A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water. The water rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of water is `sigma`. The angle of contact between water and the wall of the capillary tube is `theta`. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

Options

A. For a given material of the capillary tube, h decreases with increase in r
B. For a given material of the capillary tube, h is independent of ` sigma `
C. If this experiment is performed in a lift going up with a constant acceleration, then h decreases
D. h is proportional to contact angle ` theta `

Explanation

To solve the problem regarding the capillary rise of water in a uniform capillary tube, we will analyze the given statements based on the formula for capillary rise.

### Step-by-Step Solution:

1. **Understand the Capillary Rise Formula**:
The height \( h \) to which the liquid rises in a capillary tube is given by the formula:
\[ h = \frac{2\sigma \cos \theta}{\rho r g} \]
where:
- \( \sigma \) = surface tension of the liquid,
- \( \theta \) = angle of contact,
- \( \rho \) = density of the liquid,
- \( r \) = radius of the capillary tube,
- \( g \) = acceleration due to gravity.

2. **Analyze Statement A**:
- Statement: "For a given material capillary tube, \( h \) decreases with increase in \( r \)."
- From the formula, we see that \( h \) is inversely proportional to \( r \) (since \( \sigma, \theta, \rho, g \) are constants for a given material). Therefore, as \( r \) increases, \( h \) decreases.
- **Conclusion**: Statement A is **True**.

3. **Analyze Statement B**:
- Statement: "For a given material, \( h \) is independent of \( \sigma \)."
- From the formula, \( h \) is directly proportional to \( \sigma \). Therefore, if \( \sigma \) changes, \( h \) will also change.
- **Conclusion**: Statement B is **False**.

4. **Analyze Statement C**:
- Statement: "If this experiment is performed in a lift going up with constant acceleration, then \( h \) would decrease."
- In a lift accelerating upwards, the effective gravitational acceleration \( g' \) becomes \( g + a \) (where \( a \) is the acceleration of the lift). Since \( h \) is inversely proportional to \( g' \), if \( g' \) increases, \( h \) will decrease.
- **Conclusion**: Statement C is **True**.

5. **Analyze Statement D**:
- Statement: "H is proportional to the contact angle."
- From the formula, \( h \) is proportional to \( \cos \theta \), not \( \theta \) itself. Therefore, this statement is incorrect.
- **Conclusion**: Statement D is **False**.

### Final Conclusions:
- The true statements are:
- A: True
- B: False
- C: True
- D: False

### Summary of Correct Options:
- The correct options are A and C.