Question

A person looks into an empty glass at an angle that allows her to barely see the bottom of the glass. When looking at the same angle after filling the glass with water, she can see the center of the bottom of the glass. If the width of the glass is W=6.1 cm then the height of the glass is H= ___cm. Use n=1.33 for the refractive index of water.

Answer 1

When Glass is empty, then person can see edge of the glass then from this we can find the incidence angle of light, which will be

sin i = W/sqrt (H^2 + W^2)

When Glass is filled with water then refracted angle will be (he can see only center of the bottom of the glass which is W/2), So

sin r = (W/2)/sqrt (H^2 + (W/2)^2)

Now Using Snell's law:

ni*sin i = nr*sin r

ni = refractive index of air = 1

nr = refractive index of water = 1.33

So,

1*W/sqrt (H^2 + W^2) = (1.33*W/2)/sqrt (H^2 + W^2/4)

sqrt [(4H^2 + W^2)/(H^2 + W^2)] = 1.33

square both sides

[(4H^2 + W^2)/(4*(H^2 + W^2))] = 1.33^2

(4H^2 + W^2)/(H^2 + W^2) = 1.33^2 = 1.7689

4H^2 + W^2 = 1.7689*H^2 + 1.7689*W^2

0.7689*W^2 = 2.2311*H^2

H = W*sqrt (0.7689/2.2311)

Given that W = 6.1 cm

H = 6.1*sqrt (0.7689/2.2311)

H = 3.581 cm = 3.58 cm

Height of glass = H = 3.58 cm

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