JEE Challenger
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PhyHardAdv 2019 • Paper 1

Water Height and Meniscus Correction in Joined Capillary Tubes

A composite capillary tube with a radius of 0.2 mm0.2\text{ mm} is formed by joining two individual capillaries T1T_1 and T2T_2 made of different materials, with contact angles for water being 0∘0^\circ and 60∘60^\circ, respectively. This capillary tube is immersed vertically in water in two distinct setups, case I and case II, as depicted in the figure. Which of the following option(s) is/are correct? [Take surface tension of water =0.075 N/m= 0.075\text{ N/m}, density of water =1000 kg/m3= 1000\text{ kg/m}^3, and g=10 m/s2g = 10\text{ m/s}^2]

Question Diagram 1

Options

A

The height correction for the water column inside the tube, accounting for the weight of water in the meniscus, is distinct for both cases.

Correct
B

For case II, provided the capillary joint is located 5 cm5\text{ cm} above the water level, the height of the water column in the tube will be 3.75 cm3.75\text{ cm}. (Ignore the weight of water in the meniscus)

Correct
C

For case I, provided the joint is positioned 8 cm8\text{ cm} above the water surface, the height of the water column in the tube will be 7.5 cm7.5\text{ cm}. (Ignore the weight of water in the meniscus)

Correct
D

For case I, provided the capillary joint is positioned 5 cm5\text{ cm} above the water level, the height of the water column in the tube will exceed 8.75 cm8.75\text{ cm}. (Ignore the weight of water in the meniscus)

Topics & Concepts

Mechanical Properties of FluidsSurface Tension

Step-by-Step Solution

To determine the correct options, we analyze the capillary rise in the two individual tubes T1T_1 and T2T_2.

1. Fundamental Capillary Height Formula

The height hh to which a liquid rises in a capillary tube of radius rr is given by: h=2Tcos⁡θrρgh = \frac{2T\cos\theta}{r\rho g}

Given parameters:

  • Radius of the capillary tube, r=0.2 mm=2×10−4 mr = 0.2\text{ mm} = 2 \times 10^{-4}\text{ m}
  • Surface tension of water, T=0.075 N/mT = 0.075\text{ N/m}
  • Density of water, ρ=1000 kg/m3\rho = 1000\text{ kg/m}^3
  • Acceleration due to gravity, g=10 m/s2g = 10\text{ m/s}^2

2. Maximum Heights in Individual Tubes

  • For tube T1T_1 (contact angle θ1=0∘\theta_1 = 0^\circ): h1=2×0.075×cos⁡0∘2×10−4×1000×10=0.152=0.075 m=7.5 cmh_1 = \frac{2 \times 0.075 \times \cos 0^\circ}{2 \times 10^{-4} \times 1000 \times 10} = \frac{0.15}{2} = 0.075\text{ m} = 7.5\text{ cm}

  • For tube T2T_2 (contact angle θ2=60∘\theta_2 = 60^\circ): h2=2×0.075×cos⁡60∘2×10−4×1000×10=0.15×0.52=0.0375 m=3.75 cmh_2 = \frac{2 \times 0.075 \times \cos 60^\circ}{2 \times 10^{-4} \times 1000 \times 10} = \frac{0.15 \times 0.5}{2} = 0.0375\text{ m} = 3.75\text{ cm}

3. Evaluation of the Options

  • Option (A): The correction in height due to the weight of water in the meniscus depends on the volume of the meniscus, which is determined by the contact angle θ\theta at the top surface. Since the meniscus forms with different contact angles in Case I (θ1=0∘\theta_1 = 0^\circ or pinned at an intermediate angle) and Case II (θ2=60∘\theta_2 = 60^\circ), the meniscus shape and volume differ in both cases. Thus, the height corrections are distinct. Option (A) is correct.

  • Option (B): In Case II, tube T2T_2 is at the bottom and T1T_1 is at the top, with the joint at 5 cm5\text{ cm} above the water surface. Water rises in T2T_2 up to its maximum capacity of h2=3.75 cmh_2 = 3.75\text{ cm}. Since 3.75 cm<5 cm3.75\text{ cm} < 5\text{ cm}, the water level never reaches the joint, and the height of the column remains 3.75 cm3.75\text{ cm}. Option (B) is correct.

  • Option (C): In Case I, tube T1T_1 is at the bottom and T2T_2 is at the top, with the joint at 8 cm8\text{ cm} above the water surface. Water rises in T1T_1 up to its maximum capacity of h1=7.5 cmh_1 = 7.5\text{ cm}. Since 7.5 cm<8 cm7.5\text{ cm} < 8\text{ cm}, the water level stays inside tube T1T_1 at a height of 7.5 cm7.5\text{ cm}. Option (C) is correct.

  • Option (D): In Case I, if the joint is at 5 cm5\text{ cm}, water rises in T1T_1 up to the joint at 5 cm5\text{ cm}. It cannot ascend into tube T2T_2 because T2T_2 can only support a maximum column height of 3.75 cm3.75\text{ cm}. Instead, the meniscus becomes pinned at the boundary/joint, adjusting its contact angle to satisfy: cos⁡θ=5 cm7.5 cm=23  ⟹  0∘<θ<60∘\cos\theta = \frac{5\text{ cm}}{7.5\text{ cm}} = \frac{2}{3} \implies 0^\circ < \theta < 60^\circ Hence, the water column height remains fixed at 5 cm5\text{ cm}, which is not greater than 8.75 cm8.75\text{ cm}. Option (D) is incorrect.

Final Answer:

The correct options are A, B, and C.

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