If a `5 cm` long capillary tube with `0.1 mm` internal diameter, open at both ends, is slightly dipped in water having surface tension `75 dyn//cm`, state whether:
(i) water will rise halfway in the capillary,
(ii) water will rise up to the upper end of capillary,
(iii) water will overflow out of the upper end of capillary.
Explain your answer.

Explanation

To solve the problem, we will analyze the behavior of water in a capillary tube based on the principles of capillarity. The height to which the liquid will rise in a capillary tube can be calculated using the formula:

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

Where:
- \( h \) = height of the liquid column
- \( T \) = surface tension of the liquid (in dyn/cm)
- \( \theta \) = angle of contact (assumed to be 0 for water in a glass tube, meaning it wets the surface)
- \( r \) = radius of the capillary tube (in cm)
- \( \rho \) = density of the liquid (for water, approximately \( 1 \text{ g/cm}^3 \) or \( 1000 \text{ kg/m}^3 \))
- \( g \) = acceleration due to gravity (approximately \( 980 \text{ cm/s}^2 \))

### Step-by-Step Solution:

1. **Convert the diameter to radius:**
The internal diameter of the capillary tube is given as \( 0.1 \text{ mm} \).
\[ \text{Radius} (r) = \frac{0.1 \text{ mm}}{2} = 0.05 \text{ mm} = 0.005 \text{ cm} \]

2. **Identify the values:**
- Surface tension \( T = 75 \text{ dyn/cm} \)
- Density of water \( \rho = 1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3 \) (but we will use \( 1 \text{ g/cm}^3 \) in our calculations)
- Acceleration due to gravity \( g = 980 \text{ cm/s}^2 \)

3. **Calculate the height \( h \):**
Using the formula for capillary rise:
\[ h = \frac{2T \cos \theta}{r \rho g} \]
Assuming \( \cos \theta = 1 \) (since water wets the glass):
\[ h = \frac{2 \times 75 \text{ dyn/cm} \times 1}{0.005 \text{ cm} \times 1 \text{ g/cm}^3 \times 980 \text{ cm/s}^2} \]
\[ h = \frac{150}{0.005 \times 980} \]
\[ h = \frac{150}{4.9} \approx 30.61 \text{ cm} \]

4. **Compare the height with the length of the capillary tube:**
The length of the capillary tube is \( 5 \text{ cm} \). The calculated height \( h \approx 30.61 \text{ cm} \) is greater than the length of the capillary tube.

5. **Determine the behavior of water in the capillary tube:**
Since the height of the liquid column exceeds the length of the capillary tube, water will rise to the upper end of the capillary tube but will not overflow because it is open at both ends. The liquid will stabilize at the top of the tube.

### Conclusion:
(i) Water will not rise halfway in the capillary.
(ii) Water will rise up to the upper end of the capillary.
(iii) Water will not overflow out of the upper end of the capillary.