One end of a glass capillary tube with a radius `r=0.05cm` is immersed into water to a depth of `h=2cm`.Excess pressure required to blow an air bubble out of the lower end of the tube will be `(S.T` of water `=70"dyne"//cm`).Take `g=980cm//s^(2)`.

Options

A. `2840"dyne"//cm^(2)`
B. `5840"dyne"//cm^(2)`
C. `7840"dyne"//cm^(2)`
D. `4760"dyne"//cm^(2)`

Explanation

To solve the problem, we need to calculate the excess pressure required to blow an air bubble out of the lower end of a glass capillary tube immersed in water. The formula we will use is:

\[ P = \rho g h + \frac{2T}{r} \]

where:
- \( P \) is the excess pressure,
- \( \rho \) is the density of water,
- \( g \) is the acceleration due to gravity,
- \( h \) is the depth of immersion,
- \( T \) is the surface tension of water,
- \( r \) is the radius of the capillary tube.

### Step 1: Identify the given values
- Radius of the capillary tube, \( r = 0.05 \, \text{cm} \)
- Depth of immersion, \( h = 2 \, \text{cm} \)
- Surface tension of water, \( T = 70 \, \text{dyne/cm} \)
- Acceleration due to gravity, \( g = 980 \, \text{cm/s}^2 \)
- Density of water, \( \rho \approx 1 \, \text{g/cm}^3 = 980 \, \text{dyne/cm}^3 \)

### Step 2: Calculate the hydrostatic pressure component
Using the formula for hydrostatic pressure:
\[ P_{\text{hydrostatic}} = \rho g h \]
Substituting the known values:
\[ P_{\text{hydrostatic}} = 980 \, \text{dyne/cm}^3 \times 980 \, \text{cm/s}^2 \times 2 \, \text{cm} \]
Calculating this gives:
\[ P_{\text{hydrostatic}} = 980 \times 980 \times 2 = 1920800 \, \text{dyne/cm}^2 \]

### Step 3: Calculate the pressure due to surface tension
Using the formula for pressure due to surface tension:
\[ P_{\text{surface tension}} = \frac{2T}{r} \]
Substituting the known values:
\[ P_{\text{surface tension}} = \frac{2 \times 70 \, \text{dyne/cm}}{0.05 \, \text{cm}} \]
Calculating this gives:
\[ P_{\text{surface tension}} = \frac{140}{0.05} = 2800 \, \text{dyne/cm}^2 \]

### Step 4: Calculate the total excess pressure
Now, we sum the two pressure components to find the total excess pressure:
\[ P = P_{\text{hydrostatic}} + P_{\text{surface tension}} \]
Substituting the values we calculated:
\[ P = 1920800 + 2800 = 1923600 \, \text{dyne/cm}^2 \]

### Step 5: Final result
The excess pressure required to blow an air bubble out of the lower end of the tube is:
\[ P = 1923600 \, \text{dyne/cm}^2 \]