Water rises up to a height h in a capillary tube of certain diameter. This capillary tube is replaced by a similar tube of half the diameter. Now the water will rise to the height of

Options

A. 4 h
B. 3 h
C. 2 h
D. `h/2`

Answer

To solve the problem of how high water will rise in a capillary tube when the diameter is halved, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Condition**:
- Let the initial height of water in the capillary tube be \( h \).
- Let the radius of the initial tube be \( R \).

2. **Determine the New Tube's Radius**:
- When the diameter is halved, the new radius \( R' \) becomes:
\[ R' = \frac{R}{2} \]

3. **Apply the Capillary Rise Formula**:
- The height of liquid rise in a capillary tube is given by the formula:
\[ h \propto \frac{1}{r} \]
- This means that the height of the liquid is inversely proportional to the radius of the tube.

4. **Set Up the Proportional Relationship**:
- Let \( h' \) be the new height of water in the tube with radius \( R' \). From the proportional relationship, we can write:
\[ \frac{h'}{h} = \frac{R}{R'} \]

5. **Substitute the New Radius**:
- Substitute \( R' = \frac{R}{2} \) into the equation:
\[ \frac{h'}{h} = \frac{R}{\frac{R}{2}} = 2 \]

6. **Solve for the New Height**:
- Rearranging gives:
\[ h' = 2h \]

7. **Conclusion**:
- Therefore, when the diameter of the capillary tube is halved, the height to which water rises in the tube will be:
\[ h' = 2h \]

### Final Answer:
The water will rise to a height of \( 2h \).

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