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_{PH203KUZMASPRING2014}
Video 4.1: Stepbystep instructions on drawing ray diagrams: Concave mirror  Video 4.2: Concave lens and convex mirror 

The Law of Reflection  

$\theta_r=\theta_i$  $\theta_r=$ angle of reflection $\theta_i=$ angle of incidence 

Focal Length for a Convex Mirror of Radius R  
$f=\frac{1}{2}R$  SI unit: meter (m) 

Focal Length for a Concave Mirror of Radius R  
$f=\frac{1}{2}R$  SI unit: meter (m) 

The Mirror Equation  
$\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}$ $f$ is positive for concave mirrors. $f$ is negative for convex mirrors. Image Distance $d_i$ is positive for images in front of a mirror (real images). $d_i$ is negative for images behind a mirror (virtual images). $d_o$ is positive for objects in front of a mirror (real objects). $d_o$ is negative for objects behind a mirror (virtual objects).  $\frac{h_o}{h_i}=\frac{d_o}{d_i}$  $\frac{h_o}{h_i}=\frac{d_oR}{Rd_i}$ 

Magnification, m  
$m=\frac{h_i}{h_o}=\frac{d_i}{d_o}$  Magnification $m$ is positive for upright images. $m$ is negative for inverted images. 

The height of an image  
$ h_i= \left(\frac{d_i}{d_o}\right)h_o$  
The Refraction of Light  
a wave propagates from a medium in which its speed is $v_1$ to another in which its speed is $v_2<v_1$ $\frac{\sin \theta_1}{v_1}=\frac{\sin \theta_2}{v_2}$  
Definition of the Index of Refraction, $n$  
$v=\frac{c}{n}$  $v=$ speed of light in a given medium  
Snell's Law  
$n_1\sin \theta_1=n_2 \sin \theta_2$  
Critical Angle for Total Internal Reflection, $\theta_c$ $\sin \theta \leq 1$, $n_1 \geq n_2$ 

$\sin \theta_c=\frac{n_2}{n_1}$  
Total Polarization, Brewster's Angle, $\theta_B$  
Reflected light is completely polarized when the reflected and refracted beams are at right angles to one another. The direction of polarization is parallel to the reflecting surface. $\tan \theta_B=\frac{n_2}{n_1}$  
The ThinLens Equation  
$\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}$ Equations to the right are only used to derive the thinlens equation above in this particular case. The thinlens equation above is true for all thin lenses, however.  $\frac{h_o}{f}=\frac{h_i}{d_if}$  $\frac{h_o}{d_o}=\frac{h_i}{d_i}$ 

Magnification, $m$  
$m=\frac{d_i}{d_o}$ 

$f$ is positive for converging (convex) lenses. $f$ is negative for diverging (concave) lenses. $m$ is positive for upright images (same orientation as object). $m$ is negative for inverted images (opposite orientation of object). $d_i$ is positive for real images (images on the opposite side of the lens from the object). $d_i$ is negative for virtual images (images on the same side of the lens as the object). Object Distance $d_o$ is positive for real objects (from which light diverges). $d_o$ is negative for virtual objects (toward which light converges). 
A customer, trying a new hat and new shoes, is standing 1 m away from a mirror mounted on a vertical wall. The customer's height (including the hat) is 1.7 m, and the vertical distance from the top of the hat to the eye level is 12 cm.
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What is the ideal width of a flat rearview mirror inside a car, if the distance from the front window to the rear window of the car is L, the width of the rear window is W, and the distance from the driver's eyes to the mirror is d ?
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The object of height $h_o$ is at a distance $\;d_o\!=\!\frac{1}{2}f\;$ from the concave mirror of focal distance $f$. Find the image distance, size and other properties.
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The object of height $h_o$ is at a distance $\;d_o\!=\!\frac{3}{2}f\;$ from the concave mirror of focal distance $f$. Find the image distance, size and other properties.
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The object of height $h_o$ is at a distance $\;d_o\!=\!2\,f\;$ from the convex lens of focal distance $f$. Find the image distance, size and other properties.
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The object of height $h_o$ is at a distance $\;d_o\!=\!1.1\,f\;$ from the convex lens of focal distance $f$. Find the image distance, size and other properties.
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The object of height $h_o$ is at a distance $\;d_o\!=\!0.9\,f\;$ from the convex lens of focal distance $f$. Find the image distance, size and other properties.
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The object of height $h_o$ is at a distance $\;d_o\!=\!\big\,f\big\;$ from the concave lens of focal distance $\;f\!<\!0$. Find the image distance, size and other properties.
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Sunlight enters a room at an angle of 32$^\circ$ above the horizontal and reflects from a small mirror lying flat on the floor. The reflected light forms a spot on a wall that is 2.0 m behind the mirror. If you now place a pencil under the edge of the mirror nearer the wall, tilting it upward by 5.0$^\circ$, how much higher on the wall $(\Delta y)$ is the spot?
I was working on problem 26.5 and I set it up the following way:
So, basically, your question is: if the mirror is rotated by a small angle $\alpha$, and the incident ray is the same relative to the room, how much is the reflected ray rotated by. Try to think through the following easier questions (in terms of the angle $\alpha$):