two rules of reflection for concave mirrors

8/18/2019 Two Rules of Reflection for Concave Mirrors

1/15

Two Rules of Reflection for Concave Mirrors• The Anatomy of a Curved Mirror

• Reflection of Light and Image Formation

• Two Rules of Reflection for Concave Mirrors

• Ray Diagrams - Concave Mirrors

• Image Characteristics for Concave Mirrors

• The Mirror Euation

• !"herical A#erration

Light always refects according to the law o refection, regardless o whether

the refection occurs o a fat surace or a curved surace. Using refection

laws allows one to determine the image location or an object. The image

location is the location where all refected light appears to diverge rom. Thus

to determine this location demands that one merely needs to know how light

refects o a mirror. n theprevious section o Lesson !, the image o an

object or a concave mirror was determined by tracing the path o light as itemanated rom an object and refected o a concave mirror. The image was

merely that location where all refected rays intersected. The use o the law

o refection to determine a refected ray is not an easy task. "or each

incident ray, a normal line at the point o incidence on a curved surace must

be drawn and then the law o refection must be applied. # simpler method o

determining a refected ray is needed.

The simpler method relies on two rules o refection or concave mirrors.

They are$

• #ny incident ray traveling parallel to the principal a%is on the way tothe mirror will pass through theocal point upon refection.

• #ny incident ray passing through the ocal point on the way to themirror will travel parallel to the principal a%is upon refection.

These two rules o refection are illustrated in the diagram below.

These two rules will greatly simpliy the task o determining the image

locations or objects placed in ront o concave mirrors. n the ne%t section o

Lesson !, these two rules will be applied to determine the location,

orientation, si&e and type o image produced by a concave mirror. #s the

rules are applied in the construction o ray diagrams, do not orget the act

that the law o refection holds or each o these rays. t just so happens that

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when the law o refection is applied or a ray 'either traveling parallel to the

principal a%is or passing through "( that strikes the mirror at a location near

the principal a%is, the ray will refect in close appro%imation with the above

two rules.

Ray Diagrams - Concave Mirrors

• The Anatomy of a Curved Mirror







The theme o this unit has been that we see an object because light rom the

object travels to our eyes as we sight along a line at the object. )imilarly, we

see an image o an object because light rom the object refects o a mirror

and travel to our eyes as we sight at the image location o the object. "rom

these two basic premises, we have de*ned the image location as the location

in space where light appears to diverge rom. +ay diagrams have been a

valuable tool or determining the path taken by light rom the object to the

mirror to our eyes. n this section o Lesson !, we will investigate the method

or drawing ray diagrams or objects placed at various locations in ront o a

concave mirror.

To draw these diagrams, we will have to recall the two rules o refection or

concave mirrors$

• #ny incident ray traveling parallel to the principal a%is on the way to the

mirror will pass through theocal point upon refection.• #ny incident ray passing through the ocal point on the way to the mirror will

travel parallel to the principal a%is upon refection.

arlier in this lesson, the ollowing diagram was shown to illustrate the patho light rom an object to mirror to an eye.

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n this diagram *ve incident rays are drawn along with their corresponding

refected rays. ach ray intersects at the image location and then diverges to

the eye o an observer. very observer would observe the same image

location and every light ray would ollow the law o refection. -et only two o

these rays would be needed to determine the image location since it onlyreuires two rays to *nd the intersection point. / the *ve incident rays

drawn, two o them correspond to the incident rays described by our two

rules o refection or concave mirrors. 0ecause they are the easiest and

most predictable pair o rays to draw, these will be the two rays used

through the remainder o this lesson

Step-by-Step Method for Drawing Ray Diagrams

The method or drawing ray diagrams or concave mirror is described below.

The method is applied to the task o drawing a ray diagram or an object

located beyond the center o curvature '1( o a concave mirror. -et the same

method works or drawing a ray diagram or any object location.

2. 3ick a point on the top o the object and draw two incident rays traveling

towards the mirror.

Using a straight edge, accurately draw one ray so that it

passes e%actly through the ocal point on the way to the

mirror. 4raw the second ray such that it travels e%actly

parallel to the principal a%is. 3lace arrowheads upon the

rays to indicate their direction o travel.

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5. /nce these incident rays strike the mirror, refect them according to

thetwo rules o refection or concave mirrors.

The ray that passes through the ocal point on the way to

the mirror will refect and travel parallel to the principal

a%is. Use a straight edge to accurately draw its path. The

ray that traveled parallel to the principal a%is on the way

to the mirror will refect and travel through the ocal

point. 3lace arrowheads upon the rays to indicate their direction o travel. %tend

the rays past their point o intersection.

!. 6ark the image o the top o the object.

The image point o the top o the object is the point

where the two refected rays intersect. your were to

draw a third pair o incident and refected rays, then the

third refected ray would also pass through this point.

