Reflection at a Spherical Surface
Sign Rule (#4) for the
radius of curvature of a
spherical surface:
radius of curvature R
is + when the center of
curvature C is on the same
side as the outgoing light
(concave) and – otherwise.
concave
V
The
(CV is called the optical axis.)
convex
V
Reflection at a Spherical Surface
From  PBC ,
   
From CBP ',
   
Eliminating  between
these 2 Eqs:
    2
Reflection at a Spherical Surface
Relating the angles to the
physical distances, we have:
tan  
h
s 
tan  
tan  
h
R 
h
s ' 
Reflection at a Spherical Surface
Paraxial Assumption: We consider only rays which are nearly
parallel to the optical axis and close to it. These rays are called
paraxial rays.
With this approximation, the angles , , and  will be small and
 can be neglected with respect to s, s’, and R and
h
  tan  
s
h
  tan  
s'
  tan  
h
R
Combining with     2 and eliminating h, we then arrive at,
1 1 2
 
s s' R
(object-image relation, spherical mirror)
Focal Point and Focal Length
For an object (stars) very far away
from the mirror ( s  ) , the incoming
rays can be considered to be parallel.
What will happen to these rays?
1 1 2
 
 s' R
R
 s' 
2
Focal Point and Focal Length
All parallel rays from ( s  ) will
converge to the same image point at
s’ = R/2. This special point is called
the focal point F and the distance
from the vertex of the mirror to F is
call the focal length f,
R
f 
2
(focal length of a
spherical mirror)
Focal Point and Focal Length
Let consider the reverse situation,
light rays are emanating from the
focal point F, where will the image
be?
R
f 
2 1 2
 
R s' R

2
1
 0 or
s'
s'  
Thus, as expected, the situation
is time reversed, rays starting out
from F will be reflected out
toward infinity as parallel rays.
Spherical Aberration
Recall that this is only an approximation. The focal
point is a sharp point only if we consider paraxial rays.
For non-paraxial rays, they do not necessary converge
to a precise point. The blurring of the focal point in an
actual spherical mirror is called spherical aberration.
Parabolic vs. Circular Mirrors
With aberrations  only paraxial rays
fall on Focus
No aberrations  all parallel rays fall
on Focus
Lateral Magnification of a Spherical
Mirror
Following rays #1 and #2, we can form the following two similar
triangles: beige and blue.
1
2
Lateral Magnification of a Spherical
Mirror
The two similar triangles gives,
y y'

s
s'
Substituting this into the definition for m,
m
y ' s '

y
s
Convex Spherical Mirrors
The geometric optics formulas for a convex mirror are the same
as for a concave mirror except that R and f are negative.
Convex Spherical Mirrors
With R and f negative, parallel rays falling upon a convex mirror will
diverge as if emanating from a virtual focal point F behind the mirror.
Similar to the concave mirror, rays aiming toward this virtual focal
point will be reflected back toward infinity as parallel rays.
convex
concave
Convex Spherical Mirrors
With R and f negative, parallel rays falling upon a convex mirror will
diverge as if emanating from a virtual focal point F behind the mirror.
Similar to the concave mirror, rays aiming toward this virtual focal
point will be reflected back toward infinity as parallel rays.
convex
concave
Summary for Spherical Mirrors
The following are valid for both concave and convex spherical
mirrors if we follow the proper sign conventions.
1 1 1
 
s s' f
R
f 
2
s'
m
s
(object-image relation, spherical mirror)
(focal length, spherical mirror)
(lateral magnification, spherical mirror)
Note: these equations agree with results for a flat mirror if we take R   .
Sign Rules for Spherical Mirrors
1.
Object Distance:

2.
Image Distance:

3.
s’ is + if the image is on the same side as the outgoing light and is –
otherwise.
Object/Image Height:

