If velocity of light c, planck's constant h and gravitational constnat G are taken as fundamental quantities then the dimensions of the length will be

Options

A. `sqrt((ch)/(G))`
B. `sqrt((hG)/(c^(5)))`
C. `sqrt((hG)/(c^(3)))`
D. `sqrt((hc^(3))/(G))`

Explanation

To solve the problem of finding the dimensions of length when the velocity of light (c), Planck's constant (h), and the gravitational constant (G) are taken as fundamental quantities, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Relationship**:
We start by expressing length (L) in terms of the fundamental quantities c, h, and G. We can write:
\[ L \propto c^x h^y G^z \]
where \(x\), \(y\), and \(z\) are the powers to which each fundamental quantity is raised.

2. **Express Dimensions**:
We need to express the dimensions of each quantity:
- The dimension of velocity of light \(c\) is \([L][T]^{-1}\) or \(L T^{-1}\).
- The dimension of Planck's constant \(h\) is \([M][L]^2[T]^{-1}\) or \(M L^2 T^{-1}\).
- The dimension of gravitational constant \(G\) is \([M]^{-1}[L]^3[T]^{-2}\) or \(M^{-1} L^3 T^{-2}\).

3. **Set Up the Equation**:
Now, substituting these dimensions into our proportionality:
\[ [L] = (L T^{-1})^x (M L^2 T^{-1})^y (M^{-1} L^3 T^{-2})^z \]
This expands to:
\[ [L] = L^x T^{-x} \cdot M^y L^{2y} T^{-y} \cdot M^{-z} L^{3z} T^{-2z} \]
Combining the dimensions gives:
\[ [L] = M^{y - z} L^{x + 2y + 3z} T^{-x - y - 2z} \]

4. **Equate Dimensions**:
Since the left side is just length, its dimensions are \(M^0 L^1 T^0\). Therefore, we equate the powers:
- For mass: \(y - z = 0\) (Equation 1)
- For length: \(x + 2y + 3z = 1\) (Equation 2)
- For time: \(-x - y - 2z = 0\) (Equation 3)

5. **Solve the Equations**:
From Equation 1, we have \(y = z\). Substituting \(y = z\) into Equations 2 and 3:
- Equation 2 becomes \(x + 2y + 3y = 1\) or \(x + 5y = 1\) (Equation 4)
- Equation 3 becomes \(-x - y - 2y = 0\) or \(-x - 3y = 0\) or \(x = -3y\) (Equation 5)

Now substituting \(x = -3y\) into Equation 4:
\[ -3y + 5y = 1 \implies 2y = 1 \implies y = \frac{1}{2} \]
Then from \(y = z\), we have \(z = \frac{1}{2}\). Now substituting \(y\) back into Equation 5:
\[ x = -3 \cdot \frac{1}{2} = -\frac{3}{2} \]

6. **Final Expression**:
Now substituting \(x\), \(y\), and \(z\) back into the original proportionality:
\[ L \propto c^{-\frac{3}{2}} h^{\frac{1}{2}} G^{\frac{1}{2}} \]
This gives:
\[ L = \sqrt{\frac{h G}{c^3}} \]

### Conclusion:
Thus, the dimensions of length in terms of the fundamental quantities \(c\), \(h\), and \(G\) is:
\[ L = \sqrt{\frac{h G}{c^3}} \]