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# 6.2: Activities

## Things You Will Need

All you will need for this lab is the PhET simulator mentioned in the Background Material, which you can run now in a separate window, though you won't need it until the final part.

## Identify This Material!

A lot of visual evidence regarding a transparent rectangular prism is given below. You know that this clear brick is made of either glass ($$n\approx 1.5$$) or plastic ($$n\approx 1.4$$) – and your task is to identify it from the evidence provided. There are four "runs" of the experiment provided, which means you can determine an average value and a rough statistical uncertainty, so you will need to determine if your result is reliable enough to distinguish between the two possibilities, and if so, then announce your findings!

Run A  Run B  Run C  Run D  1. Create a table that clearly organizes the raw data you collect from above, and include rows/columns that provide the quantities you calculate for your conclusion and the related uncertainties.
2. Can you determine within the limits of uncertainty which of the two materials this brick is made from? If so, justify your conclusion with your data. If not, explain why not.

## Flat Refractor Image Location

In the text references given in the Background Material is a derivation of the (approximate) depth of an image in a material with a flat boundary, in terms of the index of refraction of that material (and the external region, which in our case is air, for which $$n=1.00$$). Our goal for this part of the lab is to confirm this relation using both ray traces and the parallax trick discussed in the Background Material.

We will be using the same brick as above, this time with the object nail placed flush against one side. Then looking through the brick, the image of the nail will exist somewhere within the brick. Below is visual data of the image from two different perspectives. Two nails are used to track the direction of the emerging ray from each perspective.

[Important note: The formula for determining the position of the image is an approximation, because the image moves farther from the object as the angle at which it is viewed increases. The formula expresses where the image is located if one looks at the image straight through the flat surface (because the sine and tangent of an angle converge as the angle goes to zero). For the angles involved below, the approximation of $$\sin\theta \approx \tan\theta$$ is off by about 5%, which means that the image viewed at this angle is about 5% closer to the viewing eye than the formula predicts. Take this into account in your analysis below.]

Run A  Run B  1. Use the data above to confirm the apparent depth equation, using the index of refraction you found above, and keeping in mind the ~5% correction mentioned above.

We can now confirm this position of the image using the parallax method: In finding this location using parallax, there were two other trials that clearly did not result in the correct position:

Trial A Trial B 1. For each of these two failed trials, determine whether the guessed position of the image (the location of the nail above the clear brick) is too close or too far from our eye to match the image position.

## Lensmaker Equation

We turn now to our simulator. We could use this application to test various examples of the thin lens equation, looking at how the object distance, image distance, and focal length are related to each other, as well as the magnification and orientation of the image. Indeed, you are encouraged to come back to this simulator while solving problems to help provide you insight. But none of these activities really capture the flavor of a lab experiment. So we are going to use the simulator exclusively to look at the relationship between the focal length, the index of refraction, and the radius of curvature of the surface of the lens (which is restricted to be the same on both sides). Specifically, we are going to experimentally derive the lensmaker equation.

This is not really something we could do in a lab, because we would have to grind several lenses at different radii to test the dependence on this variable, and we would have to create lenses from several materials to test the dependence on the index of refraction. With the simulator, we can change these parameters simply by adjusting a slide-bar. So here is how to configure the simulator:

• click the radio button "No Rays" (these only get in the way)
• drag the object far out of the way (the upper-right corner is best)
• select the box "Ruler"

The following features should also be noted:

• Both the ruler and the lens can be dragged around the screen as needed
• Unless the refractive index is set very high, the ruler can be seen through the lens (though of course it makes no sense to represent index of refraction with opacity).
• The little yellow $$\times$$'s on the optical axis designate the focal points of the lens.

We are looking for a relationship between three quantities – the radius of curvature of the faces of the lens, the index of refraction of the lens material, and the focal length of the lens. The physics that lies behind this relationship is Snell's law, and a basic feature of this is that the more sharply-curved the surface is, the greater the ray deflects. In particular, we'll start by showing that for a fixed index of refraction $$n$$, the radius of curvature $$R$$ is directly proportional to the focal length $$f$$:

$R=S\cdot f\;,\;\;\;\;\;\left(S\equiv \text{slope}\right)$

1. Select a refractive index that you will hold fixed, then use the ruler to measure focal lengths for four different values of curvature radius, filling the results into the table below. Explain how this data demonstrates the linear relationship (you don't need to graph the points). $n=\text{_____}\;\;\;\;\;\; S=\text{_____}\nonumber$

So the slope of the $$R$$-vs-$$f$$ curve is a constant, provided the index of refraction is held fixed. But certainly $$n$$ plays a role in this relation, and we can express this role by stating that the slope is actually a function of $$n$$. So if we can determine $$S\left(n\right)$$, we can plug it back into Equation 6.2.1 above, and we'll have what we are looking for.

We accomplish this by choosing a radius of curvature and holding it fixed, while changing the index of refraction. With the fixed value of $$R$$ and the various values of $$f$$ that we record, we can use Equation 6.2.1 to determine $$S$$, which we can then plot against $$n$$.

1. Select a radius of curvature that you will hold fixed, then use the ruler to measure focal lengths for four different values of refractive index, filling the results into the table below.

$R=\text{_____}\nonumber$ 1. Plot $$S$$-vs-$$n$$, label it clearly, and attach the graph to your lab report.
2. Use the graph to determine the function $$S\left(n\right)$$, construct the full formula for $$f$$ in terms of $$n$$ and $$R$$, and compare it with the lensmaker equation in the Background Material.

## Lab Report

Download, print, and complete this document, and upload your lab report to Canvas. [If you don't have a printer, then two other options are to edit the pdf directly on a computer, or create a facsimile of the lab report format by hand.]