A drop of liquid of diameter d = 5.0 mm rests on a horizontal, flat solid surface. In order to find the contact angle θ between the drop and the surface, a collimated laser beam of identical diameter

Question

A drop of liquid of diameter d = 5.0 mm rests on a horizontal, flat solid surface. In order to find the contact angle θ between the drop and the surface, a collimated laser beam of identical diameter is shone vertically onto the liquid drop, and a translucent screen is set up horizontally at a height h = 12 cm above the drop. Assume that θ < 9 0 ∘ . The diagram is not drawn to scale. Given that a reflected image of diameter D = 5.10 cm is formed on the screen, find the contact angle θ (in degree) between the liquid and solid surface.

TestHub · Accepted Answer

The drop's surface is a spherical cap acting as a convex mirror of radius $R_{s}$ . A ray parallel to the axis at height y reflects at angle $\frac{2 y}{R _{s}}$ , so the edge ray ( $y = a = \frac{d}{2} = 2.5$ mm) reaches the screen at radius $a + \frac{2 ah}{R _{s}}$ .

Diameter on the screen: $D = d (1 + \frac{2 h}{R _{s}})$ : $51 = 5 (1 + \frac{240}{R _{s}}) \Rightarrow R_{s} = 26.09$ mm.

For the spherical cap: $sin θ = \frac{a}{R _{s}} = \frac{2.5}{26.09} = 0.0958$

$\Rightarrow θ = 5. 5^{\circ}$ - within the official range 5.45-5.55.

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