Around 1968, Richard A. Thorud, an engineer at The Toro...

Around 1968, Richard A. Thorud, an engineer at The Toro Company, invented a gasoline gauge for small $$ $$ engines diagrammed in Figure $\mathrm{P} 34.30$. 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

Light Transmission and Refraction

This concept encompasses how light travels through different materials and changes direction at interfaces as dictated by Snell’s law. In optical device design, understanding refraction is essential for predicting how light is redirected when passing between media with different optical properties.

Total Internal Reflection

Total internal reflection is the phenomenon where light is completely reflected within a medium when it strikes the boundary at an angle greater than the critical angle. This principle is widely used in devices such as optical fibers and indicators to confine and direct light efficiently without losses.

Index of Refraction and Critical Angle

The index of refraction quantifies how much a material slows down light, which in turn affects the critical angle necessary for total internal reflection. In the design of optical systems, selecting a material with the proper refractive index ensures that light interacts with interfaces in a controlled manner, enabling features such as distinct brightness transitions.

Transcript

00:01 So this is a problem involving total internal reflection.

00:05 Now total internal reflection happens when you have light coming from an incident medium with a higher index refraction to a medium that has a lower index refraction.

00:19 Say like for example in this situation we have light hitting the interface in the plastic and the air beneath it.

00:28 This is the part of the gauge that does not have any.

00:31 Any gasoline underneath it.

00:33 And so up here we have n for the plastic.

00:39 And down here we have n for air, which is just usually taken to be one.

00:44 And the reason you get total internal reflection is because as the angle of incidence gets bigger, snell's law tells us that the angle of refraction gets bigger.

00:56 And if you have a situation where the incident median, index refraction is greater than the outgoing medium's index refraction, that means that the light that comes in at an angle is going to refract out at an even higher angle.

01:13 But you can then have a situation in which you pass what is called the critical angle, meaning that the light coming in is at an angle of incidence so big that the angle of refraction would be 90 degrees.

01:31 And then you can't get any bigger than that because the refraction greater than 90 degrees would just be reflecting back into the incident medium again.

01:42 The way we get the formative for this is we just use snell's law.

01:46 As i've drawn it here, we can say that the incident medium's index refraction, this is the plastic, times the sign of the angle of refraction, is equal to the outgoing mediums index refraction, which is just taken to be one since it's going out into air.

02:11 This is the part of the gas gate.

02:14 It does not have any gasoline below it, times the outgoing or the refracted angle of refraction, the sign of the refracted angle of refraction.

02:27 This is a sign of the angle of incidence, and this is a sign of the angle of refraction.

02:32 Now, with the critical angle, the angle that you have to be bigger than is given by this format.

02:39 Let's adjust this a little bit.

02:41 We'll say that for the critical angle, the refracting medium must have an angle equal to 90 degrees.

02:54 So that's the angle that you have to have the incident ray being larger than in order that the refracted ray cannot possibly refract anymore, so it's going to be reflected instead.

03:05 And of course, that means the sign of 90 degrees here, but the sign of 9 degrees, we don't note that is that's just 1.

03:16 So what we end up having is that the incident medium tenetion fraction times the sign of the critical angle just ends up being equal to 1.

03:27 If the outgoing medium is air, which means this is the way the formative is usually stated, the sign of the critical angle is going to equal one over the index refraction of the plastic.

03:48 So that's the basic operation of it.

03:51 What you want to have happened is that when light comes in and there's not air but gasoline underneath the plastic, you don't want that to be critically, you want that to be totally internally reflected.

04:07 In the words, you don't want to have a critical angle where it's exceeding that critical angle as being internally reflected.

04:13 You want all of that light to go into the refracting medium.

04:18 And i'll show you the condition for that in just a moment...

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