Concave Mirrors Worked Example | Ray Tracing 2 of 5 | Geometric Optics | Doc Physics

Summary

The video covers the principles of light reflection, focusing on the behavior of four principal rays (P, F, C, and M rays) when interacting with mirrors. The speaker explains how these rays reflect off both diffuse and specular reflectors and how they converge to form images. Through various examples, including scenarios involving real and virtual images, the speaker also demonstrates how image size and orientation change depending on the object's position relative to the mirror. The discussion concludes with a focus on virtual images and the properties of convex mirrors.

Takeaways

• 🔆 The script discusses four principal rays involved in image formation: the P Ray, F Ray, C Ray, and M Ray.
• 📏 When an object reflects light diffusely, rays scatter in all directions, unlike a specular reflector like a mirror.
• 🌟 The P Ray comes in parallel and bounces out through the focal point after hitting the mirror.
• 🎯 The F Ray travels through the focus before hitting the mirror and exits parallel to the principal axis.
• 🏹 The C Ray travels straight through the center of curvature and bounces back along the same path.
• 🔵 The M Ray hits the center of the mirror and bounces back at the same angle, following the principal axis.
• 🔍 Real images are formed where rays converge, and they can be projected on a screen since actual light is present.
• 📉 The height of the image is typically smaller than the object and can be inverted, as seen in certain examples.
• 🔄 Virtual images appear behind the mirror, but no light actually exists there, so they cannot be projected.
• 🪞 The script hints at moving to convex mirrors next, which are easier to understand than concave mirrors.

Q & A

What are the four principal rays mentioned in the script?

The four principal rays are the P Ray (Parallel Ray), the F Ray (Focus Ray), the C Ray (Center Ray), and the M Ray (Middle Ray). These rays are used to trace the path of light and determine where an image forms.

What is the key difference between a diffuse reflector and a specular reflector?

A diffuse reflector scatters light in all directions due to its rough surface, making it possible to see the object from different angles. In contrast, a specular reflector (like a mirror) reflects light at the same angle it arrives, making it difficult to see the object itself.

How does the script explain image formation using the four principal rays?

The script explains that an image is formed at the point where the outgoing rays from the mirror converge. These rays include the P Ray, which comes in parallel and exits through the focus, the F Ray, which enters through the focus and exits parallel, the C Ray, which travels straight through the center, and the M Ray, which reflects off the center of the mirror.

What happens to an object placed at the focal point of the mirror?

When an object is placed at the focal point, the outgoing rays do not converge, and the image forms at infinity, appearing infinitely large but still taking up the same fraction of the viewer’s field of vision.

What is the significance of the ‘line of action’ in the context of the mirror?

The line of action refers to the axis where the principal rays intersect the mirror. It's essential for determining where the rays meet to form an image, and it helps in tracing the correct path of the rays.

What properties does a real image have according to the script?

A real image is formed where outgoing rays converge, and it can be projected onto a screen because there is actual light at that location. The real image is inverted and can be smaller or larger than the object, depending on its distance from the mirror.

How does the size and distance of the image compare to the object in this example?

In the first example, the object is 4 cm tall and 18 cm from the mirror, while the image is 2.3 cm tall (negative because it's inverted) and 9 cm from the mirror. This shows that the image is smaller and closer to the mirror.

What changes when an object is placed between the center of curvature and the focal point?

When the object is placed between the center of curvature and the focal point, the image is larger and still inverted. In the example provided, the image is 6 cm tall and 16 cm from the mirror.

What is a virtual image, and how does it differ from a real image?

A virtual image is one that cannot be projected onto a screen because there is no actual light at the location of the image. It appears upright and is located on the opposite side of the mirror from where the object is placed, unlike a real image, which is formed by converging light rays.

Why do the rays seem to diverge when using a concave mirror in the final example?

In the final example, the rays seem to diverge because the image is virtual and located behind the mirror. The rays never physically cross, but to the eye, it appears as though they come from a point behind the mirror, creating the illusion of a larger object.

