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

Options

A. charge
B. work
C. force
D. Angular momentum

Explanation

To determine the dimension of Planck's constant and compare it with the dimensions of charge, work, force, and angular momentum, we will follow these steps:

### Step 1: Write the formula for Planck's constant
Planck's constant (H) is defined as the energy (E) divided by the frequency (ν):
\[ H = \frac{E}{ν} \]

### Step 2: Write the dimensions of energy (E)
Energy has the dimensions of work, which can be expressed as:
\[ \text{Work} = \text{Force} \times \text{Distance} \]
The dimension of force is given by:
\[ \text{Force} = \text{mass} \times \text{acceleration} = m \cdot \frac{l}{t^2} \]
Thus, the dimensions of work become:
\[ \text{Work} = m \cdot \frac{l}{t^2} \cdot l = m \cdot l^2 \cdot t^{-2} \]
So, the dimensions of energy (E) are:
\[ [E] = [m \cdot l^2 \cdot t^{-2}] \]

### Step 3: Write the dimensions of frequency (ν)
Frequency is defined as the number of cycles per unit time:
\[ ν = \frac{1}{T} \]
Thus, the dimensions of frequency are:
\[ [ν] = [t^{-1}] \]

### Step 4: Substitute the dimensions into the formula for Planck's constant
Now, substituting the dimensions of energy and frequency into the formula for Planck's constant:
\[ [H] = \frac{[E]}{[ν]} = \frac{[m \cdot l^2 \cdot t^{-2}]}{[t^{-1}]} \]
This simplifies to:
\[ [H] = m \cdot l^2 \cdot t^{-2} \cdot t = m \cdot l^2 \cdot t^{-1} \]

### Step 5: Write the dimensions of the other quantities
1. **Charge (Q)**:
\[ [Q] = [I] \cdot [t] = [A] \cdot [t] \]
(where A is the dimension of electric current)

2. **Work**:
\[ [\text{Work}] = m \cdot l^2 \cdot t^{-2} \]

3. **Force**:
\[ [\text{Force}] = m \cdot l \cdot t^{-2} \]

4. **Angular momentum (L)**:
\[ [L] = [m \cdot l^2 \cdot t^{-1}] \]

### Step 6: Compare the dimensions
Now we compare the dimensions:
- **Planck's constant**: \( [H] = m \cdot l^2 \cdot t^{-1} \)
- **Charge**: \( [Q] = [A] \cdot [t] \) (not the same)
- **Work**: \( [\text{Work}] = m \cdot l^2 \cdot t^{-2} \) (not the same)
- **Force**: \( [\text{Force}] = m \cdot l \cdot t^{-2} \) (not the same)
- **Angular momentum**: \( [L] = m \cdot l^2 \cdot t^{-1} \) (same as Planck's constant)

### Conclusion
The dimensions of Planck's constant are the same as that of angular momentum. Therefore, the correct answer is:
**Option D: Angular momentum**.

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