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The following observations were taken for determining the surface tension $T$ of water by the capillary method.:
The diameter of capillary, $D = 1.25 \times {10^{ - 2}}m$
rise of water, $h = 1.45 \times {10^{ - 2}}m$ .
Using $g = 9.8m/{s^2}$ and the simplified relation. $T = \dfrac{{rhg}}{2} \times {10^3}N/m$ , the possible error in surface tension is closest to :
(A) $10%$
(B) $0.15%$
(C) $1.5%$
(D) $2.4%$

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Last updated date: 17th Apr 2024
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Answer
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Hint Surface tension is the property of a liquid to minimize its surface area. It has units of force per unit length. Cohesive force is the main reason for surface tension. Error percentage is defined as the difference between the estimated value of a given quantity to its actual value, divided by the actual value multiplied by $100$ (to convert it into percentage).

Complete Step by step answer
Surface tension is given by the formula
$T = \dfrac{{rhg}}{2} \times {10^3}N/m$
The above equation can be written in terms of diameter as follows
$T = \dfrac{{Dhg}}{4} \times {10^3}N/m$
Taking logarithm with a base $10$ on both sides, we get
$\log T = \log D + \log h + \log g - \log 4 + 3$
Now, taking the error, we get
$\dfrac{{\Delta T}}{T} = \dfrac{{\Delta D}}{D} + \dfrac{{\Delta h}}{h}$
Substituting the measured values of the diameter of the capillary and the capillary rise of the water, we get
$\dfrac{{\Delta T}}{T} = \dfrac{{0.01}}{{1.25}} + \dfrac{{0.01}}{{1.45}}$
$ \Rightarrow \dfrac{{\Delta T}}{T} = 0.01489 \approx 0.015$
Therefore we get the error percentage as
$\dfrac{{\Delta T}}{T} \times 100 = 1.5\% $
That is, we get the percentage error in the value measured for surface tension as $1.5%$ .

Hence, we can conclude the option (C) to be the correct option.

Additional information
In a liquid that is contained in a capillary tube, the cohesive force between the molecules of liquid will be equal in all directions except the top of the liquid. Here, the force will be inward as there are no molecules above it resulting in a net inward pull, that contracts the surface area of the liquid at the top. The surface tension lets insects that are usually denser than water to float or slide on the surface of the water.

Note
You can see that in the error equation, there is no term related to the acceleration due to gravity or the values of $\log 4$ or $3$ . This is because there will be no error in the measurements of any of these terms as they are constants.