Outlines

00:00

🔍 Introduction to Light Rays and Reflection

In this paragraph, the speaker introduces the four principal rays (P Ray, F Ray, C Ray, M Ray) that are essential for understanding reflection. The object being studied is a diffuse reflector, which scatters light in all directions, unlike a specular reflector like a mirror. The paragraph explains how light reflects differently on surfaces and introduces the concept of how light behaves when hitting various objects. This serves as the foundation for exploring light rays and their interactions.

05:01

🧙‍♂️ A Mysterious Encounter with a Puzzle

The speaker recalls a childhood memory of an encounter with an old man in Wyoming who gave him mysterious scrolls. The man hinted that these scrolls would hold answers to something important in the speaker’s life. The scrolls contain drawings of objects and instructions to find out where images are formed, a challenge that the speaker now faces. This sets up a metaphorical journey where the speaker attempts to understand the deeper meanings behind these rays and mirror principles, starting with the P Ray.

10:02

🌈 Tracing the P Ray and Other Rays

Here, the speaker begins drawing rays and analyzing their behavior. The P Ray enters parallel and exits through the focus, while the F Ray passes through the focus and exits parallel. The C Ray travels through the center of curvature, and the M Ray bounces back at the same angle. By tracing these rays, the speaker explains how the image is formed at the point where the rays converge. The explanation highlights how diverging rays from an object are refocused by the mirror, creating a real image where the rays meet.

15:02

📏 Measuring Image Height and Distance

The speaker discusses how to calculate the height and distance of the image formed by the mirror. Using the object’s height (4 cm) and distance from the mirror (18 cm), the speaker determines that the image is smaller (2.3 cm), inverted, and located at a distance of 9 cm from the mirror. The explanation emphasizes that the image is real and can be captured on a screen, as it is formed by converging rays. The calculations and proportions demonstrate how the mirror alters the object’s appearance in terms of size and orientation.

🔄 Understanding Rays at the Center of Curvature

The object is placed at the center of curvature, and the speaker continues tracing the principal rays (P Ray, F Ray, C Ray, and M Ray) to locate the image. The rays converge at the center, creating an image that is inverted but of the same size as the object. This exercise demonstrates that when an object is at the center of curvature, the image formed is identical in size but flipped upside down. The speaker also notes that the image distance and object distance from the mirror are the same.

🔬 Object Between the Focal Point and Center of Curvature

The speaker examines a case where the object is between the center of curvature and the focal point. By tracing the rays, the image is determined to be larger than the object and still inverted. The speaker comments on the precision needed when drawing these rays, as slight deviations can cause noticeable errors. The experiment shows how the image becomes magnified when the object moves closer to the focal point, while still maintaining the inverted orientation. This setup leads to an image that is larger than the object.

🌌 Infinity and Focal Point Mystery

The object is now placed at the focal point, which creates an interesting scenario. The speaker explains that when the object is at the focal point, the rays emerge parallel, and the image appears to be at infinity. This means the image is infinitely large and far away, leading to an abstract conclusion where the speaker humorously notes the impossibility of truly visualizing an infinite image. This exploration emphasizes the unique behavior of rays at the focal point and how no real image is formed when the object is positioned there.

👁 Virtual Image Formation Behind the Mirror

In this final experiment, the object is placed beyond the focal point, and the speaker traces the rays that appear to intersect behind the mirror. Even though the rays do not actually cross behind the mirror, the eye perceives a virtual image as if it exists on the opposite side. The image is upright and larger than the object, forming a virtual, magnified image. The speaker concludes by explaining the properties of virtual images: they are upright and located on the opposite side of the mirror, though no light actually exists there.

Keywords

💡P Ray

The 'P Ray' refers to a principal ray in ray diagrams for mirrors and lenses. It travels parallel to the principal axis and, upon reflection, passes through the focal point of a mirror. In the script, the P Ray is described as coming in parallel and then bouncing out through the focus, demonstrating the basic behavior of light rays in mirror reflection.

💡F Ray

The 'F Ray' is another principal ray used in ray tracing. It enters through the focal point and exits the mirror parallel to the principal axis. In the video, the F Ray is used to illustrate how rays behave when entering at a specific angle, helping viewers understand how images form in concave mirrors.

