ormulas 2 m 3. A tall cup of diameter D is partially filled with a liquid with t

Question

ormulas 2 m 3. A tall cup of diameter D is partially filled with a liquid with the refraction index n to a height of Hwater. A student looks the downward just over the left of the cup at an angle with surface. the refraction of light at the waters Han surface barely allows her see the corner of the cup. A sketch of the path of light is shown at the right. Derive the formula for the height of the cup Heup. Calculate its cm, 0 40.47o. numerical value for n 1.33 (water), D 14.64 cm, Hwater 7.8 water

Explanation / Answer

From geometry we can see

H cup = H water + xtan(theta)
where x = D - r
and tan(phi) = r/Hwater

also, from snell's law
sin(theta)/sin(phi) = n
so, sin(40.47)/sin(phi) = 1.33
phi = 29.209 deg

r = Hwater*tan(phi) = 7.8*tan(29.209) = 4.36 cm
x = D - r = 14.64 - r = 10.279 cm
H cup = 7.8 + 10.279*tan(40.47) = 16.569 cm

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