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Curved mirrors have the capability to create images that are larger or smaller than the object placed in front of them. They can also create images that are upside-down and images that can be projected on a screen.

There are two types of curved mirrors, those that curve in (concave) and those that curve out (convex)

convex concave

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CONCAVE MIRRORS

Concave mirrors can make many different kinds of images. Locating the image is similar to how we did it with a plane mirror. Lets find the image of our candle.

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CONCAVE MIRRORS

Here are some of the light rays coming from the tip of the candle…

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CONCAVE MIRRORS

…reflecting off the mirror. Even though the mirror is curved, the light rays still follow the Law of Reflection. But the surface is at a different angle at each point. You can see where the image of the tip of the candle is formed.

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CONCAVE MIRRORS

How about the base of the candle, lets draw those light rays a different color to distinguish them.

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CONCAVE MIRRORS

So the image is inverted and larger than the original object.

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CONCAVE MIRRORS

This kind of image is very special, unlike the image from the plane mirror, this image is in front of the mirror and was created by light rays that ACTUALLY cross in space. That makes this image a REAL image. A REAL image can be cast upon on a screen and the light rays actually meet.

object

image

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CONCAVE MIRRORS

But there has to be a better way to do this, not only is this diagram a mess of light rays, but it is very difficult to measure the angles for the law of reflection off the surface of a curved mirror.

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CONCAVE MIRRORS

A concave mirror is sometimes called a converging mirror because it reflects light toward the center. In fact, parallel light rays directed into the mirror will all reflect through one point. That point is known as the focal point.

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CONCAVE MIRRORS

The focal length, f, is the distance measured from the center of the mirror to the focal point.

f

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CONCAVE MIRRORS

The focal length is exactly ½ the distance to the center of the circle (or center of curvature cc) that the mirror is just a portion of.

f

r

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CONCAVE MIRRORS

Finally, the principal axis is a line that goes straight through the center of the mirror and is perpendicular to the mirror at the center.

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CONCAVE MIRRORS

The reason the mirror is set up this way is because there are 3 light rays that hit the mirror that will reflect in a very predictable manor. These light rays can help up determine the position of the image. This procedure, known as ray tracing requires us to know where the focal point (f) and center of curvature (cc) of our mirror are.

f cc

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CONCAVE MIRRORS

There are 3 light rays that reflect off the mirror in a predictable and easily drawn manor. We call these light rays the principal light rays. To determine the placement of the image, you must draw at least 2 of the 3 principal light rays.

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CONCAVE MIRRORS

The 1st principal light ray is the one the goes into the mirror parallel to the principal axis. How will this light ray reflect off the mirror?

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CONCAVE MIRRORS

The 1st principal light ray is the one the goes into the mirror parallel to the principal axis. This light ray will reflect through the focal point.

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CONCAVE MIRRORS

The 2nd principal light ray is the one that goes into the mirror through the focal point. How will this light ray reflect?

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CONCAVE MIRRORS

The 2nd principal light ray is the one that goes into the mirror through the focal point. This light ray will reflect parallel to the principal axis.

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CONCAVE MIRRORS

The 3rd principal light ray is the one that goes into the mirror through the center of curvature. How will this light ray reflect?

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CONCAVE MIRRORS

The 3rd principal light ray is the one that goes into the mirror through the center of curvature. This light ray will reflect right back on top of itself.

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CONCAVE MIRRORS

Look at the 3 reflected light rays, they all cross at a single point. This is where the image of the tip of the candle will be.

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CONCAVE MIRRORS

To locate the image of the base of the candle we can do the same trick. But notice that the base of the candle is sitting on the principal axis. The image of an object on the principal axis will always be on the principal axis.

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CONCAVE MIRRORS

So if the image of the top of the candle is here.

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CONCAVE MIRRORS

And the image of the base of the candle is on the principal axis.

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CONCAVE MIRRORS

Then the image looks like this.

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CONCAVE MIRRORS

It is inverted and much smaller than the original object. It is also a real image.

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For any curved mirror the process to find the image is the same. Find the image of the top of the object by drawing 2 of the 3 principal light rays. Repeat the procedure to find the image of the base of the object. If any part of the object is on the principal axis, you know that the image will be on the principal axis as well and so do not need to draw the principal light rays. Draw only 2 light rays and only use the 3rd principal light ray when you can’t draw one of the first two.

1st principal light ray – the light ray that enters the mirror parallel to the principal axis will reflect through the focal point.

2nd principal light ray – the light ray that enters the mirror through the focal point will reflect parallel to the principal axis.

3rd principal light ray – the light ray that enters the mirror through the center of curvature will reflect back through the center of curvature.