💡C Ray

The 'C Ray' is a ray that passes through the center of curvature of the mirror. In the script, it is described as going straight through the center and bouncing back along the same path. This ray helps determine where an image will form in relation to the object and the mirror.

💡M Ray

The 'M Ray' is a principal ray that hits the center of the mirror and reflects at the same angle. In the script, the M Ray is described as being drawn in blue, hitting the mirror at the center, and reflecting back at an equal angle. It helps demonstrate how light reflects symmetrically.

💡Diffuse reflection

Diffuse reflection refers to the scattering of light in many directions after hitting a rough surface. The video explains how objects like Newton's surface, which is bumpy on a microscopic level, reflect light diffusely, allowing us to see them from various angles. This contrasts with specular reflection, where light reflects at the same angle, as with a smooth mirror.

💡Specular reflection

Specular reflection occurs when light reflects off a smooth surface, such as a mirror, at the same angle as it hits the surface. In the video, it’s mentioned as the type of reflection that doesn’t allow us to see an object’s details clearly, in contrast to diffuse reflection, which makes an object visible from different perspectives.

💡Line of action

The 'line of action' is the axis along which light rays interact with the mirror. It’s a critical reference line in ray diagrams. The video emphasizes that the line of action should intersect the mirror, serving as a guide for where rays will reflect and form an image. It plays a central role in determining the position of the rays.

💡Principal axis

The principal axis is the imaginary line that runs through the center of curvature and the focal point of a mirror or lens. It acts as a reference line for understanding how rays behave in optical systems. In the video, it is often referred to when discussing how the P Ray, F Ray, and other rays interact with the mirror.

💡Real image

A real image is formed when light rays converge and can be projected onto a screen. In the script, the real image is described as one where the rays physically intersect, and if you place a piece of paper at that point, you would see the image. Real images are inverted and can be smaller or larger than the object, depending on the ray diagram.

💡Virtual image

A virtual image occurs when light rays only appear to converge but don’t actually do so. In the script, it’s noted that virtual images are upright and appear on the wrong side of the mirror. The video uses virtual images to explain how magnifying mirrors work, as the viewer perceives the object as being larger and on the opposite side of the mirror.

Highlights

Introduction of the four principal rays: the P Ray, F Ray, C Ray, and M Ray, which are essential for studying light reflection and image formation.

Explanation of diffuse reflectors versus specular reflectors and their differences in light behavior.

The P Ray travels parallel to the principal axis and reflects through the focus of the mirror.

The F Ray enters through the focus and reflects parallel to the principal axis.

The C Ray travels through the center of curvature and reflects back along the same path.

The M Ray hits the mirror at the principal axis and reflects back at the same angle.

When the outgoing rays converge, a real image is formed, which can be captured on a screen.

An object placed at different positions relative to the mirror (e.g., between the center and focal point) affects the size and orientation of the image.

At the focal point, the rays reflect parallel and do not form an image, implying infinite magnification.

Virtual images are upright and located on the 'wrong' side of the mirror, where no actual light converges.

Explanation of how mirrors can create the illusion of a larger object by forming virtual images behind the mirror.

Illustration of how different rays interact with concave mirrors and the formation of both real and virtual images.

Detailed exploration of how the distance and height of images relate to the object’s position using ray diagrams.

Description of how light refocuses after being reflected, showing the role of the mirror in controlling image size and position.

Complex analysis of ray tracing inaccuracies, errors, and the visual interpretation of where image formation occurs.

