(II) Figure 57 shows a liquid-detecting prism device that...

(II) Figure 57 shows a liquid-detecting prism device that might be used inside a washing machine or other liquid-containing appliance. If no liquid covers the prism's hypotenuse, total internal reflection of the beam from the light source produces a large signal in the light sensor. If liquid covers the hypotenuse, some light escapes from the prism into the liquid and the light sensor's signal decreases. Thus a large signal from the light sensor indicates the absence of liquid in the reservoir. If this device is designed to detect the presence of water, determine the allowable range for the prism's index of refraction $n .$ Will the device work properly if the prism is constructed from (inexpensive) lucite? For lucite, $n=1.5 .$

Key Concepts

Total Internal Reflection

This is the phenomenon where light traveling in a medium of higher refractive index is completely reflected at the boundary when it strikes the interface with a medium of lower refractive index at an angle exceeding the critical angle. This concept underpins how the device detects the presence or absence of liquid based on the direction of the light beam.

Critical Angle

The critical angle is the smallest angle of incidence at which total internal reflection occurs at the interface between two media with different refractive indices. It is calculated using Snell's law, and its determination is key to understanding under what conditions the sensing device's beam will remain confined or leak into an adjoining medium.

Index of Refraction

The index of refraction is a measure of how much the speed of light is reduced in a medium compared to vacuum. It is crucial for predicting how light behaves at interfaces, including the extent of bending, the critical angle for total internal reflection, and ultimately the proper functioning of devices that rely on these light interactions.

Snell's Law

Snell's law governs the relationship between the angles of incidence and refraction when light passes between two media with different refractive indices. It is essential for calculating the critical angle necessary for total internal reflection and thereby for designing and evaluating optically based sensing devices.

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