Physics 4BL Lab 5

Laboratory 5

Geometric Optics

Max Woo
12/3/14
Lab 11, Michael Ip
Partners: Vir Thakor, Luis Chinchilla-garcia

Introduction

The purpose of this lab was to investigate the geometric properties of light rays and the refraction of light rays through different mediums. For the first part, we test Snell’s law by measuring refraction angles and total internal reflection angle of a prism with unknown refraction index. For the second part, we measure the focal length and magnification of various combinations of convex, plano-convex, and bi-concave lenses. Our third part involves measuring the focal length of various combination of thin lenses and of a sphere using twin lasers. For the fourth and final part, we measure the magnification of an image through a thin lens.

Experimental Data

Part 1: Snell’s Law and Total Internal Reflection

For this part, we use a ray box to produce a single ray of light. We place the prism on a clean sheet of printer paper, and position the ray box to shine the light through the prism. We then trace the prism and the path of the light ray, and measure the refraction and reflection angles relative to the normal for both the light rays entering the prism and exiting the prism. A diagram of the setup is shown below:

Fig 1: Setup for measuring refraction/reflection, with markings for the angles to be measured

Next, shined the light ray through the beveled end of the prism, and rotated the prism until there was no more light exiting the prism on the first reflection. We then measured the angles of internal reflection for that setup.

Fig 2: Diagram for measuring total internal reflection, with markings for the angles to be measured

All the measurements are shown in the table below:

Table 1: Measured refraction, reflection, and total internal reflection angles for a prism with unknown refraction index

Part 2: Thick Lenses

For this part, we first changed the ray box to emit 3 rays. Then, for each of the biconvex, plano-convex, and biconcave lenses, we positioned them so their axis was normal to the light rays, and then traced the path of the light rays through the lenses. For each of the resulting tracings, we extended the lines of the outgoing rays until they met at a single point, and we measured the distance of this point to the center of the lens to measure the focal length of each lens. The results are shown in the table below:

Table 2: Measured focal lengths for the biconvex, plano-convex, and biconcave lenses

Next, we change the ray box to emit 5 rays, and shine them through the biconvex lens. There are now two focal points, with the middle three rays converging to one focal point, and the outer ray converging to another focal point. This is because circular/spherical lenses only converge light rays to a single focal point for small angles (which is why many calculations for the focal point of a circular/spherical thick lens uses small-angle approximations), so the larger the image going through the lens, the less accurate the small-angle approximations are, and the more deviation from the focal point the light rays converge.

In our case, the distance from the two focal points was measured to be $0.69 \pm 0.05 cm$, and the distance from the lens to the first focal point was $5.41 \pm 0.05 cm$, which gives us an abberation of $12.75\% \pm 0.93\%$.

Finally, we chang the ray box back to 3 light rays, and place a biconcave lens and plano-convex lens so that the light rays first enter the biconcave lens and then the plano-convex lens, adjusting the two lenses so the final outgoing light rays are parallel. Notice that this combination of lenses magnifies the image, increasing the separation of the rays, which we can quantify by measuring the initial and final separation of the beams. We also reverse the order of the lenses to cause demagnification, and measure the initial and final ray separation for that. The results are shown below:

Table 3: Initial and final separation for biconcave-planoconvex lens setups. Note that the initial separations for magnification and demagnification are the same.

Part 3: Thin Lens Properties

For this part, we first split a laser beam using a transparent plate to get two parallel laser beams. Then we put a thin lens so that the twin laser beams pass through the center of the lens, and then we shine the outgoing beams onto a piece of paper, adjusting the distance until the outgoing beams focus to a point on the paper. The distance from the paper to the lens is the focal length of the thin lens, which we measure and record. Note that we can move the paper forward and backward a certain distance before the beams begin to diverge again. We can use this distance to approximate our uncertainty. We then use a different thin lens and again measure the focal length. Lastly, we combine the two lenses, separated by a set distance $D$, and measure the focal length of both lenses together.

We also follow the same procedure to measure the focal length of an acrylic sphere, and then, in preparation for Part 4, we measure the focal length of the 2cm diameter lens. The results are shown below:

Table 4: Measurements from part 3 (thin lenses) of the laboratory

Part 4: Image Formation with Lenses

We start with a 5V LED shining through a thin transparent film on the iris of a metal plate. The thin slide has an image of a grid with lines separated by 1mm. We then place the 2cm lens on the other side of the film, so that the distance between the lens and film is more than the focal length of the lens (measured in part 3). Finally, we place a movable screen of cardboard behind the lens so that the image of the graph projected by the lens shines onto the cardboard. We adjust the cardboard until the image is sharp, and measure both the distance from the film to the lens, and from the lens to the cardboard. We also measure the distance between the graph lines of the projected image to measure the magnification. We repeat these measurements for various distances between the film and lens, making sure to keep the distance above the focal length of the lens. The results are shown below:

Table 5: Measurements for part 4

Data Analysis

Part 1

We start by calculating the index of refraction using Snell’s law:

