A glass capillary tube is of the shape of a truncated cone with an...

A glass capillary tube is of the shape of a truncated cone with an apex angle $\alpha$ so that its two ends have cross-sections of different radii. When dipped in water vertically, the water rises in it to a height $h$ , where the radius of its cross-section is $b$ . If the surface tension of water is $S$ , its density is $ρ$ , and its contact angle with glass is $\theta$ , then the value of $h$ will be ( $g$ is the acceleration due to gravity)
A glass capillary tube is of the shape of a truncated cone with an apex angle α so that its two ends have

Options

A $\dfrac{2S}{bρg}\cos ( \theta - \alpha )$
B $\dfrac{2S}{bρg}\cos \left(\theta + \alpha\right)$
C $\dfrac{2S}{bρg}\cos \left(\theta - \dfrac{\alpha}{2}\right)$
D $\dfrac{2S}{bρg}\cos \left(\theta + \dfrac{\alpha}{2}\right)$

Correct answer

D. $\dfrac{2S}{bρg}\cos \left(\theta + \dfrac{\alpha}{2}\right)$

Step-by-step solution

If $R$ be the meniscus radius,
$⇒ Rcos \left(\theta + \dfrac{\alpha}{2}\right) = b$
Excess pressure on concave side of meniscus $= \dfrac{2S}{R}$
A glass capillary tube is of the shape of a truncated cone with an apex angle α so that its two ends have
$hρg = \dfrac{2S}{R} = \dfrac{2S}{b}\cos \left(\theta + \dfrac{\alpha}{2}\right)$
$⇒ h = \dfrac{2S}{bρg}\cos \left(\theta + \dfrac{\alpha}{2}\right)$

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