Question

On a calm, bright day, you're scuba diving in a deep lake (n = 1.33), 6.60 m from your sailboat. When you are 3.20 m below the surface, the top of the sailboat mast appears to you to be in a direction 37.0° from vertical. Calculate the height of the mast.

From your point of view, the entire sky appears to be confined to a bright disk directly above you on the surface of the water. Determine the diameter of this circle.

Answer 1

Hi,

  Draw a diagram and you will see that there will be an angle ? (which is the angle on the surface between the 6.6m distance to the boat and the line form the water to the top of the mast).

So the refracted angle as measured from the normal ?b = 90 - ?

  => ? = 90 - ?b

From Snell's Law we have nA * Sin(?a) = nB * Sin(?b)

      where nA = 1.33 (refractive index of water)

          and nB = 1.0 (refractive index of air).
          and ?a =37 deg
  

   Hence, ?b = arcsin(1.33 * Sin(37)) = 53.17 deg

     therefore ? = 90 - 53.17 = 36.83 deg.

Therefore the height of the mast = 6.6 * Tan(36.83) = 4.943m


B) The disk will be the diameter as seen from 3.2m below when the light strikes the surface at the critical angle.

  Hence nA * Sin(?crit) = nB * Sin(?b) = 1.0 * Sin(90) = 1      (here ?b = 90 deg)

  Hence ?crit = arcsin(1 / 1.33) = 48.75 deg

  => The radius of the circle will be 3.2 * Tan(48.75) = 3.649m

  Hence the diameter will be 3.649 * 2 =  7.298m