Water rises in a capillary tube upto a height 'h' so that the upward force due to S.T is balanced by the force due to the weight of the water column. If this force is 90 dyne and surface tension of water is `6xx10^(-2)` N/m then the inner circumference of the capillary must be

Options

A. `0.75 xx10^(-2)` m
B. `1.5 xx10^(-2)` m
C. `15xx10^(-2)` m
D. `0.5 xx10^(-2)`m

Explanation

To solve the problem, we need to find the inner circumference of a capillary tube given the upward force due to surface tension and the surface tension of water.

### Step-by-Step Solution:

1. **Understand the Forces Involved**:
The upward force due to surface tension (F) is balanced by the weight of the water column (mg). Therefore, we can write:
\[ F = mg \]

2. **Convert the Given Force**:
We are given that the force (F) is 90 dyne. To convert this to Newtons, we use the conversion factor \(1 \text{ dyne} = 10^{-5} \text{ N}\):
\[ F = 90 \text{ dyne} = 90 \times 10^{-5} \text{ N} = 9.0 \times 10^{-4} \text{ N} \]

3. **Use the Surface Tension Formula**:
The force due to surface tension can also be expressed as:
\[ F = T \cdot L \]
where \(T\) is the surface tension and \(L\) is the length along which the surface tension acts. In the case of a capillary tube, this length is the inner circumference of the tube, which can be expressed as:
\[ L = 2\pi r \]
Therefore, we can rewrite the equation as:
\[ F = T \cdot (2\pi r) \]

4. **Substitute Known Values**:
We know the surface tension \(T = 6 \times 10^{-2} \text{ N/m}\) and the force \(F = 9.0 \times 10^{-4} \text{ N}\). Substituting these values into the equation gives:
\[ 9.0 \times 10^{-4} = (6 \times 10^{-2}) \cdot (2\pi r) \]

5. **Solve for the Inner Circumference**:
Rearranging the equation to solve for \(2\pi r\):
\[ 2\pi r = \frac{9.0 \times 10^{-4}}{6 \times 10^{-2}} \]
\[ 2\pi r = \frac{9.0}{6} \times 10^{-2} \text{ m} \]
\[ 2\pi r = 1.5 \times 10^{-2} \text{ m} \]

6. **Final Result**:
Thus, the inner circumference of the capillary tube is:
\[ 2\pi r = 1.5 \times 10^{-2} \text{ m} \]