If power (P), surface tension (S) and Planck's constant (h) are arranged so that the dimensions of time in their dimensional formulae are in ascending order, then which of the following is correct?

Options

A. P,S,h
B. P,h,S
C. S,P,h
D. S,h,P

Explanation

To solve the problem, we need to find the dimensional formulas for power (P), surface tension (S), and Planck's constant (h), and then identify the dimensions of time in each of these formulas to arrange them in ascending order.

### Step-by-Step Solution:

1. **Find the Dimensional Formula of Power (P)**:
- Power is defined as work done per unit time.
- The dimensional formula for work (W) is given by:
\[ W = \text{Force} \times \text{Distance} = (mL T^{-2}) \times L = mL^2 T^{-2} \]
- Therefore, the dimensional formula for power is:
\[ P = \frac{W}{t} = \frac{mL^2 T^{-2}}{T} = mL^2 T^{-3} \]

2. **Find the Dimensional Formula of Surface Tension (S)**:
- Surface tension is defined as force per unit length.
- The dimensional formula for force is:
\[ \text{Force} = mL T^{-2} \]
- Hence, the dimensional formula for surface tension is:
\[ S = \frac{\text{Force}}{\text{Length}} = \frac{mL T^{-2}}{L} = mT^{-2} \]

3. **Find the Dimensional Formula of Planck's Constant (h)**:
- Planck's constant is defined in the context of energy and frequency:
\[ E = h \nu \quad \text{(where } \nu \text{ is frequency)} \]
- Rearranging gives:
\[ h = \frac{E}{\nu} \]
- The dimensional formula for energy (E) is:
\[ E = mL^2 T^{-2} \]
- The frequency (ν) has the dimension of time as:
\[ \nu = T^{-1} \]
- Thus, the dimensional formula for Planck's constant is:
\[ h = \frac{mL^2 T^{-2}}{T^{-1}} = mL^2 T^{-1} \]

4. **Extract the Dimensions of Time**:
- Now we summarize the dimensions of time from the formulas we derived:
- For Power (P): \( mL^2 T^{-3} \) → Time dimension = -3
- For Surface Tension (S): \( mT^{-2} \) → Time dimension = -2
- For Planck's Constant (h): \( mL^2 T^{-1} \) → Time dimension = -1

5. **Arrange in Ascending Order**:
- The dimensions of time in ascending order are:
- Surface Tension (S): \( T^{-2} \)
- Power (P): \( T^{-3} \)
- Planck's Constant (h): \( T^{-1} \)

Thus, the correct arrangement of the dimensions of time in ascending order is:
- Surface Tension (S) < Power (P) < Planck's Constant (h)

### Final Answer:
The correct order is: Surface Tension (S) < Power (P) < Planck's Constant (h).