4.
s is + if the object is on the same side as the incoming light (for both
reflecting and refracting surfaces) and s is – otherwise.
y (y’) is + if the image (object) is erect or upright. It is – if it is
inverted.
Radius of Curvature and Focal Length:

R and f is + when the center of curvature C is on the same side as the
outgoing light and – otherwise.
Geometric Methods: Rays Tracing
Principal rays for concave mirror
Geometric Methods: Rays Tracing
Principal rays for convex mirror
(might need to extrapolate lines to
intersect at image point)
Example 34.4: Concave Mirror
P>C>F
P=C
Example 34.4: Concave Mirror
P=F
P < F< C
Example 34.4: Concave Mirror
Mirrors & Thin Lens Applet (by Fu-Kwun Hwang)
http://www.physics.metu.edu.tr/~bucurgat/ntnujava/Lens/lens_e.html
This applet is for both mirrors and thin lens. Use the drop down menu to
choose.
Demo with Two Circular Mirrors
forming a real image here
Dsp mirror
Ddouble mirror
Refraction at a Spherical Surface
2
1
•
•
•
Ray 1 from P going through V (normal to the interface) will not
suffer any deflection.
Ray 2 from P going toward B will be refracted into nb according to
Snell’s law.
Image will form at P’ where these two rays converge.
Refraction at a Spherical Surface
At B, Snell’s law
gives,
2
1
na sin  a  nb sin b
From  PBC ,
a    
From  P ' BC ,
    b
 b    
From trigonometry, we also have,
tan  
h
s 
tan  
h
tan  
R 
h
s ' 
Refraction at a Spherical Surface
Again, consider only paraxial rays so that the incident angles are small, we
can use the small angle approximations: sin  ~ tan  ~ .
With this, Snell’s law becomes: na a  nbb
Substituting a and b from previous slide, we have,
na (   )  nb (   )
na  na  nb  nb 
na  nb   (nb  na )
With the small angle approximations, the trig relations reduce to,
h
  tan  
s
  tan  
h
s'
  tan  
h
R
Refraction at a Spherical Surface
Substituting these expressions for , and  into Eq.
common factor h, we then have,
na nb nb  na
 
s s'
R
(object-image relationship,
spherical refracting surface)
and eliminating the
Refraction at a Spherical Surface
To calculate the lateral magnification m, we consider the following rays:
1
2
•
•
Ray 1 from Q going toward C (along the normal to the interface) will
not suffer any deflection.
Ray 2 from Q going toward V will be refracted into nb according to
Snell’s law.
Refraction at a Spherical Surface
1
2
From geometry, we have the following relations,
tan  a  y s
From Snell’s law, we have,
tan b   y ' s '
na sin  a  nb sin b
Refraction at a Spherical Surface
Using the small angle approximation again (sin  ~ tan ), the
Snell’s law can be rewritten as,
na sin  a  nb sin b
y
y'

 na   nb
s
s'
sin   tan 
Substituting these into the definition for lateral
magnification, we have,
m
n s'
y'
m a
nb s
y
(lateral magnification,
spherical refracting surface)
Refraction at a Flat Surface
For a flat surface, we have R   . Then, the Object-Image
relation can be reduced simply as,
na nb nb  na
 
0
s s'


na
n
 b
s
s'

na s '
 1
nb s
virtual
Combing this with our result for lateral
magnification, we have, m  1 (upright)
so that, the image is unmagnified and
upright.
Example 34.5 & 34.6

Images formed by a spherical surface can be real or virtual
depending on na, nb, s, and R.
Ex 34.5:
na nb nb  na
 
s s'
R

1.00
1.52
0 .5 2


8.00cm
s'
2.00cm
 s '  11.3cm
Example 34.5 & 34.6

Images formed by a spherical surface can be real or virtual
depending on na, nb, s, and R.
Ex 34.6:
na nb nb  na
 
s s'
R
1.33
1.52
0 .1 9



8.00cm
s'
2.00cm
 s '  21.3cm