This is merely the point where all light rom the top o

the object would intersect upon refecting o the mirror. / course, the rest o the

object has an image as well and it can be ound by applying the same three steps to

another chosen point. ')ee note below.(

7. +epeat the process or the bottom o the object.

The goal o a ray diagram is to determine the location,

si&e, orientation, and type o image that is ormed by the

concave mirror. Typically, this reuires determining where

the image o the upper and lower e%treme o the object

is located and then tracing the entire image. #ter

completing the *rst three steps, only the image location o the top e%treme o the

object has been ound. Thus, the process must be repeated or the point on the

bottom o the object. the bottom o the object lies upon the principal a%is 'as it

does in this e%ample(, then the image o this point will also lie upon the principal

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a%is and be the same distance rom the mirror as the image o the top o the object.

#t this point the entire image can be *lled in.

)ome students have di8culty understanding how the entire image o an

object can be deduced once a single point on the image has been

determined. the object is a vertically aligned object 'such as the arrow

object used in the e%ample below(, then the process is easy. The image is

merely a vertical line. n theory, it would be necessary to pick each point on

the object and draw a separate ray diagram to determine the location o the

image o that point. That would reuire a lot o ray diagrams as illustrated

below.

"ortunately, a shortcut e%ists. the object is a vertical line, then the image is

also a vertical line. "or our purposes, we will only deal with the simpler

situations in which the object is a vertical line that has its bottom located

upon the principal a%is. "or such simpli*ed situations, the image is a vertical

line with the lower e%tremity located upon the principal a%is.

The ray diagram above illustrates that when the object is located at a

position beyond the center o curvature, the image is located at a position

between the center o curvature and the ocal point. "urthermore, the image

is inverted, reduced in si&e 'smaller than the object(, and real. This is the

type o inormation that we wish to obtain rom a ray diagram. These


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characteristics o the image will be discussed in more detail in the ne%t

section o Lesson !.

/nce the method o drawing ray diagrams is practiced a couple o times, it

becomes as natural as breathing. ach diagram yields speci*c inormation

about the image. The two diagrams below show how to determine image

location, si&e, orientation and type or situations in which the object is

located at the center o curvature and when the object is located between

the center o curvature and the ocal point.

t should be noted that the process o constructing a ray diagram is the same

regardless o where the object is located. 9hile the result o the ray diagram

'image location, si&e, orientation, and type( is dierent, the same two rays

are always drawn. The two rules o refection are applied in order to

determine the location where all refected rays appear to diverge rom 'whichor real images, is also the location where the refected rays intersect(.

n the three cases described above : the case o the object being located

beyond 1, the case o the object being located at 1, and the case o the

object being located between 1 and " : light rays are converging to a point

ater refecting o the mirror. n such cases, a real image is ormed. #s

discussed previously, a real image is ormed whenever refected light passes

through the image location. 9hile plane mirrors always produce virtual

images, concave mirrors are capable o producing both real and virtual

images. #s shown above, real images are produced when the object is

located a distance greater than one ocal length rom the mirror. # virtual

image is ormed i the object is located less than one ocal length rom the

concave mirror. To see why this is so, a ray diagram can be used.

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Watch It

# physics instructor discusses the nature o a real image using a phun physics

demonstration.

Ray Diagram for the !ormation of a "irtual Image

# ray diagram or the case in which the object is located in front of the ocal

point is shown in the diagram at the right. /bserve that in this case the light

rays diverge ater refecting o the mirror. 9hen light rays diverge ater

refection, a virtual image is ormed. #s was done with plane mirrors, the

image location can be ound by tracing all refected rays backwards until

they intersect. "or every observer, the refected rays would seem to be

diverging rom this point. Thus, the point o intersection o the e%tended

refected rays is the image point. )ince light does not actually pass through

this point 'light never travels behind the mirror(, the image is reerred to as a

virtual image. /bserve that when the object in located in front of the ocal

point, its image is an upright and enlarged image that is located on the other

side o the mirror. n act, one generali&ation that can be made about allvirtual images produced by mirrors 'both plane and curved( is that they are

always upright and always located on the other side o the mirror.

Ray Diagram for an #b$ect %ocated atthe !ocal &oint

Thus ar we have seen via ray diagrams that a real image is produced when

an object is located more than one ocal length rom a concave mirror; and a

virtual image is ormed when an object is located less than one ocal length

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8/15

rom a concave mirror 'i.e., in front of F (. 0ut what happens when the object

is located at "< That is, what type o image is ormed when the object is

located e%actly one ocal length rom a concave mirror< / course a ray

diagram is always one tool to help *nd the answer to such a uestion.

