4. Example: Multiple mirrors and virtual images: how far away are the virtual images?
1m3m
VII. MirrorsVII. Mirrors
4. Example: Multiple mirrors and virtual images: how far away are the virtual images?
2m3m
4. Example: Multiple mirrors and virtual images: how far away are the virtual images?
6m 2m
4. Example: Multiple mirrors and virtual images: how far away are the virtual images?
6m 2m 8m
4. Example: Multiple mirrors and virtual images: how far away are the virtual images?
2m 8m 10m 16m
18m
d) Example: A telescope is designed to take images of distant galaxies. If the
diameter of the telescope’s spherical mirror is 40 cm, where should a detector be placed?
*Step 1: Stars ~ infinitely far away*Step 2: Use mirror equation:
1/s + 1/s’ = 2/R => 0 + 1/f = 2/R, so
f = R/2 = 10 cm.
VII. MirrorsVII. Mirrors
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r
Exterior angle of a triangle:1 = =
VIII. LensesVIII. Lenses
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r
Snell’s Law:n1sin1 =n2sin
VIII. LensesVIII. Lenses
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r
Tangents:tan= r/(s + ).tan= r/(s’ - ).tan = r/(R - ).
VIII. LensesVIII. Lenses
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r
Small angle approximation:tan~ ( << 1).sin~
VIII. LensesVIII. Lenses
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r
•Rewrite Snell’s Law:
n11 ~ n22, so
2 = (n1/n2)().
VIII. LensesVIII. Lenses
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r
* Use exterior law:
(n1 n2) = (n2 - n1)
VIII. LensesVIII. Lenses
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r* Small angles:
tan ~ ~ r/s ( << s).tan ~ ~ r/s’ (<<s’).tan ~ ~ r/R ( << R).
VIII. LensesVIII. Lenses
*Put it all together!
n1/s + n2/s’ = (n2 - n1)/R. (VIII.A.2)
A. Refraction at a spherical surface 1. Images
a) Object distance, Image distance, and Radius of curvature
s
n1 < n2
s’
1
2R
r
VIII. LensesVIII. Lenses
A. Refraction at a spherical surface 1. Images
b) Magnification
s
n1 < n2
s’
1
2
h’
tan1 = h/s.tan2 = -h’/s’.
M = h’/h = -s’tan2/(stan1).
h
VIII. LensesVIII. Lenses
* Use Snell’s Law & small angle
M = h’/h ~ -s’2/(s1) = -n1s’/(n2s). (VIII.A.3)
A. Refraction at a spherical surface 1. Images
b) Magnification
s
n1 < n2
s’
1
2
h’h
VIII. LensesVIII. Lenses
3. Example: An insect is centrally embedded in a spherical globule of amber. The index of refraction for the amber is 2 and the diameter of the globule is 4 cm. How far from the surface does the insect appear?
n1/s + n2/s’ = (n2 -n1)/R;2/(2 cm) + (1/s’) = (-1)1/(2 cm);1/s’ = 1/(4 cm) - 1/(1 cm) = -3/4 cm-1.
s’ = -4/3 = -1.333 cm: virtual image
VIII. LensesVIII. Lenses
Magnification:M = -n1s/n2s’ = -2*(-4/3)/1*2 = 4/3
B. Refraction at a flat surface1. Let R => infinity. Sphere => plane
n1/s = -n2/s’, or s’ = -(n2 /n1)s. (VIII.B.1)
n1 > n2.
VIII. LensesVIII. Lenses
B. Refraction at a flat surface2. Example: A frog sits at the bottom of a
reflecting pool. The pool is .5 m deep with n = 4/3. Where does the frog appear?
s’ = -(n2 /n1)s = -(1/[4/3])(0.5 m) = 3/8 m.
n1 > n2.
VIII. LensesVIII. Lenses
C. Thin Lenses2. Ray tracing: single lens
VIII. LensesVIII. Lenses
C. Thin Lenses2. Ray tracing: double lens
VIII. LensesVIII. Lenses
C. Thin Lenses6. Example: What is the image distance for a
thin glass lens with a focal length of 4/3 cm and an object at 10 cm?
1/s’ =1/f - 1/s = 1/(4/3 cm) - 1/10cm = .65 cm-1.Thus, s’ = +1.5 cm.
How is the image magnified? M = -s’/s = -1.5/10;M = -.15
VIII. LensesVIII. Lenses
7. Example: What is the image distance and magnification for the following pair of lenses:Lens 1: R1 = infinity, R2 = -10 cm, n = 1.5
Lens 2: R1 = 10 cm, R2 = -10 cm, n = 1.5The lenses are separated by 50 cm, and the object is at a distance of 45 cm from the first lens.
Step 1: Find the focal lengths1/f1 = (1/2)(0 - 1/(-10cm) => f1 = 20 cm.1/f2 = (1/2)(1/10cm - 1/(-10cm)) => f2 = 10 cm.
VIII. LensesVIII. Lenses
7. Example: What is the image distance and magnification for the following pair of lenses:Lens 1: R1 = infinity, R2 = -10 cm, n = 1.5
Lens 2: R1 = 10 cm, R2 = -10 cm, n = 1.5The lenses are separated by 50 cm, and the object is at a distance of 45 cm from the first lens.
Step 2: Find image distance for lens 1:1/s’1 = 1/f1 - 1/s1 = 1/20cm - 1/45cm => s’1 = 36 cm.
VIII. LensesVIII. Lenses
7. Example: What is the image distance and magnification for the following pair of lenses:Lens 1: R1 = infinity, R2 = -10 cm, n = 1.5
Lens 2: R1 = 10 cm, R2 = -10 cm, n = 1.5The lenses are separated by 50 cm, and the object is at a distance of 45 cm from the first lens.
Step 3: Find image magnification for lens 1:M1 = -s’1 = -36 cm/45 cm = -0.8
VIII. LensesVIII. Lenses
7. Example: What is the image distance and magnification for the following pair of lenses:Lens 1: R1 = infinity, R2 = -10 cm, n = 1.5
Lens 2: R1 = 10 cm, R2 = -10 cm, n = 1.5
Step 4: Find image distance for lens 2 using image 1 as object:
1/s’2 = 1/10 cm - 1/14 cm => s’2 = 35 cm.
The image is to the right of the second lens.
VIII. LensesVIII. Lenses
7. Example: What is the image distance and magnification for the following pair of lenses:Lens 1: R1 = infinity, R2 = -10 cm, n = 1.5
Lens 2: R1 = 10 cm, R2 = -10 cm, n = 1.5
Step 5: Find total magnificationM2 = -s’2/s2 = -35 cm/14 cm = -2.5.
Total magnification: M = M1* M2 = (-0.8)*(-2.5) = 2
VIII. LensesVIII. Lenses
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