The dimensional formula for Planck's constant and gravitational constant G respectively are :-

Options

A. `[ML^(3)T^(-2)], [M^(-1)L^(2)T^(-3)]`
B. `[ML^(2)T^(-1)], [M^(-1)L^(3)T^(-2)]`
C. `[ML^(3)T^(-2)], [M^(-1)L^(2)T^(2)]`
D. `[MLT^(-3)], [M^(-1)L^(3)T^(-3)]`

Explanation

To find the dimensional formulas for Planck's constant (h) and the gravitational constant (G), we can follow these steps:

### Step 1: Dimensional Formula for Planck's Constant (h)

1. **Understanding the relationship**: Planck's constant is defined through the equation \( E = h \nu \), where:
- \( E \) is the energy of a photon.
- \( \nu \) (nu) is the frequency of radiation.

2. **Rearranging the equation**: We can express Planck's constant as:
\[ h = \frac{E}{\nu} \]

3. **Finding the dimensions of energy (E)**: The dimensional formula for energy is:
\[ [E] = [\text{mass}] \times [\text{length}]^2 \times [\text{time}]^{-2} = M^1 L^2 T^{-2} \]

4. **Finding the dimensions of frequency (\( \nu \))**: Frequency is the reciprocal of time:
\[ [\nu] = T^{-1} \]

5. **Substituting the dimensions into the formula for h**:
\[ [h] = \frac{[E]}{[\nu]} = \frac{M^1 L^2 T^{-2}}{T^{-1}} = M^1 L^2 T^{-1} \]

### Step 2: Dimensional Formula for Gravitational Constant (G)

1. **Understanding the relationship**: The gravitational force \( F \) between two masses \( M \) and \( m \) separated by a distance \( r \) is given by:
\[ F = \frac{G M m}{r^2} \]

2. **Rearranging the equation**: We can express \( G \) as:
\[ G = \frac{F r^2}{M m} \]

3. **Finding the dimensions of force (F)**: The dimensional formula for force is:
\[ [F] = [\text{mass}] \times [\text{length}] \times [\text{time}]^{-2} = M^1 L^1 T^{-2} \]

4. **Finding the dimensions of \( r^2 \)**: The dimensional formula for distance squared is:
\[ [r^2] = [\text{length}]^2 = L^2 \]

5. **Finding the dimensions of mass (M and m)**: The dimensional formula for mass is:
\[ [M] = M^1 \quad \text{and} \quad [m] = M^1 \]

6. **Substituting the dimensions into the formula for G**:
\[ [G] = \frac{[F] [r^2]}{[M][m]} = \frac{(M^1 L^1 T^{-2})(L^2)}{(M^1)(M^1)} = \frac{M^1 L^3 T^{-2}}{M^2} = M^{-1} L^3 T^{-2} \]

### Final Result

- 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:
\[ [G] = M^{-1} L^3 T^{-2} \]