=owever, when a ray diagram is used or this case, an immediate di8culty is

encountered. The incident ray that begins rom the top e%tremity o the

object and passes through the ocal point does not meet the mirror. Thus, a

dierent incident ray must be used in order to determine the intersection

point o all refected rays. #ny incident light ray would work as long as it

meets up with the mirror. +ecall that the only reason that we have used the

two we have is that they can be conveniently and easily drawn. The diagram

below shows two incident rays and their corresponding refected rays.

"or the case o the object located at the ocal point '"(, the light rays neither

converge nor diverge ater refecting o the mirror. #s shown in the diagramabove, the refected rays are traveling parallel to each other. )ubseuently,

the light rays will not converge on the object>s side o the mirror to orm a

real image; nor can they be e%tended backwards on the opposite side o the

mirror to intersect to orm a virtual image. )o how should the results o the

ray diagram be interpreted< The answer$ there is no image?? )urprisingly,

when the object is located at the ocal point, there is no location in space at

which an observer can sight rom which all the refected rays appear to be

diverging. #n image is not ormed when the object is located at the ocal

point o a concave mirror.

Ray Diagrams - Concave Mirrors

• The Anatomy of a Curved Mirror


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9/15






The theme o this unit has been that we see an object because light rom the

object travels to our eyes as we sight along a line at the object. )imilarly, we

see an image o an object because light rom the object refects o a mirror

and travel to our eyes as we sight at the image location o the object. "rom

these two basic premises, we have de*ned the image location as the location

in space where light appears to diverge rom. +ay diagrams have been a

valuable tool or determining the path taken by light rom the object to the

mirror to our eyes. n this section o Lesson !, we will investigate the method

or drawing ray diagrams or objects placed at various locations in ront o a

concave mirror.

To draw these diagrams, we will have to recall the two rules o refection or

concave mirrors$

• #ny incident ray traveling parallel to the principal a%is on the way to the

mirror will pass through theocal point upon refection.• #ny incident ray passing through the ocal point on the way to the mirror will

travel parallel to the principal a%is upon refection.

arlier in this lesson, the ollowing diagram was shown to illustrate the path

o light rom an object to mirror to an eye.

n this diagram *ve incident rays are drawn along with their corresponding

refected rays. ach ray intersects at the image location and then diverges to

the eye o an observer. very observer would observe the same image

location and every light ray would ollow the law o refection. -et only two o

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10/15

these rays would be needed to determine the image location since it only

reuires two rays to *nd the intersection point. / the *ve incident rays

drawn, two o them correspond to the incident rays described by our two

rules o refection or concave mirrors. 0ecause they are the easiest and

most predictable pair o rays to draw, these will be the two rays used

through the remainder o this lesson

Step-by-Step Method for Drawing Ray Diagrams

The method or drawing ray diagrams or concave mirror is described below.

The method is applied to the task o drawing a ray diagram or an object

located beyond the center o curvature '1( o a concave mirror. -et the same

method works or drawing a ray diagram or any object location.

2. 3ick a point on the top o the object and draw two incident rays traveling

towards the mirror.

Using a straight edge, accurately draw one ray so that it

passes e%actly through the ocal point on the way to the

mirror. 4raw the second ray such that it travels e%actly

parallel to the principal a%is. 3lace arrowheads upon the

rays to indicate their direction o travel.

5. /nce these incident rays strike the mirror, refect them according to

thetwo rules o refection or concave mirrors.

The ray that passes through the ocal point on the way to

the mirror will refect and travel parallel to the principal

a%is. Use a straight edge to accurately draw its path. The

ray that traveled parallel to the principal a%is on the way

to the mirror will refect and travel through the ocal

point. 3lace arrowheads upon the rays to indicate their direction o travel. %tend

the rays past their point o intersection.

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11/15

!. 6ark the image o the top o the object.

The image point o the top o the object is the point

where the two refected rays intersect. your were to

draw a third pair o incident and refected rays, then the

third refected ray would also pass through this point.

This is merely the point where all light rom the top o

the object would intersect upon refecting o the mirror. / course, the rest o the

object has an image as well and it can be ound by applying the same three steps to

another chosen point. ')ee note below.(

7. +epeat the process or the bottom o the object.

The goal o a ray diagram is to determine the location,

si&e, orientation, and type o image that is ormed by the

concave mirror. Typically, this reuires determining where

the image o the upper and lower e%treme o the object

is located and then tracing the entire image. #ter

completing the *rst three steps, only the image location o the top e%treme o the

object has been ound. Thus, the process must be repeated or the point on the

bottom o the object. the bottom o the object lies upon the principal a%is 'as it

does in this e%ample(, then the image o this point will also lie upon the principal

a%is and be the same distance rom the mirror as the image o the top o the object.

#t this point the entire image can be *lled in.

