Curved Mirrors

Chapter 14 Section 3

Spherical Mirrors A Spherical mirror has the shape of a

sphere’s surface. It has some kind of curve The mirror is not flat.

Examples: Passenger side rear view mirror Some dressing table mirrors

Used to magnify or shrink the size of a reflected image of an object.

Concave Spherical Mirrors

Concave Spherical Mirror – An inwardly curved mirrored surface that is a portion of sphere and that converges incoming light rays. Concave spherical mirrors are used

whenever a magnified image of an object is needed.

Curvature of the mirror

One factor that determines where the image will appear and how large that image will be is the amount by which the mirror is curved.

This depends on the radius of curvature

Radius of Curvature

The radius of curvature is the same as the radius of the sphere whose curvature would be identical to that of the mirror.

Radius of curvature variable – Uppercase (R) The radius is the distance from the

mirror’s surface to the center of curvature (C)

Principal Axis

The Principal axis is the line that extends infinitely from the center of the mirrors surface through the center of curvature.

Concave Mirror Diagram

Image Point

The image point in the previous picture forms below the principal axis and forms the image in front of the mirror.

If a piece of paper was placed at the image point, a clear image would form on the paper of the pencil. Move the paper back and forth and the

image will become blurry and out of focus.

Real Image

Real Image – An image formed when rays of light actually intersect at a single point.

Real images can be displayed on a surface Example:

Movie screen

Image Location Can Be Predicted

The object’s distance, image distance and radius of curvature are all interdependent.

Change one of the three can affect one of the others.

Mirror Equation

p – Object Distance

q – Image Distance

R – Radius of Curvature

Rqp

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Focal Point

Focal Point - Point where rays from lens or curved mirrors converge. Denoted by capital letter (F)

At this point on a curved mirror the light rays converge to a single point. The reflected rays of light from a source

emitted at the focal point will emerge parallel to each other.

Focal Point Diagram

Focal Length

Focal Length – The distance from the curved mirror or lens to the focal points. Denoted as lower case (f)

The focal length is half the distance of the radius of curvature. Or the radius of curvature is double that

of the focal length.

Mirror Equation

p – Object Distance

q – Image Distance

f – focal Length fqp

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Front and Back of a Concave Mirror

In order to use the mirror equation, a set of sign conventions for the three variables must be established. The region where light rays reflect and real

images form is called the front side of the mirror.

The region where light rays do not exist and imaginary images are formed is called the back side of the mirror.

Mirror images are usually drawn so the front side is to the left of the mirror’s surface.

Positive and Negative Concave Mirror

Object and image distances can be positive and negative. The distance has a positive sign when

measured from the center of the mirror to any point in front side of the mirror.

Distances for images that form on the back side of the mirror will always be a negative distance.

Since the focal point is in front on the mirror, the focal length will always be positive.

Positive and Negative Heights

The object and image heights are positive when they are above the principal axis.

The object and image heights are negative when they are below the principal axis.

Magnification

Unlike flat mirrors, curved mirrors form images that are not the same size as the object.

To measure how much larger or smaller the image is compared to the object's size is called the magnification of the image.

Magnification is defined as the ratio of the height of the pencil’s image to the pencil’s actual height. Magnification is denoted by an

uppercase letter (M) It is also defined as the negative of

the ratio of the image distance over the object distance.

If the image is smaller then the object, M should be less than 1.

If the image is greater then the object, M should be greater than 1. Magnification is a unitless quantity.

Magnification Equation

M – Magnification h’ – Image height h – Object height q – image distance p – object height

p

q

h

hM

'

Magnification Clarification

For an image in front of the mirror, M is negative and is inverted.

For an image in the back of the mirror, M is positive and the image is upright.

Sign Convention for Magnification

Orientation of image with respect to

object

Sign of M Types of image this applies to

Upright + Virtual

Inverted - Real

Ray Diagrams

Ray diagrams can be used for concave spherical mirrors.

The rules for making a ray diagram of a flat mirror are the same for making ray diagrams of concave spherical mirrors. Measure all distances along the principal

axis and mark the center of curvature (C) and focal point (F).

Must be drawn to scale.

Ray Diagram for a Spherical Mirror

For a spherical mirror, three reference rays are used to find the image point. The intersection of any two rays locates

the image. The third ray should intersect the same

point and is used to check the diagram.

Rules For Drawing Reference Rays

Ray Line drawn from object to mirror

Line drawn from mirror to image after reflection

1 Parallel to principal axis

Through focal point (F)

2 Through focal point (F)

Parallel to principal axis

3 Through center of curvature (C)

Back along itself through (C)

Object Distance Greater Than Focal Length

Object Distance Equal to Focal Length

Object Distance Less Than Focal Length

Example Problem

When an object is placed 30.0cm in front of a concave mirror, a real image is formed 60.0cm from the mirror’s surface. Find the focal length.

Example Problem Answer

f = 20.0 cm

Convex Spherical Mirror Convex Spherical Mirror – An

outwardly curved, mirrored surface that is a portion of a sphere and that diverges incoming light rays.

Examples: Mirrors to dangerous driveway

entrances. Passenger Side rearview Mirror

“Objects in the mirror are closer than they appear”

Diverging Mirror

A convex spherical mirror is sometimes called a “Diverging Mirror” The incoming rays of light diverge after

reflecting off the surface as though they were coming from some point behind the mirror.

Virtual Image

The image that is produced on a convex spherical mirror (diverging mirror) will always be a virtual image. Imaginary image.

Convex Spherical Mirror Terminology

The image distance (q) will always be negative.

The focal length (f) will always be negative cause the mirrored surface is on the opposite side of the radius.

Ray Diagrams for a Convex Mirror

Drawing Ray Diagrams for a convex mirror are slightly different then for a concave mirror. The three rays are still needed going to

the focal point, center of curvature and one parallel to the principal axis

The focal point and center of curvature are located behind the mirror. Dotted lines are extended along the

reflected reference rays to points behind the mirror.

An upright imaginary image forms where the three rays intersect.

Ray Diagram of a Convex Spherical Mirror

Magnification for a Convex Spherical Mirror

The magnification for a convex spherical mirror will always be less than 1. The image always appears smaller than

the object. This gives the appearance that the object is

further away from the mirror. Hence the warning sticker on the bottom of

the mirror on the passenger side rearview mirror.

Convex Mirror Applications

Convex spherical mirrors take the objects in a large field of view and produce a small image.

They are well suited for providing a fixed observer with a complete view of a large area.

Different Applications of Convex Spherical Mirrors

Symbol Situation Sign

p The object is in front of the mirror +

q The image is in front of the mirror (real image) +

q The image is behind the mirror (virtual image) -

R,f The center of curvature is in front of the mirror (Concave Spherical Mirror)

+

R,f The center of curvature is behind the mirror (Convex Spherical Mirror)

-

R,f The mirror has no curvature (Flat Mirror) ∞

h’ The image is above the principal axis +

h’ The image is below the principal axis -

Example Problem

The radius of curvature of a convex mirror is 12.0cm. Where is the focal point?

Example Problem Answer

6.00 cm behind the mirror, therefore the value needs to be negative.

f = -6.00cm