A submarine periscope uses two totally reflecting 45-45-90 prisms with total internal reflection on the sides adjacent to the 45 angles. Explain why the periscope will no longer work if it springs a leak and the bottom prism is covered with water.
The Correct Answer and Explanation is:
Correct Answer:
If the bottom prism of the submarine periscope is covered with water due to a leak, total internal reflection (TIR) will no longer occur at the prism’s reflecting surfaces. This happens because the critical angle for TIR changes when the prism is in contact with water instead of air, causing the light to refract out of the prism rather than being fully reflected internally. As a result, the periscope will fail to properly direct the light path, and the image will be lost or significantly degraded.
Detailed Explanation
A submarine periscope commonly uses 45-45-90 right-angle prisms designed for total internal reflection (TIR) at their hypotenuse faces. The key principle behind TIR is that light traveling inside a denser medium (the glass prism) hitting an interface with a less dense medium (air) at an angle greater than the critical angle is completely reflected back inside the prism. This allows the periscope to efficiently bend the light path without using mirrors or coatings, ensuring high brightness and image clarity.
The critical angle, θc\theta_c, for TIR is determined by the refractive indices of the prism n1n_1 and the surrounding medium n2n_2, given by Snell’s law: θc=sin−1(n2n1)\theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right)
For typical glass (n1≈1.5n_1 \approx 1.5) in air (n2≈1.0n_2 \approx 1.0), the critical angle is about 42°. Since the prism is designed so light hits the internal reflecting surfaces at 45°, which is greater than 42°, TIR occurs and light is fully reflected.
However, if the bottom prism is flooded and covered with water (n2≈1.33n_2 \approx 1.33), the critical angle increases to: θc=sin−1(1.331.5)≈62.5∘\theta_c = \sin^{-1} \left( \frac{1.33}{1.5} \right) \approx 62.5^\circ
Now, the incident angle of 45° is less than this new critical angle of 62.5°. Therefore, instead of total internal reflection, some light refracts out into the water, resulting in partial reflection and transmission. The internal reflection is no longer “total.” This loss causes the light beam to escape the prism rather than being redirected efficiently.
As a result, the periscope’s optical path is broken, the image brightness drops sharply, and the periscope effectively stops working. This explains why a leaking periscope prism covered with water will fail to function properly.
