Question:

The radii of the two columns in a U - tube are r1 and r2. When a liquid of density p (angle of contact is 0 0{}^\circ ) is filled in it, the level difference of the liquid in the two arms is h. The surface tension of the liquid is: (g = acceleration due to gravity)

Updated On: Sep 3, 2024
  • ρghr1r22(r2r1) \frac{\rho gh{{r}_{1}}{{r}_{2}}}{2({{r}_{2}}-{{r}_{1}})}
  • ρgh(r1r2)r2r1 \frac{\rho gh({{r}_{1}}-{{r}_{2}})}{{{r}_{2}}{{r}_{1}}}
  • 2(r1r2)ρghr1r2 \frac{2({{r}_{1}}-{{r}_{2}})}{\rho gh{{r}_{1}}{{r}_{2}}}
  • ρgh2(r2)r1 \frac{\rho gh}{2({{r}_{2}})-{{r}_{1}}}
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The Correct Option is A

Solution and Explanation

Rise of liquid in the capillary tube is given by h=2Trρg h=\frac{2T}{r\rho g} Suppose, h1 {{h}_{1}} is the height in tube of radius r1 {{r}_{1}} and h2 {{h}_{2}} is the height in tube of radius of r2 {{r}_{2}} So, h1=2Tr1ρg {{h}_{1}}=\frac{2T}{{{r}_{1}}\rho g} ?(i) and h2=2Tr2ρg {{h}_{2}}=\frac{2T}{{{r}_{2}}\rho g} ?(ii) Level difference of liquid in the two arms is given by h=h1h2=2Tr1ρg2Tr2ρg h={{h}_{1}}-{{h}_{2}}=\frac{2T}{{{r}_{1}}\rho g}-\frac{2T}{{{r}_{2}}\rho g} \Rightarrow h=2Tρg[1r11r2] h=\frac{2T}{\rho g}\left[ \frac{1}{{{r}_{1}}}-\frac{1}{{{r}_{2}}} \right] \Rightarrow h=2Tρg[r2r1r1r2] h=\frac{2T}{\rho g}\left[ \frac{{{r}_{2}}-{{r}_{1}}}{{{r}_{1}}{{r}_{2}}} \right] Hence, T=hρgr1r22(r2r1) T=\frac{h\rho g{{r}_{1}}{{r}_{2}}}{2({{r}_{2}}-{{r}_{1}})}
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Concepts Used:

Surface Tension

The amount of energy required to increase the liquid's surface area by one unit area is known as surface tension. In other words, it is a property of the liquid surface to resist force.

Surface tension is defined as,

The ratio of the surface force F to the length L along which the force acts.

Mathematically, the surface tension formula can be expressed as follows:

T=F/L

Where,

  • F is the force per unit length
  • L is the length in which force act
  • T is the surface tension of the liquid

Read More: Detergents and Surface Tension

Factors affecting surface tension:

  • Impurities: The surface tension decreases with the addition of impurities.
  • Surfactants: Adding surfactants in liquids lowers the tension of water making it interrupt aside or get susceptible.
  • Temperature: The surface tension of a liquid reduces as the temperature rises.

The Unit of Surface Tension:

The SI unit of Surface Tension is Newton per Meter or N/m.