7.5.2: Explorations
- Page ID
- 33417
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exploration 1: Camera
This animation can be used to demonstrate the basic operation of a camera (position is given in arbitrary units and angle is given in degrees). Various lenses and light sources can be added by clicking on the appropriate links. The camera is "focused" by dragging the lens to change the lens-to-film distance.
- Click on the link for "Normal Lens" and add a near source. What is the closest position an object could be and still be focused on the film?
- Remove the near source and add an object source. When the object source is at its original location (\(x = 2.3,\: h = 1.2\)), where should a normal lens be placed to focus this object? Where must a telephoto lens be placed? Where should a wide-angle lens be placed?
- Rank the height of the images from (b), from smallest to largest.
- Based on your answer for (c), which lens would you use if you wanted to take a picture in which the object took up most of the photographed area (zoom in)? Explain.
- Based on your answer for (c), which lens would you use if you wanted to take a picture of the object and much of its surroundings (zoom out)? Explain.
- Rank, from smallest to largest, the focal lengths of the three lenses.
Exploration authored by Melissa Dancy and Wolfgang Christian.
Exploration 2: Telescope
In order to understand the magnification of a telescope, it is helpful to understand the idea of the angle that an object or image subtends. So, before exploring a telescope, we need to understand the idea of "the angle that an object subtends." Restart.
- If an eye is located where the image of the eye is (position is given in centimeters and angle is given in radians), the object subtends an angle of \(6^{\circ}\). You should check this by moving the protractor along the optical axis, putting its vertex at the front of the eye, and measuring the angle a light ray from the top of the arrow would make with the optical axis (remember that your protractor measures in radians).
- Now, add a simple magnifying glass. The angle that the image subtends is the angle at which the light exits the lens crossing the optical axis. What angle does the image subtend? The magnification is the ratio of the height of the image to the height of the object, but with small enough angles, it is also the ratio of the angles the image and object subtend.
- Two lenses, an eyepiece and an objective, are used to make a telescope. What good is the telescope if it takes essentially parallel light (from essentially infinity) and turns it back into essentially parallel light (for the relaxed eye)? The answer is in the difference in the angle the object and image subtend. Consider the angle that the object (very far away) subtends. Measure the angle between the beams from infinity and the optical axis. What is that angle?
- Now, measure the angle the light exiting the eyepiece makes with the optical axis (the angle the image subtends). What is that angle?
- The ratio of these angles is the magnification. What is the magnification of this telescope?
- Change the focal length of the eyepiece and focus it (move it so that the light exiting is again parallel light). What is the magnification of this new lens?
- Verify that the magnification of the telescope is equal to the ratio of the focal lengths of the two lenses.
Exploration authored by Anne J. Cox.
Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.