17'1 Reflection at a spherical surface

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17.1 Reflection at a spherical surface

• 17.2.1 Spherical mirror
• is defined as a reflecting surface that is part
  of a sphere.
• There are two types of spherical mirror. It is
  convex (curving outwards) and concave (curving
  inwards) mirror.
• Figures 1.7a and 1.7b show the shape of concave
  and convex mirrors.

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• Terms of spherical mirror
• Centre of curvature (point C)
• is defined as the centre of the sphere of which a
  curved mirror forms a part.
• Radius of curvature, r
• is defined as the radius of the sphere of which a
  curved mirror forms a part.
• Pole or vertex (point P)
• is defined as the point at the centre of the
  mirror.
• Principal axis
• is defined as the straight line through the
  centre of curvature C and pole P of the mirror.
• AB is called the aperture of the mirror.

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17.2.2 Focal point and focal length, f

• Consider the ray diagram for a concave and convex
  mirrors as shown in Figures 1.8a and 1.8b.
• Point F represents the focal point or focus of
  the mirrors.
• Distance f represents the focal length of the
  mirrors.
• The parallel incident rays represent the object
  infinitely far away from the spherical mirror
  e.g. the sun.

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• Focal point or focus, F
• For concave mirror is defined as a point where
  the incident parallel rays converge after
  reflection on the mirror.
• Its focal point is real (principal).
• For convex mirror is defined as a point where
  the incident parallel rays seem to diverge from a
  point behind the mirror after reflection.
• Its focal point is virtual.
• Focal length, f
• is defined as the distance between the focal
  point (focus) F and pole P of the spherical
  mirror.
• The paraxial rays is defined as the rays that are
  near to and almost parallel to the principal axis.

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17.2.3 Relationship between focal length, f and
radius of curvature, r

• Consider a ray AB parallel to the principal axis
  of concave mirror as shown in Figure 1.9.

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• From the Figure 1.9,
• ?BCD
• ?BFD
• By using an isosceles triangle CBF, thus the
  angle ? is given by
• then
• Because of AB is paraxial ray, thus point B is
  too close with pole P then
• Therefore

BD


Taken the angles are ltlt small by considering the
ray AB is paraxial ray.
i
i
tan
CD
BD
q
q


tan
FD
ö
æ
BD
BD


ç
2
ø
è
CD
FD
OR
This relationship also valid for convex mirror.
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17.2.4 Ray diagrams for spherical mirrors

• is defined as the simple graphical method to
  indicate the positions of the object and image in
  a system of mirrors or lenses.
• Figures 1.10a and 1.10b show the graphical method
  of locating an image formed by concave and convex
  mirror.

(a) Concave mirror
(b) Convex mirror
1
1
1
3
2
2
3
2
C
P
F
F
2
3
1
Figure 1.10b
Figure 1.10a
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• Ray 1 - Parallel to principal axis, after
  reflection, passes through the focal point
  (focus) F of a concave mirror or appears to
  come from the focal point F of a convex
  mirror.
• Ray 2 - Passes or directed towards focal point
  F reflected parallel to principal axis.
• Ray 3 - Passes or directed towards centre of
  curvature C, reflected back along the same
  path.
• Images formed by a convex mirror
• Figure 1.11 shows the graphical method of
  locating an image formed by a convex mirror.

At least any two rays for drawing the ray
diagram.
C
P
F
Figure 1.11
back
front
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• The characteristics of the image formed are
• virtual
• upright
• diminished (smaller than the object)
• formed at the back of the mirror (behind the
  mirror)
• Object position ?? any position in front of the
  convex mirror.
• Convex mirror always being used as a driving
  mirror because it has a wide field of view and
  providing an upright image.
• Images formed by a concave mirror
• Concave mirror can be used as a shaving and
  makeup mirrors because it provides an upright and
  virtual images.
• Table 1.1 shows the ray diagrams of locating an
  image formed by a concave mirror for various
  object distance, u.

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• Real
• Inverted
• Diminished
• Formed between point C and F.

u gt r

• Real
• Inverted
• Same size
• Formed at point C.

u r
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• Real
• Inverted
• Magnified
• Formed at a distance greater than CP.

f lt u lt r

• Real or virtual
• Formed at infinity.

u f
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• Linear (lateral) magnification of the spherical
  mirror, m is defined as the ratio between image
  height, hi and object height, ho

• Virtual
• Upright
• Magnified
• Formed at the back of the mirror

u lt f
Table 1.1
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17.2.5 Derivation of Spherical mirror equation

• Figure 1.12 shows an object O at a distance u and
  on the principal axis of a concave mirror. A ray
  from the object O is incident at a point B which
  is close to the pole P of the mirror.

• From the figure,
• ?BOC
• ?BCI
• then, eq. (1)?(2)
• By using ?BOD, ?BCD and ?BID thus

(1)
(2)
D
Figure 1.12
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• By considering point B very close to the pole P,
  hence
• then
• therefore

Substituting this value in eq. (3)
ö
æ
BD
BD
BD



ç
2

ø
è
r
v
u
2
1
1



where
r
v
u
Spherical mirrors equation
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• Table 1.2 shows the sign convention for spherical
  mirrors equation .
• Note
• Real image is formed by the actual light rays
  that pass through the image.
• Real image can be projected on the screen.

Virtual object
Real object
Object distance, u
(in front of the mirror)
(at the back of the mirror)
Real image
Virtual image
Image distance, v
(same side of the object)
(opposite side of the object)
Focal length, f
Concave mirror
Convex mirror
Table 1.2
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Example 3
A dentist uses a small mirror attached to a thin
rod to examine one of your teeth. When the tooth
is 1.20 cm in front of the mirror, the image it
forms is 9.25 cm behind the mirror. Determine a.
the focal length of the mirror and state the type
of the mirror used, b. the magnification of
the image.
(Concave mirror)
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Example 4

• An upright image is formed 20.5 cm from the real
  object by using the spherical mirror. The images
  height is one fourth of objects height.
• a. Where should the mirror be placed relative to
  the object?
• AnsThe mirror should be placed 16.4 cm in front
  of the object.
• b. Calculate the radius of curvature of the
  mirror and describe the
• type of mirror required.
• c. Sketch and label a ray diagram to show the
  formation of the
• image.

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