A glass rod of radius `1 mm` is inserted symmetrically into a glass capillary tube with inside radius `2 mm`. Then the whole arrangement is brought in contact of the surface of water. Surface tension of water is `7 xx 10^(-2) N//m`. To what height will the water rise in the capillary? (`theta = 0^@`)

Options

A. `1.4cm`
B. `4.2cm`
C. `2.1cm`
D. `6.8cm`

Explanation

To determine the height to which water will rise in the capillary tube when a glass rod is inserted, we can use the capillary rise formula. The formula for the height of liquid rise in a capillary tube is given by:

\[ h = \frac{2 \gamma \cos \theta}{\rho g r} \]

Where:
- \( h \) = height of the liquid column
- \( \gamma \) = surface tension of the liquid (water in this case)
- \( \theta \) = angle of contact (given as \( 0^\circ \))
- \( \rho \) = density of the liquid (density of water is approximately \( 1000 \, kg/m^3 \))
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, m/s^2 \))
- \( r \) = effective radius of the capillary tube

### Step 1: Identify the parameters
- Given:
- Surface tension of water, \( \gamma = 7 \times 10^{-2} \, N/m \)
- Angle of contact, \( \theta = 0^\circ \) (thus, \( \cos \theta = 1 \))
- Density of water, \( \rho = 1000 \, kg/m^3 \)
- Acceleration due to gravity, \( g = 9.81 \, m/s^2 \)
- Radius of the capillary tube, \( R = 2 \, mm = 2 \times 10^{-3} \, m \)

### Step 2: Calculate the effective radius
Since a glass rod of radius \( r = 1 \, mm = 1 \times 10^{-3} \, m \) is inserted into the capillary tube, the effective radius \( r \) for the capillary action will be the radius of the capillary tube minus the radius of the rod:

\[ r = R - r_{rod} = (2 \times 10^{-3} \, m) - (1 \times 10^{-3} \, m) = 1 \times 10^{-3} \, m \]

### Step 3: Substitute values into the formula
Now, substituting the values into the capillary rise formula:

\[ h = \frac{2 \times (7 \times 10^{-2}) \times 1}{1000 \times 9.81 \times (1 \times 10^{-3})} \]

### Step 4: Calculate the height
Calculating the numerator:

\[ 2 \times (7 \times 10^{-2}) \times 1 = 0.14 \]

Now calculating the denominator:

\[ 1000 \times 9.81 \times (1 \times 10^{-3}) = 9.81 \]

Now substituting back into the height formula:

\[ h = \frac{0.14}{9.81} \approx 0.01426 \, m \]

### Step 5: Convert to centimeters
To convert meters to centimeters:

\[ h \approx 0.01426 \, m \times 100 \approx 1.426 \, cm \]

### Final Result
The height to which water will rise in the capillary tube is approximately:

\[ h \approx 1.43 \, cm \]

### Conclusion
The final answer is approximately **1.4 cm**.

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