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. 4h
B. 3h
C. 2h
D. h

Answer

To solve the problem of how high water will rise in a capillary tube when the diameter is halved, we can use the formula related to capillary action. The height \( h \) of the liquid column in a capillary tube is given by the formula:

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

where:
- \( T \) is the surface tension of the liquid,
- \( \theta \) is the contact angle,
- \( \rho \) is the density of the liquid,
- \( g \) is the acceleration due to gravity,
- \( r \) is the radius of the capillary tube.

### Step 1: Understand the relationship between height and radius
From the formula, we can see that the height \( h \) is inversely proportional to the radius \( r \). This means:

\[ h \propto \frac{1}{r} \]

### Step 2: Set up the initial conditions
Let’s denote the initial height of water in the first capillary tube as \( h_1 \) and the radius of the first tube as \( r_1 \). Therefore, we can write:

\[ h_1 = \frac{2T \cos \theta}{\rho g r_1} \]

### Step 3: Set up the conditions for the second tube
Now, we replace the first tube with a second tube that has half the diameter. Since the diameter is halved, the radius \( r_2 \) of the second tube will be:

\[ r_2 = \frac{r_1}{2} \]

### Step 4: Calculate the new height
Using the same formula for the second tube, we can express the height \( h_2 \) as:

\[ h_2 = \frac{2T \cos \theta}{\rho g r_2} \]

Substituting \( r_2 \):

\[ h_2 = \frac{2T \cos \theta}{\rho g \left(\frac{r_1}{2}\right)} = \frac{2T \cos \theta \cdot 2}{\rho g r_1} = \frac{4T \cos \theta}{\rho g r_1} \]

### Step 5: Relate the new height to the initial height
Now, we can relate \( h_2 \) to \( h_1 \):

\[ h_2 = 2 \cdot \frac{2T \cos \theta}{\rho g r_1} = 2h_1 \]

### Conclusion
Thus, if the height of water in the first capillary tube is \( h \), in the second capillary tube with half the diameter, the height will be:

\[ h_2 = 2h \]

### Final Answer
The water will rise to a height of \( 2h \) in the capillary tube with half the diameter.

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