When a capillary is dipped in water, water rises 0.015 m in it. If the surface tension of water is `75 xx 10^(-3) N//m` , the radius of capillary is

Options

A. `0.1 mm`
B. `0.5 mm`
C. `1mm`
D. `2 mm`

Answer

To find the radius of the capillary tube when water rises in it, we can use the formula for capillary rise:

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

Where:
- \( h \) = height of the liquid column (in meters)
- \( T \) = surface tension of the liquid (in N/m)
- \( r \) = radius of the capillary tube (in meters)
- \( \rho \) = density of the liquid (in kg/m³)
- \( g \) = acceleration due to gravity (in m/s²)

Given:
- \( h = 0.015 \, \text{m} \)
- \( T = 75 \times 10^{-3} \, \text{N/m} \)
- \( \rho = 1000 \, \text{kg/m}^3 \) (density of water)
- \( g = 10 \, \text{m/s}^2 \)

### Step 1: Rearranging the formula to find the radius \( r \)

We can rearrange the formula to solve for \( r \):

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

### Step 2: Substitute the known values into the equation

Now we can substitute the known values into the equation:

\[ r = \frac{2 \times (75 \times 10^{-3})}{0.015 \times 1000 \times 10} \]

### Step 3: Calculate the numerator

Calculating the numerator:

\[ 2 \times (75 \times 10^{-3}) = 150 \times 10^{-3} = 0.150 \, \text{N/m} \]

### Step 4: Calculate the denominator

Calculating the denominator:

\[ 0.015 \times 1000 \times 10 = 150 \, \text{N/m} \]

### Step 5: Substitute the results back into the equation

Now substituting the results back into the equation for \( r \):

\[ r = \frac{0.150}{150} \]

### Step 6: Simplify the fraction

Now we simplify the fraction:

\[ r = 0.001 \, \text{m} \]

### Step 7: Convert to millimeters

To convert meters to millimeters:

\[ r = 0.001 \, \text{m} = 1 \, \text{mm} \]

### Final Answer

The radius of the capillary tube is \( 1 \, \text{mm} \).

---