RAY TRACING DIAGRAM RULES

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SPHERICAL ABERRATION

Spherical mirrors do not perfectly reflect all parallel light rays entering the mirror though the focal point. To have a well defined focal point you will need to use a parabolic shaped mirror. Most of our problems will involve spherical mirrors. Light rays entering a spherical mirror near the edges of the mirror will not reflect perfectly through the focal point. For this reason, ray tracings will not be a perfect indicator of image placement since image made by spherical mirrors are a little blurry at times.

SPHERICAL ABERRATION

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MIRROR (lens) EQUATION

1/do + 1/di = 1/f

do = distance from center of mirror to

object

di = distance from center of mirror to image

f = focal length

Given 2 of the variables the 3rd can be found.

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si/so = -di/do

Where si = size of image

so = size of object

Given 3 you can calculate the 4th.

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Magnification

When you take the ratio of image over object you are finding the magnification

- di/do = Magnification = si/so

If the image is smaller than the object the ratio will be less than one but greater than 0.

If the image is bigger than the object then the ratio will be greater than 1

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If we have a 1.5 cm object located 12 cm from a concave mirror that

has a radius of 12 cm - where is the image located and how big is it?

Where is the focal point?

1/2 of the cc so f = 6 cm

What does the ray diagram look like?

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Ray diagram for an object placed at cc or 2f.

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Solve for di (image distance)

Given so = 1.5 cm do = 12 cm cc = 12 cm f = 6 cm

1/do + 1/di = 1/f 1/di = 1/f - 1/do

1/di = 1/6cm -1/12cm = 2/12cm -1/12cm

1/di =1/12cm Cross Multiply

(di) 1/12 = (di) 1/di di/12 =1

(12)di/12 =1(12) di =1(12)

di = 12 cm

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Solve for si (image size)

si/so = -di/do si = (-di)(so)/do

si = -12cm (1.5cm)/12cm = - 1.5 cm

If the object is located at the center of curvature (cc) then the image will be located at the cc, it is the same size and inverted.

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If an object located between f and 2f (cc) the image will be located beyond 2f (cc),

inverted and enlarged.

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Where will the image be located if we place an object 4 cm from a concave mirror that has a radius (cc) of 12

cm?

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Solving for image distanceGiven do = 4 cm cc 12 cm f =6

1/do + 1/di = 1/f 1/di = 1/f - 1/do

1/di = 1/6 - 1/ 4 = 2/12 - 3/12 = -1/12

1/di = -1/12 cross multiply

(-12)1/di = (-12) (-1/12) = -12/di = 1

(di) (-12)/di = (di) 1

-12 cm = di Image distance is negative

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Lets draw the ray diagram for this one.

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CONCAVE MIRRORS

1st principal light ray – the light ray that enters the mirror parallel to the principal axis will reflect through the focal point.

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CONCAVE MIRRORS

2nd principal light ray – the light ray that enters the mirror through the focal point will reflect parallel to the principal axis.

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CONCAVE MIRRORS

Because of spherical aberration we will only draw the first two principal light rays. Where is the image of the top of the candle?

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CONCAVE MIRRORS

An observer sees diverging light rays coming from the mirror. Similar to the plane mirror, to the eye, these light rays appear to have come from a point behind the mirror. This is where the image of the tip of the arrow is formed.

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CONCAVE MIRRORS

Notice the image is upright and larger than the original object. This image is known as a virtual image. Unlike a real image, a virtual image is not formed by real light rays crossing in space and so virtual images cannot be projected on screens.

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so – height or size of the object

so

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so – height or size of the objectsi – height or size of the image

si

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so – height or size of the objectsi – height or size of the image do – distance to the objectdi – distance to the image

di do

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so – height or size of the objectsi – height or size of the imagedo – distance to the objectdi – distance to the imagef – focal lengthcc– radius of curvature or center of curvature

cc

f

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What if the object is placed at f The rays are reflected parallel so there is

no image

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 Object

Distance 

 

Image Distance

 Image

Orientation

 Image

Magnification

 Inside f

 

 behind the mirror

 upright

 larger

 At f

 

 no image

 no image

 no image

 Between f and 2f

 

 beyond 2f

 inverted

 larger

 At 2f

 

 at 2f

 inverted

 no magnification

 Beyond 2f

 

 between f and 2f

 inverted

 smaller

Concave Mirror

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A convex mirror is sometimes called a diverging mirror because it tends to diverge incoming parallel light rays.

CONVEX MIRRORS

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CONVEX MIRRORS

The light rays from a convex mirror will be reflected outward in such a way that they all appear to have come from a single point, the focal point, which for a convex mirror is on the back side of the mirror.

