Question

A ray of light enters a glass block from air and comes out from the opposite surface. If the angle of refraction at the first surface is not the same as the angle of incidence at the second surface, then:
(a) What is the product of the ratio \(\frac{\sin i}{\sin r}\) at the first surface and at the second surface?
(b) State whether the opposite surfaces are parallel or not parallel.
(c) How did you reach the conclusion in (b) above?

Hint:

For a rectangular (parallel-sided) glass slab, the emergent ray is parallel to the incident ray. This only happens if \(r_1 = i_2\). If this condition is not met, the surfaces cannot be parallel.

Answer 1

(a) Product of the ratios:
Let \(n_a\) be the refractive index of air (\(\approx 1\)) and \(n_g\) be the refractive index of glass.
- At the first surface (air to glass): Let the angle of incidence be \(i_1\) and the angle of refraction be \(r_1\). According to Snell's law: \[ \frac{\sin i_1}{\sin r_1} = \frac{n_g}{n_a} = n_g \] - At the second surface (glass to air): Let the angle of incidence be \(i_2\) and the angle of refraction (emergence) be \(r_2\). According to Snell's law: \[ \frac{\sin i_2}{\sin r_2} = \frac{n_a}{n_g} = \frac{1}{n_g} \] The question asks for the product of the ratio \(\frac{\sin i}{\sin r}\) at the first surface and at the second surface. \[ \text{Product} = \left( \frac{\sin i_1}{\sin r_1} \right) \times \left( \frac{\sin i_2}{\sin r_2} \right) = n_g \times \frac{1}{n_g} = 1 \] The product is 1.
(b) Parallel or not parallel:
The opposite surfaces are not parallel.
(c) Conclusion:
If the opposite surfaces of the glass block were parallel, then the normal to the first surface would be parallel to the normal to the second surface. In this case, the refracted ray from the first surface and the incident ray at the second surface would form alternate interior angles with the parallel normals. Therefore, the angle of refraction at the first surface (\(r_1\)) would be equal to the angle of incidence at the second surface (\(i_2\)).
The question explicitly states that \(r_1 \neq i_2\). Since this geometric condition for parallel lines is not met, we can conclude that the surfaces are not parallel.