The dimension of Planck's constant are the same as that of

Options

A. Energy
B. Momentum
C. Angular momentum
D. Power

Answer

To determine the dimensions of Planck's constant and compare it with the given options, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of energy**:
- The dimension of energy (E) is given by the formula:
\[ [E] = [M][L^2][T^{-2}] \]
- Therefore, the dimension of energy is \( [M^1 L^2 T^{-2}] \).

2. **Identify the dimensions of momentum**:
- Momentum (p) is defined as mass times velocity:
\[ [p] = [M][V] = [M][L][T^{-1}] \]
- Thus, the dimension of momentum is \( [M^1 L^1 T^{-1}] \).

3. **Identify the dimensions of angular momentum**:
- Angular momentum (L) is defined as mass times velocity times radius:
\[ [L] = [M][V][R] = [M][L][T^{-1}][L] = [M][L^2][T^{-1}] \]
- Therefore, the dimension of angular momentum is \( [M^1 L^2 T^{-1}] \).

4. **Identify the dimensions of power**:
- Power (P) is defined as the rate of work done, which is energy per unit time:
\[ [P] = \frac{[E]}{[T]} = \frac{[M^1 L^2 T^{-2}]}{[T]} = [M^1 L^2 T^{-3}] \]
- Hence, the dimension of power is \( [M^1 L^2 T^{-3}] \).

5. **Calculate the dimensions of Planck's constant**:
- Planck's constant (h) can be derived from the equation for the energy of a photon:
\[ E = h \nu \]
- Rearranging gives:
\[ h = \frac{E}{\nu} \]
- The dimension of frequency (ν) is the reciprocal of time:
\[ [\nu] = [T^{-1}] \]
- Thus, the dimension of Planck's constant is:
\[ [h] = \frac{[E]}{[\nu]} = \frac{[M^1 L^2 T^{-2}]}{[T^{-1}]} = [M^1 L^2 T^{-1}] \]

6. **Compare dimensions**:
- From the calculations, we have:
- Energy: \( [M^1 L^2 T^{-2}] \)
- Momentum: \( [M^1 L^1 T^{-1}] \)
- Angular Momentum: \( [M^1 L^2 T^{-1}] \)
- Power: \( [M^1 L^2 T^{-3}] \)
- The dimension of Planck's constant \( [M^1 L^2 T^{-1}] \) matches with the dimension of angular momentum.

### Conclusion:
The dimension of Planck's constant is the same as that of angular momentum. Therefore, the correct option is **option c**.