Using dimensional analysis, find out the correct formula for the capillary force acting on a spider leg with length l in contact with the lake water surface.

Steps to Solve

1. Understanding the Relationship

   The energy $E$ required to form new liquid surface $S$ is given by the equation: $$ E = \sigma \cdot S $$ where $\sigma$ is the surface tension and $S$ is the surface area.

2. Identify the Dimensions

   The dimensions of energy $E$ are $[E] = [M][L^2][T^{-2}]$. The dimensions of surface tension $\sigma$ are $[\sigma] = [M][T^{-2}][L^{-1}]$. The dimensions of surface area $S$ are $[S] = [L^2]$.

3. Relate Linear Dimensions to the Force

   The capillary force $F$ can be related to the surface tension and length $l$: $$ F = R \cdot \sigma $$ where $R$ is a length scale. From the dimensional consistency, we can set $R = l$, leading to: $$ F \propto \sigma \cdot l $$

4. Establish the Final Formula

   Considering that we can have a prefactor (which will be determined empirically or theoretically) and the dimensions, we have: $$ F = k \cdot \sigma \cdot l $$ where $k$ could be a constant factor based on geometrical considerations.

5. Select the Correct Answer

   From the given choices, we can assess the options. The option (A) $F = 2 \cdot \sigma \cdot l$ represents the relationship we derived.