A quantity f is given by `f=sqrt((hc^(5))/(G))` where c is speed of light, G universal gravitational constant and h is the Planck's constant. Dimension of f is that of:

Options

A. Volume
B. energy
C. area
D. momentum

Answer

To determine the dimensions of the quantity \( f = \sqrt{\frac{hc^5}{G}} \), we will analyze the dimensions of each component involved in the formula.

### Step 1: Identify the dimensions of each constant

1. **Planck's constant \( h \)**:
- The dimension of Planck's constant is given by:
\[ [h] = [E][T] = [M][L^2][T^{-1}] \]
- Where \( [E] \) is energy, which has dimensions \( [M][L^2][T^{-2}] \), and \( [T] \) is time.

2. **Speed of light \( c \)**:
- The dimension of speed is:
\[ [c] = \frac{[L]}{[T]} \]

3. **Universal gravitational constant \( G \)**:
- The dimension of \( G \) is:
\[ [G] = \frac{[L^3]}{[M][T^2]} \]

### Step 2: Substitute the dimensions into the expression for \( f \)

Now, substituting these dimensions into the expression for \( f \):

\[ f = \sqrt{\frac{hc^5}{G}} \]

### Step 3: Calculate the dimensions of \( hc^5 \)

1. **Calculate \( c^5 \)**:
- The dimension of \( c^5 \) is:
\[ [c^5] = \left(\frac{[L]}{[T]}\right)^5 = \frac{[L^5]}{[T^5]} \]

2. **Combine \( h \) and \( c^5 \)**:
- Now, we can find the dimensions of \( hc^5 \):
\[ [hc^5] = [h][c^5] = [M][L^2][T^{-1}] \cdot \frac{[L^5]}{[T^5]} = [M][L^{7}][T^{-6}] \]

### Step 4: Calculate the dimensions of \( \frac{hc^5}{G} \)

Now we need to divide \( hc^5 \) by \( G \):

\[ \frac{hc^5}{G} = \frac{[M][L^{7}][T^{-6}]}{[G]} = \frac{[M][L^{7}][T^{-6}]}{\frac{[L^3]}{[M][T^2]}} = [M][L^{7}][T^{-6}] \cdot \frac{[M][T^2]}{[L^3]} = [M^2][L^{4}][T^{-4}] \]

### Step 5: Calculate the dimensions of \( f \)

Finally, we take the square root of the dimensions obtained:

\[ [f] = \sqrt{[M^2][L^{4}][T^{-4}]} = [M][L^{2}][T^{-2}] \]

### Conclusion

Thus, the dimension of \( f \) is:

\[ [f] = [M][L^{2}][T^{-2}] \]

This is the dimension of energy.