A cylindrical capillary tube of $0.2\mathrm{ }\mathrm{m}\mathrm{m}$ radius is made by joining two capillaries ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ of different materials having water contact angles of $0^{\mathrm{o}}$ and ${60}^{\mathrm{o}},$ respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct?
(Surface tension of water $=0.075\mathrm{ }\mathrm{N}/\mathrm{m},$ density of water $=1000\mathrm{ }\mathrm{k}\mathrm{g}/{\mathrm{m}}^3,$ take $\mathrm{g}=10\mathrm{m}/{\mathrm{s}}^2)$

• Correct answer
  A

  For case II, if the capillary joint is $5\mathrm{ }\mathrm{c}\mathrm{m}$ above the water surface, the height of water column raised in the tube will be $3.75\mathrm{ }\mathrm{c}\mathrm{m}.$ (Neglect the weight of the water in the meniscus)
• Correct answer
  B

  For case I, if the joint is kept at $8\mathrm{ }\mathrm{c}\mathrm{m}$ above the water surface, the height of water column in the tube will be $7.5\mathrm{ }\mathrm{c}\mathrm{m}.$ (Neglect the weight of the water in the meniscus)
• Correct answer
  C

  The correction in the height of water column raised in the tube, due to weight of water contained in the meniscus, will be different for both cases.
• D

  For case I, if the capillary joint is $5\mathrm{ }\mathrm{c}\mathrm{m}$ above the water surface, the height of water column raised in the tube will be more than $8.75\mathrm{ }\mathrm{c}\mathrm{m}.$ (Neglect the weight of the water in the meniscus)

Solution

Heights if only single material tubes are used of sufficient length, For table $T_1\Rightarrow h_1=\frac{2T\mathrm{c}\mathrm{o}\mathrm{s}\theta }{prg}=\frac{2\times 0.075\times \mathrm{c}\mathrm{o}\mathrm{s}0Kd}{1000\times 2\times {10}^{-4}\times 10}=7.5\mathrm{c}\mathrm{m}$ For table $T_2\Rightarrow h_2=\frac{2T\mathrm{c}\mathrm{o}\mathrm{s}{\theta }^{\prime }}{prg}=\frac{2\times 0.075\times \mathrm{c}\mathrm{o}\mathrm{s}60°}{1000\times 2\times {10}^{-4}\times 10}=3.75\mathrm{c}\mathrm{m}$ Weight of water in meniscus will be different in two cases because angle of contact is different $\Rightarrow \left(\mathrm{C}\right)$ is correct. Liquid will rise only upto height of $5\mathrm{ }\mathrm{c}\mathrm{m}$ and meniscus will adjust by changing is radius of curvature. If the liquid goes up in tube $2$ then it will not be able to support the weight of the liquid $\Rightarrow \left(\mathrm{D}\right)$ is incorrect. For option $\left(\mathrm{B}\right)$ Refer diagram below - As $K_4$ maximum height upto which liquid may rise in capillary is $h_1=7.5\mathrm{c}\mathrm{m}$ And length of tube available is $8\mathrm{ }\mathrm{c}\mathrm{m}$ $\Rightarrow$ Water will rise upto $7.5\mathrm{ }\mathrm{c}\mathrm{m}$ $\Rightarrow \left(\mathrm{B}\right)$ is correct. For option $\left(\mathrm{A}\right)$ , Refer figure below - Here maximum height upto which water can rise in case- $2$ , $h_2=3.75\mathrm{c}\mathrm{m}$ . Also the length of tube available is $5\mathrm{ }\mathrm{c}\mathrm{m}$ . $\Rightarrow$ Water will rise upto $3.75\mathrm{c}\mathrm{m}$ $\Rightarrow \left(\mathrm{A}\right)$ is correct.

Hence, the correct answer is options: 'A', 'B', 'C'.