$n_1 \sin(\theta_1) = n_2 \sin(\theta_2$ (Eq. 1, Snell’s law)
$n_2 = n_1 \frac{\sin(\theta_1)}{\sin(\theta_2)}$ (Eq. 2)

Where $n_1$ and $n_2$ represent the index of refraction for the air and prism respectively, and $\theta_1$ and $\theta_2$ represent the angle of incidence for the air and prism respectively. The uncertainty formula is derived from the propagation of uncertainty formula, which gives:

$\sigma_{n_2} = \sqrt{ (n_1 \frac{\cos(\theta_1)}{\sin(\theta_2)} \sigma_{\theta_1})^2 + (n_1 \sin(\theta_1) \csc(\theta_2) \cot(\theta_2) \sigma_{\theta_2})^2 }$ (Eq. 3)

Note that $n_1$, the refraction index of air, is an accepted constant and thus does not have an uncertainty value large enough to be significant. From Eq. 2 and 3, we get an index of refraction of $n_2 = 1.531 \pm 0.144$.

Next, we use the index of refraction and calculate the predicted total-internal-reflection angle, using the formula

$n_2 \sin(\theta_2) = n_1 \sin(\pi/2)$ (Eq. 4)
$\theta_2 = \sin^{-1}(n_1 / n_2)$ (Eq. 5)

with the uncertainty formula derived using propagation of error:

$\sigma_{\theta_2} = \frac{1}{\sqrt{1 - (n_1 / n_2)^2}} \frac{n_1}{n_2} \frac{\sigma_{n_2}}{n_2}$ (Eq. 6)

This gives us a predicted total internal reflection angle of $40.774° \pm 0.281°$, which our measured value of $42.5° \pm 0.5°$ falls within the error bounds of, verifying Snell’s law and the formula for total internal reflection.

Part 2

For this part, we calculate the magnification factor of the biconcave-planoconvex lens combination by first denoting initial and final separation with $d_1$ and $d_2$ respectively, then using the formula

$M = \frac{d_2}{d_1}$ (Eq. 7)

with uncertainty derived from propagation of error formula:

$\sigma_M = \frac{d_2}{d_1} \sqrt{(\frac{\sigma_{d_2}}{d_2})^2 + (\frac{\sigma_{d_1}}{d_1})^2 }$ (Eq. 8)

This gives us a $M = 2.00 \pm 0.117$ for magnification and $M = 0.237 \pm 0.0139$ for reduction.

Part 3

For this part, we first denote the measured focal lengths of the thin lenses individually with $f_1$ and $f_2$, and denote the focal length of the two combined with $f_{total}$, with the separation distance as $e$. We can predict $f_{total}$ using the equation:

$\frac{1}{f_{total}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{e}{f_1 f_2}$ (Eq. 9)

with uncertainty derived from propagation of error formula:

$\sigma_{f_{total}} = f_{total} \sqrt{(\sigma_{f_1} (\frac{1}{f_1} - \frac{1}{S}))^2 + (\sigma_{f_2} (\frac{1}{f_2} - \frac{1}{S}))^2 + (\frac{e}{S})^2 }$
$S = \frac{1}{f_1 + f_2 - e}$ (Eq. 10)

Combining these two equations gives us a value of $f_{total} = 4.129 \pm 0.272 cm$, which our measured value of $4.2 \pm 0.5 cm$ falls within the error bounds of, verifying the formula in Eq. 9.

Part 4

For this part, we start by denoting the focal length of the lens with $f$, the distance between the film and lens with $o$, and the distance between the lens and image as $i$. We know that

$\frac{1}{i} = \frac{1}{f} - \frac{1}{o}$ (Eq. 11)

with the uncertainty derviced from the propagation of error formula:

$\sigma_i = i \sqrt{ (\sigma_f (\frac{1}{f} - \frac{1}{f-o}))^2 + (\sigma_o (\frac{1}{o} - \frac{1}{f-o}))^2 }$ (Eq. 12)

From this, we calculate our predicted values for $i$ for our various values of $o$, shown alongside the measured results in the table below:

Table 6: Comparison between predicted and measured values for image distance

As can be seen in the table, our measured values all fell within the error bounds of our predicted values, confirming our formula in Eq. 11.

Also, we noticed that while all of our object distances were between $f$ and $2f$, all of our projected image sizes were magnified, confirming that when the object is within $f$ and $2f$, the resulting image is magnified. In addition, we also noticed that the resulting projections were always inverted compared to the original image.

Conclusion

To conclude, we were able to achieve all of the goals in this experiment. In part 1, we were able to verify Snell’s law by measuring the angles of refraction and total internal reflection for a prism. In part 2, we investigated the refraction of different lens shapes and found the magnification factors for a biconcave-planoconvex lens combination. For part 3, we successfully verified the formula for the focal length of a combination of two thin lenses separated by a finite distance, given in Eq. 9. And finally, in part 4 we verified the equation for image distance relative to focal length and object distance, shown in Eq. 11.