The speed of light ( c), gravitational constant (G) and plank's constant (h) are taken as fundamental units in a system. The dimensions of time in this new system should be.

Options

A. `G^(1//2) h^(1//2) c^(1//2)`
B. `G^(1//2) h^(1//2) c^(1//2)`
C. `G^(1//2) h^(1//2) c^(-3//2)`
D. `G^(1//2) h^(1//2) c^(1//2)`

Explanation

To find the dimensions of time in a system where the speed of light (c), gravitational constant (G), and Planck's constant (h) are taken as fundamental units, we can follow these steps:

### Step-by-Step Solution:

1. **Establish the relationship**: We start by expressing time (T) in terms of the fundamental units c, G, and h. We can write:
\[ T \propto c^x \cdot G^y \cdot h^z \]

2. **Write down the dimensions**:
- The dimensions of time (T) are \( [M^0 L^0 T^1] \).
- The dimensions of the speed of light (c) are \( [L^1 T^{-1}] \).
- The dimensions of the gravitational constant (G) are \( [M^{-1} L^3 T^{-2}] \).
- The dimensions of Planck's constant (h) are \( [M^1 L^2 T^{-1}] \).

3. **Substitute dimensions into the equation**:
\[ [M^0 L^0 T^1] \propto (L^1 T^{-1})^x \cdot (M^{-1} L^3 T^{-2})^y \cdot (M^1 L^2 T^{-1})^z \]

4. **Expand the dimensions**:
\[ [M^0 L^0 T^1] \propto M^{-y + z} \cdot L^{x + 3y + 2z} \cdot T^{-x - 2y - z} \]

5. **Set up equations by equating powers**:
- For mass (M):
\[ -y + z = 0 \quad \text{(1)} \]
- For length (L):
\[ x + 3y + 2z = 0 \quad \text{(2)} \]
- For time (T):
\[ -x - 2y - z = 1 \quad \text{(3)} \]

6. **Solve the equations**:
- From equation (1): \( z = y \).
- Substitute \( z = y \) into equation (2):
\[ x + 3y + 2y = 0 \implies x + 5y = 0 \implies x = -5y \quad \text{(4)} \]
- Substitute \( z = y \) and \( x = -5y \) into equation (3):
\[ -(-5y) - 2y - y = 1 \implies 5y - 2y - y = 1 \implies 2y = 1 \implies y = \frac{1}{2} \]
- Now, substitute \( y = \frac{1}{2} \) back to find \( z \):
\[ z = \frac{1}{2} \]
- Substitute \( y = \frac{1}{2} \) into equation (4) to find \( x \):
\[ x = -5 \cdot \frac{1}{2} = -\frac{5}{2} \]

7. **Final expression for time**:
- Now we have \( x = -\frac{5}{2}, y = \frac{1}{2}, z = \frac{1}{2} \).
- Therefore, the dimensions of time in this new system can be expressed as:
\[ T \propto c^{-\frac{5}{2}} \cdot G^{\frac{1}{2}} \cdot h^{\frac{1}{2}} \]

### Conclusion:
The dimensions of time in this new system are given by:
\[ T \propto c^{-\frac{5}{2}} \cdot G^{\frac{1}{2}} \cdot h^{\frac{1}{2}} \]