An open cappillary tube contains a drop of water. The internal diameter of the capillary tube is `1mm`. Daternube tge radii if curvature of the upper and lower meniscuses in each case. Consider the wetting to be complete. Surface tension of water `= 0.073 N//m`. `(g = 9.8 m//s^(2))`
When the tube is in its vertical position, the drop froms a column witha length of `2.98 cm`.

Options

A. `0.5 mm, 1.52 mm`
B. `0.5 mm, 1.46 mm`
C. `0.5 mm`, lower surface will be flat
D. `0.4 mm, 1.46 mm`

Explanation

To determine the radii of curvature of the upper and lower meniscuses in a capillary tube containing water, we can follow these steps:

### Step 1: Understand the Geometry of the Meniscus
In a capillary tube, when wetting is complete, the upper meniscus (the surface of the water at the top) is concave downwards, and the lower meniscus (the surface of the water at the bottom) is flat.

### Step 2: Identify the Given Data
- Internal diameter of the capillary tube, \( D = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Radius of the capillary tube, \( r = \frac{D}{2} = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \)
- Surface tension of water, \( T = 0.073 \, \text{N/m} \)
- Length of the water column, \( h = 2.98 \, \text{cm} = 2.98 \times 10^{-2} \, \text{m} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

### Step 3: Calculate the Pressure due to Surface Tension at the Upper Meniscus
The pressure difference due to surface tension at the upper meniscus is given by:
\[ P_s = \frac{2T}{r_1} \]
where \( r_1 \) is the radius of curvature of the upper meniscus.

### Step 4: Calculate the Hydrostatic Pressure at the Height of the Water Column
The hydrostatic pressure at the height of the water column can be calculated using:
\[ P_2 = \rho h g \]
where \( \rho \) (density of water) is approximately \( 1000 \, \text{kg/m}^3 \).

Substituting the values:
\[ P_2 = 1000 \times (2.98 \times 10^{-2}) \times 9.8 \]
Calculating \( P_2 \):
\[ P_2 = 1000 \times 0.0298 \times 9.8 = 292 \, \text{N/m}^2 \]

### Step 5: Set the Pressures Equal
Since the pressure due to surface tension at the upper meniscus must equal the hydrostatic pressure:
\[ P_s = P_2 \]
\[ \frac{2T}{r_1} = 292 \]

### Step 6: Solve for the Radius of Curvature of the Upper Meniscus
Substituting the value of surface tension \( T \):
\[ \frac{2 \times 0.073}{r_1} = 292 \]
\[ r_1 = \frac{2 \times 0.073}{292} \]
Calculating \( r_1 \):
\[ r_1 = \frac{0.146}{292} \approx 0.0005 \, \text{m} = 0.5 \, \text{mm} \]

### Step 7: Determine the Radius of Curvature of the Lower Meniscus
Since the lower meniscus is flat, the radius of curvature \( r_2 \) is considered to be infinite:
\[ r_2 = \infty \]

### Final Results
- Radius of curvature of the upper meniscus, \( r_1 = 0.5 \, \text{mm} \)
- Radius of curvature of the lower meniscus, \( r_2 = \infty \)