Reflection at a Spherical Surface. Concave Mirrors

ReflectionReflectionat aat a

Spherical SurfaceSpherical Surface

Concave Mirrors

Concave Mirrors

The center of curvature ofthe surface is at point “c”

Point ”p” is the Objectp v

Point ”v” is the vertex of the mirror

The line “pcv” is called the optic axis.It is sometimes referred to as the principle axis.

♫

Concave Mirrors

Concave Mirrors

p’

p’ is known as the real image

o

p is known as the objectp

Concave Mirrors

p’

ooo

p

s’

s

There exists a relationship between the distances s and s’ relative to the curvature or the reflecting surface, R.

That Relationship is”

1 1 2

------- + ------- = ------

s s’ R

R

Paraxial Approximations

The trigonometric calculations for spherical mirrors in

determining focal points, radii and such (as in the last

slide) are based on accepting that the tangent of small

angles are nearly equal to the angle itself (in radians)

and that rays are assumed parallel to the optic axis at

distances far from the mirror. The term “paraxial

approximations” is used when referring to these

calculations.

Planar Spherical Mirrors

p’

ooo

p

s’

s

As “R” approaches infinity, the mirror becomes a plan mirror and the above equation reduces to s = -s’

1 1 2

------- + ------- = ------

s s’ R

R

spherical aberration

p’

ooo

p

s’

s

As “θ” increases point p’ moves closer to the vertex reducing the sharpness of the image. this is known as “spherical aberration”.

1 1 2

------- + ------- = ------

s s’ R

θ

When the center of curvature “C” is on the same side as

the outgoing (reflected) light, the radius of curvature is

positive: otherwise it is negative.

R will always be positive for a concave mirror and negative for a convex mirror

ooo

Sign Rules . . . No. 3

Lateral Magnification(spherical mirror)

p’

ooo

p

s’

s

The formula for Lateral Magnification is: y’ M = -------- y

R

The ratio of the height of the object to the height of the real image is know as lateral magnification

( y vs. y’ ) y

y’

If m is negative the image is inverted, relative to the object

Lateral Magnification and Distance(spherical mirror)

p’

ooo

p

s’

s

This relationship is:

y - y’ -------- = ------- s s’

Since the two triangles are similar, there exists a proportionality between the magnification and the distance.

y

y’

Therefore:

y’ - s’m = -------- = ------- y s

Again, If m is negative the image is inverted, relative to the object

Lateral magnification in a spherical mirror

• Can you mathematically determine the magnification in a planar mirror ?

s = - s’ and y = y’

y’ - s’ -s m = -------- = ------- = ------- = 1 y s -s

• If an object is very far away from a spherical mirror ( s = ∞ ) what is the formula for the radius of the mirror in terms of the image distance (s’) ?

1 1 2 R--- + --- = --- s’ = ----

∞ s’ R 2

Given:

A concave mirror forms an image, on a wall 3 m from the mirror, of a

filament of a headlight lamp 10 cm in front of the mirror.

(a) what is the radius of curvature of the mirror?

(b) what is the height of the image if the height of the object is 5

mm?

(a) both object distance and image distance are positive.

s = 10 cm and s’ = 300 cm

1 / 10 cm + 1 / 300 cm = 2 / R

R = 19.4 cm

(b) m = y’ / y = - s’ / s = -300 cm / 10 cm = -30

The image is therefore inverted (m is negative) and is 30 times the height of the object, or 30 x 5 mm = 150 mm

☺

Focal Point

The point “F” at which the incident parallel rays converge is called the Focal Point

The distance from the vertex to the focal point “F” is called the Focal Length

F

The focal length (f) is related to R by f = R/2

The focal length (f) is related to the distance by:

1 1 1 --- + --- = ---

s s’ f

(object-image relation – spherical mirror)

• Do problems 34.3 and 34.5