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 of finding the excess pressure required to blow an air bubble out of the lower end of a capillary tube, we can follow these steps:

### Step 1: Understand the Given Data
- Radius of the capillary tube, \( r = 0.05 \, \text{cm} \)
- Depth of immersion in water, \( h = 2 \, \text{cm} \)
- Surface tension of water, \( T = 70 \, \text{dyne/cm} \)
- Acceleration due to gravity, \( g = 980 \, \text{cm/s}^2 \)

### Step 2: Calculate the Hydrostatic Pressure
The hydrostatic pressure at the depth \( h \) can be calculated using the formula:
\[ P_{\text{hydrostatic}} = \rho g h \]
Where:
- \( \rho \) (density of water) is approximately \( 1 \, \text{g/cm}^3 \) or \( 1 \, \text{dyne/cm}^3 \).

Substituting the values:
\[ P_{\text{hydrostatic}} = 1 \, \text{dyne/cm}^3 \times 980 \, \text{cm/s}^2 \times 2 \, \text{cm} = 1960 \, \text{dyne/cm}^2 \]

### Step 3: Calculate the Pressure Due to Surface Tension
The excess pressure required to create a bubble at the lower end of the tube can be calculated using the formula:
\[ P_{\text{excess}} = \frac{2T}{r} \]
Substituting the values:
\[ P_{\text{excess}} = \frac{2 \times 70 \, \text{dyne/cm}}{0.05 \, \text{cm}} = \frac{140 \, \text{dyne/cm}}{0.05 \, \text{cm}} = 2800 \, \text{dyne/cm}^2 \]

### Step 4: Calculate the Total Excess Pressure
The total excess pressure required to blow the air bubble out of the tube is the sum of the hydrostatic pressure and the pressure due to surface tension:
\[ P_{\text{total}} = P_{\text{hydrostatic}} + P_{\text{excess}} \]
Substituting the values:
\[ P_{\text{total}} = 1960 \, \text{dyne/cm}^2 + 2800 \, \text{dyne/cm}^2 = 4760 \, \text{dyne/cm}^2 \]

### Final Answer
The excess pressure required to blow an air bubble out of the lower end of the tube is:
\[ \boxed{4760 \, \text{dyne/cm}^2} \]