The rise in the water level in a capillary tube of radius 0.07 cm when dipped vertically in a beaker containing water of surface tension `0.07 N m^(-1)` is (g = `10 m s^(-2)`)

Options

A. 2 cm
B. 4 cm
C. 1.5 cm
D. 3 cm

Explanation

To find the rise in the water level in a capillary tube, we can use the formula for capillary rise:

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

Where:
- \( h \) = height of the liquid column (rise in water level)
- \( \gamma \) = surface tension of the liquid (N/m)
- \( \theta \) = angle of contact (degrees)
- \( \rho \) = density of the liquid (kg/m³)
- \( g \) = acceleration due to gravity (m/s²)
- \( r \) = radius of the capillary tube (m)

### Step 1: Identify the given values
- Surface tension, \( \gamma = 0.07 \, \text{N/m} \)
- Radius of the capillary tube, \( r = 0.07 \, \text{cm} = 0.07 \times 10^{-2} \, \text{m} = 0.0007 \, \text{m} \)
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
- Angle of contact, \( \theta = 0^\circ \) (thus, \( \cos \theta = 1 \))

### Step 2: Substitute the values into the formula
Now we can substitute the values into the formula:

\[ h = \frac{2 \times 0.07 \times 1}{1000 \times 10 \times 0.0007} \]

### Step 3: Calculate the denominator
Calculate the denominator:

\[ 1000 \times 10 \times 0.0007 = 7 \]

### Step 4: Calculate the numerator
Calculate the numerator:

\[ 2 \times 0.07 \times 1 = 0.14 \]

### Step 5: Calculate the height \( h \)
Now substitute the numerator and denominator back into the equation:

\[ h = \frac{0.14}{7} = 0.02 \, \text{m} \]

### Step 6: Convert to centimeters
Convert the height from meters to centimeters:

\[ h = 0.02 \, \text{m} = 2 \, \text{cm} \]

### Final Answer
The rise in the water level in the capillary tube is \( 2 \, \text{cm} \).

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