)ome students have di8culty understanding how the entire image o an

object can be deduced once a single point on the image has been

determined. the object is a vertically aligned object 'such as the arrow

object used in the e%ample below(, then the process is easy. The image is

merely a vertical line. n theory, it would be necessary to pick each point on

the object and draw a separate ray diagram to determine the location o the

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12/15

image o that point. That would reuire a lot o ray diagrams as illustrated

below.

"ortunately, a shortcut e%ists. the object is a vertical line, then the image is

also a vertical line. "or our purposes, we will only deal with the simpler

situations in which the object is a vertical line that has its bottom located

upon the principal a%is. "or such simpli*ed situations, the image is a vertical

line with the lower e%tremity located upon the principal a%is.

The ray diagram above illustrates that when the object is located at a

position beyond the center o curvature, the image is located at a position

between the center o curvature and the ocal point. "urthermore, the image

is inverted, reduced in si&e 'smaller than the object(, and real. This is thetype o inormation that we wish to obtain rom a ray diagram. These

characteristics o the image will be discussed in more detail in the ne%t

section o Lesson !.

/nce the method o drawing ray diagrams is practiced a couple o times, it

becomes as natural as breathing. ach diagram yields speci*c inormation

about the image. The two diagrams below show how to determine image

location, si&e, orientation and type or situations in which the object is

located at the center o curvature and when the object is located between

the center o curvature and the ocal point.

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t should be noted that the process o constructing a ray diagram is the same

regardless o where the object is located. 9hile the result o the ray diagram

'image location, si&e, orientation, and type( is dierent, the same two rays

are always drawn. The two rules o refection are applied in order to

determine the location where all refected rays appear to diverge rom 'whichor real images, is also the location where the refected rays intersect(.

n the three cases described above : the case o the object being located

beyond 1, the case o the object being located at 1, and the case o the

object being located between 1 and " : light rays are converging to a point

ater refecting o the mirror. n such cases, a real image is ormed. #s

discussed previously, a real image is ormed whenever refected light passes

through the image location. 9hile plane mirrors always produce virtual

images, concave mirrors are capable o producing both real and virtual

images. #s shown above, real images are produced when the object is

located a distance greater than one ocal length rom the mirror. # virtual

image is ormed i the object is located less than one ocal length rom the

concave mirror. To see why this is so, a ray diagram can be used.

Watch It

# physics instructor discusses the nature o a real image using a phun physics

demonstration.

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14/15

Ray Diagram for the !ormation of a "irtual Image

# ray diagram or the case in which the object is located in front of the ocal

point is shown in the diagram at the right. /bserve that in this case the light

rays diverge ater refecting o the mirror. 9hen light rays diverge ater

refection, a virtual image is ormed. #s was done with plane mirrors, the

image location can be ound by tracing all refected rays backwards until

they intersect. "or every observer, the refected rays would seem to be

diverging rom this point. Thus, the point o intersection o the e%tended

refected rays is the image point. )ince light does not actually pass through

this point 'light never travels behind the mirror(, the image is reerred to as a

virtual image. /bserve that when the object in located in front of the ocal

point, its image is an upright and enlarged image that is located on the other

side o the mirror. n act, one generali&ation that can be made about all

virtual images produced by mirrors 'both plane and curved( is that they are

always upright and always located on the other side o the mirror.

Ray Diagram for an #b$ect %ocated atthe !ocal &oint

Thus ar we have seen via ray diagrams that a real image is produced when

an object is located more than one ocal length rom a concave mirror; and a

virtual image is ormed when an object is located less than one ocal length

rom a concave mirror 'i.e., in front of F (. 0ut what happens when the objectis located at "< That is, what type o image is ormed when the object is

located e%actly one ocal length rom a concave mirror< / course a ray

diagram is always one tool to help *nd the answer to such a uestion.

=owever, when a ray diagram is used or this case, an immediate di8culty is

encountered. The incident ray that begins rom the top e%tremity o the

object and passes through the ocal point does not meet the mirror. Thus, a

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dierent incident ray must be used in order to determine the intersection

point o all refected rays. #ny incident light ray would work as long as it

meets up with the mirror. +ecall that the only reason that we have used the

two we have is that they can be conveniently and easily drawn. The diagram

below shows two incident rays and their corresponding refected rays.

"or the case o the object located at the ocal point '"(, the light rays neither

converge nor diverge ater refecting o the mirror. #s shown in the diagram

above, the refected rays are traveling parallel to each other. )ubseuently,

the light rays will not converge on the object>s side o the mirror to orm a

real image; nor can they be e%tended backwards on the opposite side o the

mirror to intersect to orm a virtual image. )o how should the results o the

ray diagram be interpreted< The answer$ there is no image?? )urprisingly,

when the object is located at the ocal point, there is no location in space at

which an observer can sight rom which all the refected rays appear to bediverging. #n image is not ormed when the object is located at the ocal

point o a concave mirro

two rules of reflection for concave mirrors

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