Planck constant has the same dimensions as

Options

A. force `xx` time
B. force `xx` distance
C. force `xx` speed
D. force `xx` distance `xx` time.

Answer

To determine which of the given options has the same dimensions as Planck's constant, we need to analyze the dimensions of Planck's constant and each of the options provided.

### Step-by-Step Solution:

1. **Identify the Dimensions of Planck's Constant:**
- The unit of Planck's constant is Joules second (J·s).
- We know that 1 Joule (J) is equivalent to 1 kg·m²/s².
- Therefore, the dimensions of Planck's constant (h) can be expressed as:
\[ [h] = [J] \cdot [s] = \left[ \text{kg} \cdot \text{m}^2/\text{s}^2 \right] \cdot [\text{s}] = [\text{kg} \cdot \text{m}^2/\text{s}] = [M L^2 T^{-1}] \]

2. **Analyze Each Option:**
- **Option 1: Force × Time**
- The dimension of force (F) is:
\[ [F] = [M L T^{-2}] \]
- Therefore, the dimension of force multiplied by time is:
\[ [F] \cdot [T] = [M L T^{-2}] \cdot [T] = [M L T^{-1}] \]
- This does not match Planck's constant.

- **Option 2: Force × Distance**
- The dimension of distance is [L].
- Thus, the dimension of force multiplied by distance is:
\[ [F] \cdot [L] = [M L T^{-2}] \cdot [L] = [M L^2 T^{-2}] \]
- This does not match Planck's constant.

- **Option 3: Force × Speed**
- The dimension of speed is:
\[ [\text{Speed}] = [L T^{-1}] \]
- Therefore, the dimension of force multiplied by speed is:
\[ [F] \cdot [\text{Speed}] = [M L T^{-2}] \cdot [L T^{-1}] = [M L^2 T^{-3}] \]
- This does not match Planck's constant.

- **Option 4: Force × Distance × Time**
- The dimension of time is [T].
- Thus, the dimension of force multiplied by distance and time is:
\[ [F] \cdot [L] \cdot [T] = [M L T^{-2}] \cdot [L] \cdot [T] = [M L^2 T^{-1}] \]
- This matches the dimension of Planck's constant.

3. **Conclusion:**
- The correct option that has the same dimensions as Planck's constant is **Option 4: Force × Distance × Time**.