Chapter 31: Images and Optical Instruments Reflection at a plane surface Image formation The reflected rays entering eyes look as though they had come from image P’. P P’ Light rays radiate from a point object at P in all directions. virtual image
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Chapter 31: Images and Optical Instruments
Reflection at a plane surface
Image formationThe reflected rays entering eyes lookas though they had come from image P’.
P
P’Light rays radiate from a point objectat P in all directions.
virtualimage
Reflection and refraction at a plane surface
Image formation
Reflection and refraction at a plane surfacei (or s’) is the image distances is the object distance:
|s| =|i|Image formation
Sign Rules:(1) Sign rule for the object distance:
When object is on the same side of the reflectingor refracting surface as the incoming light, the objectdistance s is positive. Otherwise it is negative.
(2) Sign rule for the image distance:When image is on the same side of the reflecting orrefracting surface as the outgoing light, the imagedistance i ( or s’) is positive. Otherwise it is negative.
(3) Sign rule for the radius of curvature of a sphericalsurface:When the center of curvature C is on the same sideas the outgoing light, the radius of the curvature ispositive. Otherwise it is negative.
s’
Reflection at a plane surface
Image formationMultiple image due to multipleReflection by two mirrors
h h’
m = h’/h=1lateral magnification
image is erectimage is virtual
Reflection at a plane surface
Image formationWhen a flat mirror is rotated, howmuch is the image rotated?
Reflection at a spherical mirrorConcave and convex mirror
Reflection at a spherical mirrorFocal points at concave and convex mirror
Focal point or focus: Point F at which rays from a source point arebrought together (focused) to form an image.
Focal length: Distance f from mirror where focus occurs.f=R/2 where R is the radius of a spherical mirror.
Reflection at a spherical mirrorFocal points at a concave mirror
αγδαβγ
+=+=
γδβ 2=+
h )/(tan)'/(tan)/(tan
dRhdshdsh
−=−=−=
γδβ
d
Rhssdifsh
sh
/','/
/
=<<=
=
γδβ
fRss12
'11
==+
object
image
If 2/', Rss =∞=
s’
Reflection at a spherical mirrorImage of an extended object at a concave mirror
real image
Principle rays: Light rays that can be traced (more easily) from the source to the image:1. Parallel to optical axis2. Passing through the focal point3. Passing through the center of curvature4. Passing through the center of the mirror surface or lens
Reflection at a spherical mirrorMagnification of image at a concave mirror
hh’
sff
ss
hhm
−=−==
''When s,s’ >0 , m<0 inverted
s/s’<0, m>0 upright orerect
Reflection at a spherical mirrorExample with a concave mirror
real image real image
real image virtual image
Reflection at a spherical mirrorExample with a concave mirror
Reflection at a spherical mirrorImage at a convex mirror
fRss12
'11
==+
s s’f f
Rsf
fssm
−=−=
'
s positives’ negative (virtual image)R negativef negative
Reflection at a spherical mirrorMagnification of image at a convex mirror
s’
''
ssatheight
ssatheight=
For a convex mirror f < 0
sff
ssm
fssss
satheightsatheightm
−=−=
→
=+==
'
1'
11,''
m > 1 magnified m < 1 minimizedm > 0 image uprightm < 0 image inverted
Refraction at a spherical surfaceRefraction at a convex spherical surface
For small angles θθ ≈sin 112211 θφθθ ==→ nn
)/()/()/()(
1222111
2121
nnRnRRABABfABBF
−≅−→=−=→≅−
θθθθθθθθ R
nnnf )(
12
2
−=
θ1 θ1−θ2
Refraction at a spherical surfaceRefraction at a concave spherical surface
For a concave surface, we can use the same formula
Rnn
nf )(12
2
−=
But in this case R < 0 and f < 0. Therefore the image is virtual.
Refraction at a spherical surfaceRelation between source and image distanceat a convex spherical surface
s’
Rnn
sn
sn
sAB
RABn
sAB
RABn
ssABRABnn
122121
2121
')
'()(
')()(
−=+⇒−=+⇒
===−=+−=+=
αγβαβγβαβθγβθ Snell’s law
For a convex (concave) surface, R >(<) 0.
Refraction at a spherical surfaceExample of a convex surface
|s’|
Refraction at a spherical surfaceExample of a concave surface
|s’|
Refraction at a spherical surfaceExample of a concave surface
Refraction at a spherical surfaceExample of a concave surface
Convex LensSign rules for convex and concave lens:
Sign Rules:(1) Sign rule for the object distance:
When object is on the same side of the reflectingor refracting surface as the incoming light, the objectdistance s is positive. Otherwise it is negative.
(2) Sign rule for the image distance:When image is on the same side of the reflecting orrefracting surface as the outgoing light, the imagedistance i (or s’) is positive (real image). Otherwise it is negative(virtual image).
