A capillary of radius 0.15mm is dipped in liquid of density 'rho' = 667 kg/m^3. If surface tension of liquid is `1/20 N/m` then find height upto which liquid rises in capillary. Angle of contact between liquid and capillary tube is `60^@`. `(g=10m/s^2)`

Options

A. `0.01m`
B. `0.02m`
C. `0.04m`
D. `0.05m`

Answer

To solve the problem of how high the liquid rises in a capillary tube, we can use the formula for capillary rise:

\[ h = \frac{4 \sigma \cos \theta}{\rho g r} \]

where:
- \( h \) = height of the liquid column in the capillary
- \( \sigma \) = surface tension of the liquid
- \( \theta \) = angle of contact
- \( \rho \) = density of the liquid
- \( g \) = acceleration due to gravity
- \( r \) = radius of the capillary tube

### Step 1: Identify the given values
- Radius of the capillary tube, \( r = 0.15 \, \text{mm} = 0.15 \times 10^{-3} \, \text{m} \)
- Density of the liquid, \( \rho = 667 \, \text{kg/m}^3 \)
- Surface tension of the liquid, \( \sigma = \frac{1}{20} \, \text{N/m} \)
- Angle of contact, \( \theta = 60^\circ \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 2: Convert the angle to cosine
Using the angle of contact:
\[ \cos(60^\circ) = \frac{1}{2} \]

### Step 3: Substitute the values into the formula
Now substituting the values into the capillary rise formula:
\[ h = \frac{4 \times \left(\frac{1}{20}\right) \times \left(\frac{1}{2}\right)}{667 \times 10 \times (0.15 \times 10^{-3})} \]

### Step 4: Simplify the numerator
Calculating the numerator:
\[ 4 \times \frac{1}{20} \times \frac{1}{2} = \frac{4}{40} = \frac{1}{10} \]

### Step 5: Simplify the denominator
Calculating the denominator:
\[ 667 \times 10 \times (0.15 \times 10^{-3}) = 6670 \times 0.15 \times 10^{-3} = 1000.5 \times 10^{-3} = 1.0005 \]

### Step 6: Calculate height \( h \)
Now substituting back into the equation:
\[ h = \frac{\frac{1}{10}}{1.0005} \approx 0.0009995 \, \text{m} \]

### Step 7: Convert to centimeters
Converting to centimeters:
\[ h \approx 0.9995 \, \text{cm} \approx 0.01 \, \text{m} \]

### Final Answer
Thus, the height up to which the liquid rises in the capillary tube is approximately:
\[ h \approx 0.01 \, \text{m} \text{ or } 1 \, \text{cm} \]