Answer to Question #126477 in Mechanics | Relativity for Stefanie Hernandez-Mendez

Question #126477
A beam of light enters a glass tank filled with water at an incident angle of θa = 72° (see figure). The index of refraction of air is na = 1, the index of refraction of glass is ng = 1.5, and the index of refraction of water is nw = 1.3.

Calculate the value of θw, in degrees.
Expert's answer

Explanations & Calculations

  • Refer to the figure above.
  • Since no any figure shown previously , a justified calculation which includes all the indices of refraction is provided below.
  • Considering the air-glass interface & applying "n =\\Large \\frac{\\sin\\theta_i}{\\sin\\theta_r}" ; n is the refractive index of the medium —in which the refractive angle is present— with respect to adjoining medium; air, for the first calculation.

"\\qquad\\qquad\n\\begin{aligned}\n\\small \\frac{n_g}{n_a}&= \\small \\frac{1.5}{1} = \\frac{\\sin\\theta_a}{\\sin\\theta_g}=\\frac{\\sin72}{\\sin\\theta_g}\\\\\n\\small \\sin\\theta_g &= \\small 0.634\\cdots\\cdots\\cdots(1)\n\\end{aligned}"

  • Now applying the same relationship at the glass-water interface to calculate the angle in water.

"\\qquad\\qquad\n\\begin{aligned}\n\\small \\frac{n_w}{n_g}&= \\small \\frac{1.3}{1.5}=\\frac{\\sin\\theta_g}{\\sin\\theta_w}=\\frac{0.634}{\\sin\\theta_w}\\\\\n\\small \\sin\\theta_w &= \\small 0.7315\\\\\n\\small \\theta_w &= \\small \\bold{ 47.02^0}\n\\end{aligned}"

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