Transcripts

all right here are the ground rules one

these are the four rays that we're going

to be addressing the P

Ray the F Ray the C Ray and the M Ray

excellent so you also need to know

ground rule number two that when there

is an object and my object's going to

kind of be like this thing right here it

represents any possible thing I guess if

this object gets bigger it would look

like that and if it flipped over it

would look like that uh and I guess it

got shorter there also but remember that

rays of light are coming out from this

object if it is a diffuse reflector and

I hope that your object is if you want

to look at something that's a specular

reflector then you'll have trouble

really seeing what it looks like what do

like what does a mirror really look like

right

so Newton though is lovely because you

can see him from every angle because

he's getting hit by light from

everywhere and he's also uh reflecting

that light from everywhere and that's

because the surface of Newton is really

bumpy like on a microscopic scale the

surface of a mirror is like that though

so every Ray that comes in comes out at

the same angle where those guys are

equal but here you got a ray coming in

and there's a chance it's going to go

that direction and if it's a little bit

over it's going to go perhaps that

direction if it's right here it's going

to go that direction this is a diffuse

reflector cuz lights's coming out every

possible Direction but we're going to be

studying only the four principal rays

that we discussed it just a moment ago

the P Ray the F Ray the C Ray and the M

Ray here we go when I was a small

boy my parents took me to Wyoming and at

a quick trip an old decrepit man came up

to me on my way to the bathroom and he

said

Son take these Scrolls they may be

useful and he handed me this

packet and it's time it's that point in

my life when I'm ready to understand

what he meant and why he gave them to me

and uh oh there was one more thing that

he said he said always remember the line

of action of the mirror is not the

mirror surface itself it has to be drawn

where the principal axis intersects the

mirror itself all right so he clearly

has drawn for me an object on each one

of these Scrolls and we'll have to find

out where the image is formed and

perhaps it will be the key to a puzzle

I'll start with the pay each time and

the pay comes in parallel and it's going

to be like

this and then it will bounce out through

the focus and the focus is here so it

will go like this and I am going to put

arrows on the outgoing Rays so that I

become clear on how this is working the

next Ray is the F Ray the Fay comes in

through the focus watch it wait for it

it hits the line of action of the mirror

and comes out

parallel there it is coming out parallel

and that is our F Ray leaving and then

the next one that we like to discuss is

the C Ray which is coming straight

through the

center and going back out straight

through the center oh boy we've got

ourselves a little problem Houston it's

going to be like

this whoa not like that it'll be like

that where that's our C Ray hitting the

line of action of the mirror coming

right back out and then uh well I guess

there one more right let's just go crazy

and draw ourselves an mray I like to use

blue for my mray the mray will be

hitting the center of the

mirror where it hits the principal axis

and bouncing back out at the same angle

looks pretty good there's my M Ray where

all four of my Rays intersect it is

there that an image is formed cuz see

it's like this this is like here's a

little bit of image Theory this thing is

clearly a source of rays and they're

diverging they're all going out here and

they're spreading apart from each other

and we've only drawn four of them there

are really a multitude an infinite

number of rays coming off in all

possible directions but as these Rays go

and hit the mirror the mirror refocuses

them they were spreading apart and now

they're converging again look at them

all coming in here and they're ready to

say hey and if you put a piece of paper

right there I can't use that color I

need to use the color for um for the

object I think so I don't get confused

if you put a piece of paper

right here a screen as it were then you

would find an image so let's label these

things right here you've got your

object and you've got your image and

wherever two rays outgoing intersect

then you will have a real image it's

real because there's actually light

there and if you put a piece of paper

there you would see something you would

see the arrow or the flower pot or

whatever you had here as your object I

hope it's a flower pot

all right so if you've got a real image

right here you can ask questions like

how tall is that image and so the height

of the image could be a question that

you want to know and the distance of the

image from the lens as well I already

wrote down the height of the object was

4 cm and it's 18 cm away from the mirror

so we expect the height of the image to

be a little bit less and we expect its

distance from the mirror to be a little

bit less also I am getting that it is

2.3 CM 2.3 CM but guess what it's

negative so I'm going to quickly change

this so it says height of the image is -

2.3 CM cuz it's below the line all right

and the distance of the image well the

distance of the image it's on the proper

side of the mirror and that seems to be

9 cm 9 cm interesting this is like a

factor of two and that's like a factor

of two with a minus sign in there but

clearly the image is smaller than the

object so we go to his next clue his

next clue is an object at the center oh

dang we put a line of action down here's

the line of action of the mirror and

we're going to be acting on that line of

action and we need to make four Rays the

pay pay comes in parallel ready wait for

it here's our P Ray parallel to the axis

and it goes out through the focus

Pew mhm that is our P Ray and then we've

got to have a um dang I lost that cap

sorry

sorry FR frame comes in through the

focus comes out this direction

parallel to the

axis okay and oh shoot this is so badly

done unbelievable all right well we're

going to have to fudge it a little bit

on the next ones uh we've got a Cay that

is coming through the center and bounc

ing out at the same angle at which it

came in and it looks like it ought to be

something like

this that's our Cay and then an M Ray

comes in through the middle and oh gosh

did I say that was the C Ray that's

definitely not the C Ray that's oh no

that's the M Ray the C Ray to switch

colors on you goes straight through C

and then hits the mirror or does it no

it doesn't really so I've got these

three that are intersecting right here

you see that I've got those three that

are intersecting right there and I've

definitely got an errant um what is this

this green one that's a pay that pay

really sucks and I'm not sure why well

anyway don't worry about that if you've

got three that intersect at a certain

place that's definitely telling you

something and I'm going to say then that

my image is right here where these guys

are hitting so there's the object and

here's the image the outgoing rays are

converging right there and the cool

thing is they're at C which is really

cool so uh notice it's just about the

same height I'm not going to bother to

measure it it ought to be the same

height the height of the image is also 4

cm but it's negative it's down below the

axis and the distance of the image is 13

cm so cool that's again the same ratio

everything's equal with a minus sign

there next up they've put the object in

between the center of curvature and the

focal point so okay we're going to have

to be careful we need a pay

the pay comes in parallel and oh line of

action my apologies line of action right

[Music]

