Question

Water rises to a height of 2 cm in a capillary tube. If cross-sectional area of the tube is reduced to $\frac{1}{16}$ of initial area, then water will rise to a height of

• A. 4 cm
• B. 8 cm
• C. 12 cm
• D. 16 cm

Hint:

When dealing with capillary tubes, remember that height scales inversely with the linear dimension ($r$) but inversely with the square root of the two-dimensional property ($A$). If area drops by a factor of 16, the radius drops by a factor of $\sqrt{16} = 4$, causing the height to become exactly 4 times greater ($2\ \text{cm} \times 4 = 8\ \text{cm}$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem presents a capillary tube where water rises to a specific initial height. We need to determine the new height to which the water rises when the cross-sectional area of the tube is scaled down to a fraction of its original value.

Step 2: Key Formula or Approach:
According to Capillary Rise (Jurin's Law), the height $h$ to which a liquid rises in a capillary tube of radius $r$ is given by:
$$h = \frac{2T\cos\theta}{r\rho g}$$ Since the surface tension ($T$), contact angle ($\theta$), density ($\rho$), and gravitational acceleration ($g$) are constant, the height is inversely proportional to the radius of the tube:
$$h \propto \frac{1}{r}$$ The cross-sectional area of a cylindrical capillary tube is $A = \pi r^2$, which implies $r \propto \sqrt{A}$. Substituting this back gives:
$$h \propto \frac{1}{\sqrt{A}}$$

Step 3: Detailed Explanation:
Let the initial height be $h_1 = 2\ \text{cm}$ and the initial area be $A_1$. The final area is given as $A_2 = \frac{A_1}{16}$. Using our proportionality relationship, we set up the ratio of final height to initial height:
$$\frac{h_2}{h_1} = \sqrt{\frac{A_1}{A_2}}$$ Substitute the given relationship for $A_2$ into the ratio:
$$\frac{h_2}{h_1} = \sqrt{\frac{A_1}{\frac{A_1}{16}}} = \sqrt{16} = 4$$ Now, substitute the value of $h_1$:
$$h_2 = 4 \times h_1 = 4 \times 2\ \text{cm} = 8\ \text{cm}$$

Step 4: Final Answer:
The water will rise to a height of 8 cm, which perfectly corresponds to option (B).