(3) Sign rule for the radius of curvature of a sphericalsurface:When the center of curvature C is on the same sideas the outgoing light, the radius of the curvature ispositive. Otherwise it is negative.
Convex LensLens-makers (thin lens) formula
surface 1
surface 2
Image due to surface 1:11
'11
'11
11111nsnR
nsR
nsn
s−
−=→
−=+
s’1 becomes source s2 for surface 2:2
'2
'12
'22
11111R
nssR
nss
n −=+
−→
−=+
2'211
11)11(R
nsnsnR
nn −=++
−−
s’
R1>0 R2<0s1 = s and s’2 = s’:
fRRn
ss1)11)(1(
'11
21
=−−=+Parallel rays (s=inf.)w.r.t. the axis convergeat the focal point
Convex LensMagnification
s’
PIIPSSSSIIm
'''/'
∆=∆=
ss
m'
=
sff
ssm
−=−=
'same as for mirrors
Convex LensObject between the focal point and lens
A virtual image
Convex LensObject position, image position, and magnification
real inverted imagem < 1
real inverted imagem >1
virtual erect imagem >1
LensTypes of lens
LensTwo lens systems
LensTwo lens systems (cont’d)
LensTwo lens systems (cont’d)
LensTwo lens systems (cont’d)
EyesAnatomy of eye
EyesNear- and far-sightedness and corrective lenses
farsightedness
nearsightedness
Angular size
h
d
s
is m
dh
θθθ θ ≡≅
In general the minimum distance d=dmin~25 cm atwhich an eye can see image of an object comfortablyand clearly.
Magnifying glass
cmdf
ddh
fhmM
fh
sh
smh
sh
s
i
fsi
ii
25,//
)(|'||'|
tan
minmin
min
=====
====≈ =
θθ
θθ
θ
sfs11
'1
−= when but−∞=→= 'sfs ∞=−
=ssm '
for human eye.
the minimum distance atwhich an eye can see imageof an object comfortably and clearly.
virtual image
s’
the eye is most relaxed
s
θihi h
Microscope
)/()(]//[)]/()[(/)magnifier afor //()/()(/
)///||(/||
21minmin02102
210212
1111001
ffLddhffLhmMfhihffLhfhfLsLsimfLhhmh
objecti
iii
====→==≈≈≈=≈=
θθθθ
θ
Q
Q
21
21
,,
ffffiL >>≈
smallObject is placednear F1 (s1~f1).Image by lens1is close to thefocal point of lens2 at F2.
magnifier
image ang. size
θ2i
Refracting telescope
21
2121
1001
/////
)('/'
ffmfffh
sfsshsmhh
siobjecti
si
ss
====≈
∞=====
θθθθθθ
θθ
θ
Qangular size of image by lens2; eyeis close to eyepiece
image height by lens1 at its focal point
Image by lens1 is at its focal point which isthe focal point of lens 2
image distanceafter lens1
magnifier
Reflecting telescope
21 // ffm objecti == θθθ
Aberration
sphere paraboloid
Chromatic aberration
Gravitational lens
ExercisesProblem 1
What is the size of the smallest vertical plane mirror in which a womanof height h can see her full-length?
Solutionx x/2
The minimum length of mirror fora woman to see her full height hIs h/2 as shown in the figure right.
(h-x)/2
h-x
ExercisesProblem 2 (focal length of a zoom lens)
f2=-|f2|f1
I’r0 Q r’0
d (variable)< s’2f1
f
ray bundle
dxr0
f1
s2
|f2|>f1-d
(a) Show that the radius of the ray bundle decreases to
1101000'
0'
00 /)()/( fdfrfdrrxrrrrx −=−=−=→−=
110'
0 /)( fdfrr −=
ExercisesProblem 2 (focal length of a zoom lens)
f2=-|f2|f1
I’r0 Q r’0
d (variable)< s’2
ray bundle
dxr0
f1
s2
f1
f
(b) Show that the final image I’ is formed a distanceto the right of the diverging lens.
)/()( 1212'2 dffdffs +−−=
dffdff
sdffdff
fdfffd
sfsfdfds
+−−
=→−+−
=−−−
=→=+−
→−=12
12'2
12
12
12
21'22
'21
12)(
)()(1111
ExercisesProblem 2 (focal length of a zoom lens)
f2=-|f2|f1
I’r0 Q r’0
d (variable)< s’2
ray bundle
dxr0
f1
s2
f1
f
(c) If the rays that emerge from the diverging lens and reach the finalimage point are extended backward to the left of the diverging lens,they will eventually expand to the original radius r0 at some point Q.The distance from the final image I’ to the point Q is the effective focallength of the lens combination. Find the effective focal length.
dffff
fdff
dffdf
fsrrf
fr
sr
+−=→
+−−
−==→=
12
21
12
12
1
1'2'
0
00'2
'0 )(
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