Step 1: Understanding the Concept:
Dimensional analysis is a tool used to express a physical quantity in terms of fundamental quantities like Mass (\(M\)), Length (\(L\)), and Time (\(T\)).
Surface tension (\(T\) or \(\sigma\)) is physically defined as the force acting per unit length along an imaginary line drawn on the surface of a liquid.
Alternatively, in terms of thermodynamics, surface tension is the energy required to increase the surface area of a liquid by one unit.
Both definitions lead to the same dimensional formula, which we will use to test the options. Step 2: Key Formula or Approach:
1. Surface Tension formula: \( \text{Surface Tension} = \frac{\text{Force}}{\text{Length}} \).
2. Dimensional formula of Force: \( [MLT^{-2}] \).
3. Dimensional formula of Length: \( [L] \).
4. Dimensional formula of Surface Tension: \( \frac{[MLT^{-2}]}{[L]} = [MT^{-2}] \).
We must check each option to find which one results in \( [MT^{-2}] \). Step 3: Detailed Explanation:
- Option (A): Force \(\times\) Length
Dimensions: \( [MLT^{-2}] \times [L] = [ML^2T^{-2}] \).
This is the dimensional formula for Torque, Work, or Energy. It does not match.
- Option (B): Energy / Area
Dimensions of Energy: \( [ML^2T^{-2}] \).
Dimensions of Area: \( [L^2] \).
Dividing them: \( \frac{[ML^2T^{-2}]}{[L^2]} = [MT^{-2}] \).
This matches the dimensions of surface tension exactly. Physically, this represents Surface Energy density.
- Option (C): Pressure \(\times\) Length
Dimensions of Pressure: \( \frac{\text{Force}}{\text{Area}} = \frac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}] \).
Dimensions of Pressure \(\times\) Length: \( [ML^{-1}T^{-2}] \times [L] = [MT^{-2}] \).
Dimensionally, this is correct, but "Energy per unit Area" is the standard definition related to surface tension in thermodynamics textbooks. In multi-choice exams, the more conceptually fundamental definition is preferred. However, we note the dimensional equivalence.
- Option (D): Work \(\times\) Distance
Dimensions: \( [ML^2T^{-2}] \times [L] = [ML^3T^{-2}] \).
This does not match. Step 4: Final Answer:
By calculation, Energy per unit Area has the dimensions \( [MT^{-2}] \), which is identical to the dimensions of surface tension.
Therefore, the correct option is (B).
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