here the pay comes in

parallel and goes out through the

focus you should be drawing these at

home do

it and then we're going to get you an F

Ray and the F Ray is supposed to come

through the focus and go out

parallel notice there's a lot of room

for air here if I'm just slightly off

then everything's going to be

crazy but there

is my f

Ray and uh let's keep going we need a

cray and the cray

whoa the cay doesn't really make much

sense does it so the cray comes as if it

were coming from C but it leaves the

object really that's what's supposed to

happen it's supposed to leave the object

and go that way that's the beginning of

my C Ray it will ultimately hit the line

of action up here way up here and then

it will come back down

through the center and it'll be going

that direction when it comes back down

so that's my C Ray a little bit weird

Okay and then I want to give you an mray

and the mray is supposed to come off of

here here hit the middle and bounce out

at the same angle let's see what the

mray does for us I'm thinking that the

mray is probably going to be shooting

somewhere like right here uh we could

aim there if you want or we could aim

there I don't really care I'm going to

kind of split the difference and say

that the image must be somewhere in

between all of these things so these

four lines should be intersecting at a

single point but they're not go figure

they never quite do and this is

approximately where my image is so we've

got an object here and we've got an

image here notice that the image is

still inverted it's upside down but it's

bigger now than the object is so if I'm

in between C and F then I get this oh

man the image is inverted so I'm going

to say the height of the image is

negative something let's find out how

big it is it looks like it's

-6 and the distance of the image is

16 the distance of the image is 16 cm is

on the proper side so it kind of looks

like we're doing a uh 50% increase kind

of thing these ratios all kind of make

sense so let's go to the next puzzle and

uhoh it's at the focal

point let's begin we need a p p comes in

parallel oh we need a line of action

first line of

action if you're getting bored you can

stop watching this because I'm doing the

same thing every single time the P comes

in parallel hits the line of action and

goes out through the focus oh

interesting

uhuh uhhuh wait a second I probably

should continue this a little bit more

all right and uh the next one is the Fay

the Fay is supposed to go out through

the focus and that can't really happen

it's never going to hit the mirror so

I'm not going to draw the Fay my C Ray

though my C is supposed to start as if

it had come from

C so it's going to be like this it's

going to be kind of kind of like this

it's going to be kind

of and then it's going to come back out

in the same way and go try to go back

through C again it's supposed to be

straight here uhhuh and that guy was

leaving also and there's one more Ray

that I can do my M Ray is supposed to go

through I didn't quite hit see there but

I'm trying to make it look pretty so I

have to cheat a tiny bit there and then

it's going to come back out at the same

angle at which it entered and it's going

to be doing that so we've got three rays

and these Rays well where are they going

to meet haha I'll tell you they're going

to meet at Infinity which means my

object is at infinity and it is

infinitely big wait a second you're

saying right now that the height of the

image is negative infinity and you're

saying that the distance of the image is

Infinity cool that means we have some

very interesting magnification going on

I guess it's infinite magnification but

the issue is if you looked at the

image on a screen way over there you

would find that it is well still taking

up the same fraction of your field of

view because it's infinitely far away

but it's infinitely big so you can still

see it lovely doesn't make much sense

not really putting things on the focal

point is kind of weird there is no

converging that happens because that's

the essence of a focal point at the

