c), acceleration due to gravity (g), and atmospheric pressure (p) are taken as fundamental units, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the quantities:**
- The velocity of light \( c \) has dimensions of length per time:
\[ [c] = [L][T^{-1}] \]
- The acceleration due to gravity \( g \) has dimensions of length per time squared:
\[ [g] = [L][T^{-2}] \]
- Atmospheric pressure \( p \) is defined as force per unit area. The dimensions of force are mass times acceleration:
\[ [p] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

2. **Express length \( L \) in terms of \( c \), \( g \), and \( p \):**
- Assume the dimensional formula of length can be expressed as:
\[ [L] = c^x g^y p^z \]
- Substitute the dimensions we found:
\[ [L] = ([L][T^{-1}])^x ([L][T^{-2}])^y ([M][L^{-1}][T^{-2}])^z \]

3. **Combine the dimensions:**
- Expanding the right-hand side gives:
\[ [L] = [L^x][T^{-x}][L^y][T^{-2y}][M^z][L^{-z}][T^{-2z}] \]
- Combining like terms:
\[ [L] = [L^{x+y-z}][M^z][T^{-x-2y-2z}] \]

4. **Set up equations based on dimensional analysis:**
- For the dimensions of length:
\[ x + y - z = 1 \quad \text{(1)} \]
- For the dimensions of mass:
\[ z = 0 \quad \text{(2)} \]
- For the dimensions of time:
\[ -x - 2y - 2z = 0 \quad \text{(3)} \]

5. **Substituting \( z = 0 \) into the equations:**
- From equation (2), substitute \( z = 0 \) into (1):
\[ x + y = 1 \quad \text{(4)} \]
- Substitute \( z = 0 \) into (3):
\[ -x - 2y = 0 \quad \text{(5)} \]

6. **Solve the system of equations (4) and (5):**
- From equation (5), we have:
\[ x = -2y \]
- Substitute \( x = -2y \) into equation (4):
\[ -2y + y = 1 \implies -y = 1 \implies y = -1 \]
- Substitute \( y = -1 \) back into equation (4):
\[ x - 1 = 1 \implies x = 2 \]

7. **Final expression for length:**
- Now we have \( x = 2 \), \( y = -1 \), and \( z = 0 \).
- Therefore, the dimensional formula for length is:
\[ [L] = c^2 g^{-1} \]

### Conclusion:
The dimensional formula of length in terms of \( c \), \( g \), and \( p \) is:
\[ [L] = \frac{c^2}{g} \]