Chapter 36. Image Formation

36.1 Images Formed by Flat Mirrors p: object distance q: image distance O: object I: image (1) Type of image

Real image is one in which rays converge at the image point. Real images can be displayed on screens.

Virtual image is one in which the light ray do not converge to the image point but appear to emanate from that point. Virtual images cannot be displayed on screens.

The images seen in flat mirror are always virtual.

(2) Location of image

To find out where the image is formed, follow at least two rays of light as they reflect from the mirror.

The image is as far behind the mirror as the object is in front (p = q). The image is upright.

(3) Lateral magnification M

Define M !image heightobject height

=h'

h

For flat mirror, M = 1. The image is unmagnified.

Q: How tall the mirror needs to be in order to view the whole body.

36.2 Images Formed by Spherical Mirrors Concave Mirrors

Reflection Characteristics: (1) Any light ray passing through the center of curvature, C,

will be reflected back on itself. (2) A point object O placed on the principle axis that is farther

than R from the mirror surface, forms a real image at I. (3) Light ray from a distant object reflected from a concave

mirror will go through the focal point, F.

Focal length f = R/2

Image formation: Let p be the object distance, and q be the image distance. (1) If p > R, the image is real, inverted, and reduced in size. (2) If p < R/2, the image is virtual, upright, and enlarged.

Mirror equation 1p+1q=1f

Magnification M =h'h= !

qp

Example: A concave mirror has a focal length of 10 cm. Find

the location of the image for object distances of (a) 25 cm, (b) 10 cm, and (c) 5 cm.

Solution f = 10 cm, R = 2f =20 cm. (a) p = 25 cm. Thus p > R.

The image is real, inverted, and reduced in size. 1p+1q=1f

! 125cm

+1q=

110cm

! q = 16.7 cm

M = !qp= !16.7cm25cm

= !0.668 (negative means inverted)

(b) p = 10 cm The object is located at the focal point. The reflected rays travel parallel to the principal axis. No image form. 1p+1q=1f

! 110cm

+1q=

110cm

! q = "

(c) p = 5 cm, p < R The image is virtual, upright, and enlarged. 1p+1q=1f

! 15cm

+1q=

110cm

! q = -10 cm

(negative: image is behind the mirror and is virtual)

M = !qp= !

(!10cm)5cm

= 2

Convex mirror • The image formed by the real object is virtual and upright. • Equations for concave mirror also can be applied for convex

mirror with following sign convention: Sign Convention for Mirrors q > 0 if the image is in front of the mirror (real image). q < 0 if the image is in back of the mirror (virtual image). Both f and R > 0 for concave mirror. Both f and R < 0 for convex mirror. M > 0, the image is upright. M < 0, the image is inverted. Example: An object 3 cm high is placed 20 cm from a convex mirror with focal length of 8 cm.

(1) Find the position of the image. Convex mirror: R and f are negative. Thus f = -8 cm.

1q+1p=1f

! 1q+

120cm

=1

!8cm

q = -5.71 cm (a virtual image behind the mirror)

(2) Find the magnification:

M = !qp

= !!5.71cm20cm

" # $

% & ' = 0.286

The image is upright and reduced in size.

36.3 Images formed by Refraction Paraxial rays: light rays that diverge from the object make a small angle with the principal axis.

n1p+n2q=n2 ! n1R

Sign Convention for Spherical Refraction Surface and thin lenses: P > 0, if the object is in front of the surface (real image) P < 0, if the object is in back of the surface (virtual image) q > 0 if the image is in back of the surface (real image). q < 0 if the image is in front of the surface (virtual image). R > 0 if the center of curvature is in back of surface. R < 0 if the center of curvature is in front of surface. M > 0, the image is upright. M < 0, the image is inverted.

Example: A coin 2 cm in diameter is embedded in a solid glass ball of radius 30 cm. The index of refraction of the ball is n1 = 1.5, and the coin is 20 cm from the surface. Find the position of the image. Solution:

n1p+n2q=n2 ! n1R

1.520cm

+1q=1!1.5!30cm

q = -17.1 cm

The image is formed in the glass and is virtual. Flat Refracting Surfaces If the refracting surface is flat, R # "

n1p= !

n2q

! q = !n2n1p

The image formed by a flat refracting surface is on the same side of the surface as the object (virtual, upright image).