If C the velocity of light, h planck's constant and G Gravitational constant are taken as fundamental quantities, then the dimensional formula of mass is

Options

A. `h^(-1//2)G^(1//2)C^0`
B. `h^(1//2)C^(1//2)G^(-1//2)`
C. `h^(-1//2)C^(1//2)G^(-1//2)`
D. `h^(-1//2)C^(-1//2)G^(-1//2)`

Answer

To derive the dimensional formula of mass using the fundamental quantities: the velocity of light (C), Planck's constant (h), and the gravitational constant (G), we can follow these steps:

### Step-by-Step Solution:

1. **Express Mass in Terms of Fundamental Quantities**:
We can express mass (M) in terms of the fundamental quantities as follows:
\[ M = h^a \cdot G^b \cdot C^c \]
where \(a\), \(b\), and \(c\) are the powers we need to determine.

2. **Write the Dimensional Formulas**:
- The dimensional formula for Planck's constant \(h\) is:
\[ [h] = M^1 L^2 T^{-1} \]
- The dimensional formula for the gravitational constant \(G\) is:
\[ [G] = M^{-1} L^3 T^{-2} \]
- The dimensional formula for the speed of light \(C\) is:
\[ [C] = L^1 T^{-1} \]

3. **Substitute the Dimensional Formulas**:
Substitute the dimensional formulas into the expression for mass:
\[ [M] = (M^1 L^2 T^{-1})^a \cdot (M^{-1} L^3 T^{-2})^b \cdot (L^1 T^{-1})^c \]
This simplifies to:
\[ [M] = M^{a - b} L^{2a + 3b + c} T^{-a - 2b - c} \]

4. **Set Up the Equations**:
Since we know that the dimensional formula for mass is \(M^1 L^0 T^0\), we can equate the exponents:
- For mass (M):
\[ a - b = 1 \quad \text{(1)} \]
- For length (L):
\[ 2a + 3b + c = 0 \quad \text{(2)} \]
- For time (T):
\[ -a - 2b - c = 0 \quad \text{(3)} \]

5. **Solve the Equations**:
From equation (1), we can express \(a\) in terms of \(b\):
\[ a = b + 1 \quad \text{(4)} \]
Substitute equation (4) into equations (2) and (3):
- Substituting into (2):
\[ 2(b + 1) + 3b + c = 0 \implies 2b + 2 + 3b + c = 0 \implies 5b + c = -2 \quad \text{(5)} \]
- Substituting into (3):
\[ -(b + 1) - 2b - c = 0 \implies -b - 1 - 2b - c = 0 \implies -3b - c = 1 \quad \text{(6)} \]

6. **Solve Equations (5) and (6)**:
Now we have a system of equations:
\[ 5b + c = -2 \quad \text{(5)} \]
\[ -3b - c = 1 \quad \text{(6)} \]
Adding equations (5) and (6):
\[ 5b + c - 3b - c = -2 + 1 \implies 2b = -1 \implies b = -\frac{1}{2} \]
Substitute \(b\) back into equation (4):
\[ a = -\frac{1}{2} + 1 = \frac{1}{2} \]
Substitute \(b\) into equation (5) to find \(c\):
\[ 5(-\frac{1}{2}) + c = -2 \implies -\frac{5}{2} + c = -2 \implies c = -2 + \frac{5}{2} = \frac{1}{2} \]

7. **Final Expression for Mass**:
Now we have:
\[ a = \frac{1}{2}, \quad b = -\frac{1}{2}, \quad c = \frac{1}{2} \]
Thus, the dimensional formula for mass is:
\[ M = h^{\frac{1}{2}} \cdot G^{-\frac{1}{2}} \cdot C^{\frac{1}{2}} \]

### Conclusion:
The dimensional formula of mass when C, h, and G are taken as fundamental quantities is:
\[ M = h^{\frac{1}{2}} \cdot G^{-\frac{1}{2}} \cdot C^{\frac{1}{2}} \]