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CONVEX MIRRORS

Determining the placement of the image formed when an object is placed in front of a convex mirror is very similar to concave mirrors.

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CONVEX MIRRORS

1st principal light ray – the light ray that enters the mirror parallel to the principal axis will reflect through the focal point.

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CONVEX MIRRORS

2nd principal light ray – the light ray that enters the mirror through the focal point will reflect parallel to the principal axis.

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CONVEX MIRRORS

We will not draw the 3rd for now since two is enough to determine the placement of our image. Again we are observing that the reflected light rays are diverging. Where do you think the image is?

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CONVEX MIRRORS

The observer sees the diverging light rays and senses that they appear to have come from a spot behind the mirror. To determine where, trace the reflected light rays back until they cross. This marks the image of the tip of the candle.

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CONVEX MIRRORS

The image formed from a convex mirror will always be an erect or upright, virtual image (located behind or inside of the mirror). It will also always be smaller than the object itself. This kind of mirror is also known as a wide angle mirror because it can “view” a wide angle.

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So between the 3 types of mirrors we can create all the possible image sizes and orientations.

Plane mirror – upright, same size

Convex mirror – upright, smaller

Concave mirror – upright, larger inverted, smaller

inverted, same size inverted, larger

Review

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Lets review some terminology.

term positive negative

f - focal length

so – height of object

si – height of image

do – distance to object

di – distance to image

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term positive negative

f - focal length concave mirrors convex mirrors

so – height of object

si – height of image

do – distance to object

di – distance to image

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term positive negative

f - focal length concave mirrors convex mirrors

so – height of object always never

si – height of image

do – distance to object

di – distance to image

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term positive negative

f - focal length concave mirrors convex mirrors

so – height of object always never

si – height of image upright inverted

do – distance to object

di – distance to image

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term positive negative

f - focal length concave mirrors convex mirrors

so – height of object always never

si – height of image upright inverted

do – distance to object always never

di – distance to image

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term positive negative

f - focal length concave mirrors convex mirrors

so – height of object always never

si – height of image upright inverted

do – distance to object always never

di – distance to imagein front of mirror -

realbehind mirror -

virtual

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For all curved mirrors the relationship between the focal length and the distances to the object and image is

1/f = 1/do + 1/di

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For all curved mirrors the relationship between the focal length and the distances to the object and image is

1/f = 1/do + 1/di

To calculate how magnified an image is we will use the following ratio.

Magnification = si/so

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For all curved mirrors the relationship between the focal length and the distances to the object and image is

1/f = 1/do + 1/di

To calculate how magnified an image is we will use the following ratio.

Magnification = si/so

So magnification will always be + when you have a virtual image since virtual images are upright and will always be – when you have a real image since real images are inverted.

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For all curved mirrors the relationship between the focal length and the distances to the object and image is

1/f = 1/do + 1/di

To calculate how magnified an image is we will use the following ratio.

Magnification = si/so

So magnification will always be + when you have a virtual image since virtual images are upright and will always be – when you have a real image since real images are inverted.

The absolute value of magnification will be = 1 when the image is not magnified. It will be larger than 1 when the image is magnified and between 0 and 1 when the image is smaller.

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For all curved mirrors the relationship between the focal length and the distances to the object and image is

1/f = 1/do + 1/di

To calculate how magnified an image is we will use the following ratio.

Magnification = si/so = - di/do

So magnification will always be + when you have a virtual image since virtual images are upright and will always be – when you have a real image since real images are inverted.

The absolute value of magnification will be = 1 when the image is not magnified. It will be larger than 1 when the image is magnified and between 0 and 1 when the image is smaller.

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One more example. A 6 ft tall man stands 3 ft from a large convex mirror. His image is formed 9 inches behind the mirror. What is the focal length of the mirror and how tall is his image?

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One more example. A 6 ft tall man stands 3 ft from a large convex mirror. His image is formed 9 inches behind the mirror. What is the focal length of the mirror and how tall is his image?

Given so = 6 ft do = 3 ft (36 in) di = 9 in f = ? si = ? 1/f = 1/do + 1/di

1/f = 1/36 in + 1/-9in1/f = -0.08333 inf = - 12 in

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One more example. A 6 ft tall man stands 3 ft from a large convex mirror. His image is formed 9 inches behind the mirror. What is the focal length of the mirror and how tall is his image?

Given so = 6 ft do = 3 ft (36 in) di = -9 in f = ? si = ? 1/f = 1/do + 1/di

1/f = 1/36 in + 1/-9 in1/f = -0.08333 inf = - 12 in

si/so = - di/do

si/6 ft = -(-9 in)/36 insi = 1.5 ft