focal point if you send something in

you're going to get out parallel rays so

you don't get an image formed but really

there is an image it's just infinitely

far away I don't want to talk about this

anymore let's go on there's oh dang and

the final scroll we put more space over

on this side let's see what he's going

to do we need a pay the pay comes in

here parallel and then goes out through

the focus like this the p eay goes out

through the focus like that and then uh

what's the next one we do the F Ray the

F Ray comes as if it had started at the

focus see that's what it's all about

it's like it's not going through the

focus it's as if it started at the focus

and it's going to hit the line of action

of the mirror oh no I forgot my line of

action never forget your line of action

students never never never never never

never do

that so I'm supposed to go out through

here as if I came from the focus and

then when I hit the line of action of

the mirror I'm going to go out parallel

watch this I'll be

like so it's all that direction and such

and then oh this is kind of weird we've

got a couple more that we can draw the C

Ray is as if it had started at C and

then it will go back towards C so it'll

come order of doing this is pretty

important it's going to go towards the

mirror and then it's going to hit the

mirror here and it's going to bounce

back that direction so when it gets to

here that's an outgoing Ray again and

then there's finally one more Ray the M

Ray is going through the

middle mhm one more principal Ray going

through the middle and coming back out

at that same angle which is probably

something like this and you notice that

these rays will absolutely never cross

each other but something very

interesting is happening it's as if they

once crossed each other that's a little

bit weird but if you put your eye over

here you could look here put that pretty

blue eye right there you're looking this

direction and you're thinking wait a

second those Rays seem to be coming from

an object there might be something over

there shall we consider whether there's

something over here behind the mirror

there's nothing here but

Darkness all the light is over

here but it's as if something is over

here causing an image whoa causing these

Rays to come out from each other so

let's Trace them back I'm going to trace

where it looks like this blue one came

from it looks like the blue one came

from over

here somewhere and I'm going to trace

where this orange one came from it looks

like the orange one came from right here

oh sorry about

that and it looks to me like the purple

one came

from I'm gonna oh gosh that's really

disappointing okay well we'll just deal

with

it the purple one looks like it came

from over here and the green one seems

to have come

from right in the middle of all of

these good I got three lines that worked

out and my C is just the absolutely

terrible one I think I maybe didn't

measure where my F and C were in

relation to each other quite well so

there is a apparently something here

there's apparently something

here is there something there of course

not it's dark on the other side of the

mirror but your eye thinks that there's

something there and that's how we can

use this kind of a mirror as a magnifier

because when I look at it I see not the

object I see the image as even bigger

than it actually is so the distance well

let's see first the height of the image

first of all it's positive that's

Pleasant

and where the object was 4 cm this

sucker is actually 9 cm oh cool what

about the distance of the image well

it's on the wrong side of the mirror so

that's offensive to me and I'm going to

say that that uh distance is about 4

cm hm cool so what we've got right here

is not real it's a virtual

image two properties of virtual images

that I want to note

one it's upright

two it's on the wrong side of the mirror

there's nothing there if I put a piece

of paper right there I would not see

anything cuz there's no light there it's

Darkness that's all I have to say with

this we're going to do some convex

mirrors next they're a lot easier