Around 1965, engineers at the Toro Company invented a...

Around 1965, engineers at the Toro Company invented a gasoline gauge for small engines diagrammed in Figure P35.44. The gauge has no moving parts. It consists of a flat slab of transparent plastic fitting vertically into a slot in the cap on the gas tank. None of the plastic has a reflective coating. The plastic projects from the horizontal top down nearly to the bottom of the opaque tank. Its lower edge is cut with facets making angles of $45^{\circ}$ with the horizontal. A lawn mower operator looks down from above and sees a boundary between bright and dark on the gauge. The location of the boundary, across the width of the plastic, indicates the quantity of gasoline in the tank. (a) Explain how the gauge works. (b) Explain the design requirements, if any, for the index of refraction of the plastic.

Key Concepts

Refraction

Refraction is the change in direction of a light ray when it passes from one medium to another with a different optical density. This concept is governed by Snell’s law and is critical in understanding how light behaves at the boundary between materials like air, plastic, and gasoline. By observing changes in how light bends as it moves through these media, engineers can design devices that visually indicate the level of a substance without moving parts.

Total Internal Reflection

Total internal reflection occurs when light attempts to move from a medium with a higher refractive index to one with a lower refractive index at an angle greater than the critical angle. Under these conditions, all the light is reflected back into the original medium rather than refracted out. This phenomenon is essential in the operation of the gauge, as the change in the incident medium (from air to gasoline) alters whether light is internally reflected or escapes, creating a visible boundary.

Critical Angle

The critical angle is the minimum angle of incidence within a medium of higher refractive index at which total internal reflection occurs when light passes into a medium of lower refractive index. In the context of the gauge, the facets angled at 45° are designed so that, depending on whether they are in contact with air or gasoline, the incident light meets or exceeds the critical angle, thereby switching between reflection and transmission modes. This change is then visually detected as a boundary between bright and dark regions.

Refractive Index

The refractive index is a measure of how much the speed of light is reduced in a specific medium compared to its speed in a vacuum. It is a fundamental property that influences both the refraction and internal reflection of light. For the gauge design, the plastic’s refractive index must be chosen carefully to ensure that the 45° facets will produce the desired bright versus dark boundary under different contacting fluids (e.g., gasoline versus air). The precise value determines the critical angle and thus is central to the proper functioning of the device.

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