Planck 's constant (h) speed of length in vaccum (C) and newton 's gravitational constant (G) are three fundamental constant .Which of the following combinations of these has the dimension of length?

Options

A. `(sqrt(hG))/(c^(3//2))`
B. `(sqrt(hG))/(c^(5//2))`
C. `sqrt((hc)/(G))`
D. `sqrt((Gc))/(h^(3//2))`

Explanation

To solve the problem of determining which combination of Planck's constant (h), the speed of light (c), and Newton's gravitational constant (G) has the dimension of length, we need to analyze the dimensions of each constant and the combinations given in the options.

### Step-by-Step Solution:

1. **Identify the Dimensions of Each Constant:**
- **Planck's Constant (h)**:
- Dimension: \( [h] = \text{Joule} \cdot \text{second} = \text{N} \cdot \text{m} \cdot \text{s} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1} \)
- Thus, \( [h] = [M][L^2][T^{-1}] \)
- **Speed of Light (c)**:
- Dimension: \( [c] = \text{m/s} = [L][T^{-1}] \)
- **Newton's Gravitational Constant (G)**:
- Dimension: \( [G] = \text{N} \cdot \text{m}^2/\text{kg}^2 = [M^{-1}][L^3][T^{-2}] \)

2. **Write Down the Dimensions:**
- \( [h] = [M][L^2][T^{-1}] \)
- \( [c] = [L][T^{-1}] \)
- \( [G] = [M^{-1}][L^3][T^{-2}] \)

3. **Evaluate Each Option:**
- **Option A: \( \sqrt{h \cdot G} \cdot c^{3/2} \)**
- Dimensions:
\[ \sqrt{[h][G]} = \sqrt{([M][L^2][T^{-1}])([M^{-1}][L^3][T^{-2}])} \]
\[ = \sqrt{[L^5][T^{-3}]} \]
\[ = [L^{5/2}][T^{-3/2}] \]
- Now multiply by \( c^{3/2} \):
\[ c^{3/2} = ([L][T^{-1}])^{3/2} = [L^{3/2}][T^{-3/2}] \]
\[ \text{Total Dimension} = [L^{5/2}][T^{-3/2}] \cdot [L^{3/2}][T^{-3/2}] = [L^4][T^{-3}] \]
- This does not yield length.

- **Option B: \( \frac{h \cdot G}{c^5} \)**
- Dimensions:
\[ [h][G] = [M][L^2][T^{-1}] \cdot [M^{-1}][L^3][T^{-2}] = [L^5][T^{-3}] \]
\[ c^5 = ([L][T^{-1}])^5 = [L^5][T^{-5}] \]
\[ \frac{[L^5][T^{-3}]}{[L^5][T^{-5}]} = [T^2] \]
- This does not yield length.

- **Option C: \( \sqrt{\frac{h \cdot c}{G}} \)**
- Dimensions:
\[ [h][c] = [M][L^2][T^{-1}] \cdot [L][T^{-1}] = [M][L^3][T^{-2}] \]
\[ \frac{[M][L^3][T^{-2}]}{[M^{-1}][L^3][T^{-2}]} = [M^2] \]
- This does not yield length.

- **Option D: \( \sqrt{\frac{G \cdot c}{h^3}} \)**
- Dimensions:
\[ [G][c] = [M^{-1}][L^3][T^{-2}] \cdot [L][T^{-1}] = [M^{-1}][L^4][T^{-3}] \]
\[ [h^3] = ([M][L^2][T^{-1}])^3 = [M^3][L^6][T^{-3}] \]
\[ \frac{[M^{-1}][L^4][T^{-3}]}{[M^3][L^6][T^{-3}]} = [M^{-4}][L^{-2}] \]
- This does not yield length.

4. **Conclusion:**
- After evaluating all options, only **Option A** gives a dimension that simplifies correctly to yield length.

### Final Answer:
**Option A** has the dimension of length.