! A lensball is a polished spherical ball of radius $R=5 cm$, made of glass of refractive index value $n_{glass}=1.5$. The cathedral is 90 meters high, and you stand with the lensball 400 meters from the cathedral. You look at the cathedral through the lens, your eye being at 20 cm from the center of the lens. What would you expect to see?
! A lensball is a simple physical system: a sphere of glass of refractive index $`n=1.5`$ and of radius $`R=5\;cm`$.
!
! A ball lensball is placed in front of a painting. Depending on the position of the observer or the camera,
! the optical system (the sequence of simple optical elements crossed by light between the physical object
! and the observed image) that forms the image differs.
!
! Observe the 3 images of the painting given by the lensball :
!
! Image 1
!
! * _The resolution time is the typical expected time to be allocated to this problem if it was part of an examen for an optics certificate_.
! 
!
! * _The discovery time is the expected time you require to prepare this challenge if you don't have practice. However, this is just an indication, take as much time as you need. The time to question yourself serenely about how to handle the problem, about the method of resolution and its validity, about some possible approximations if they can be justified, and the time you need to check the equations if you have not previously memorised them and to perfom the calculation, are important._
! Images 2 (the smallest) and 3
!
! 
!
! For each image of the painting, can you identify the optical system, then specify `
! the characteristics of the various simple elements that constitute the system and their relative distances?
!
! * _The resolution time is the typical expected time to be allocated to this problem if it was part of an examen for an optics certificate._
! * _The discovery time is the expected time required to prepare this challenge if you don't have practice. But take as much time as you need._
!
! <\details>
! <details markdown=1>
! <summary>
! Ready to answer M3P2 team questions ?
! Ready to answer M3P2 team questions for image 1?
! </summary>
!<!--question 1-->
!<detailsmarkdown=1>
!<summary>
! What is the scientifical framework you choose to study this problem ?
!</summary>
! * All the characteristic sizes in this problem are much bigger than the wavelength of the visible radiation ($\lambda\approx5\mu m$), so I deal with this problem in the framework of geometrical optics, and in the paraxial approximation in order to characterize the image.
!
! * The cathedral sustains an angle of $arctan\dfrac{90}{400}=13°$ from the lensball. This value seems reasonable to justify at first order the use of the paraxial approximation (_we usually consider that angles of incidence would not exceed 10° on the various simple optical element encountered between the objet (here the cathedral) and the final image (retina of the eye or matrix sensor of a camera_).
!</details>
!<!--question 2-->
!<detailsmarkdown=1>
!<summary>
! <details markdown=1>
! <summary>
! Where is the painting located?
! </summary>
! * The painting is located on the other side of the lens, in relation to you.
! </details>
! <details markdown=1>
! <summary>
! What is the optical system giving the image of the painting?
! </summary>
! <br>
! * The optical system is composed of two spherical refracting surfaces, centered on the same optical axis.<br>
! <br>
! </details>
! <details markdown=1>
! <summary>
! How do you characterize each of the single optical elements that make up this optical system,
! and their relative distances?
! </summary>
! <br>
! * The optical axis is oriented positively in the direction of light propagation
! (from the painting towards the lensball).<br>
! <br>
! * The first spherical refracting surface
! $`DS1`$ encountered by the light has
! the follwing characteristics :<br>
! $`\overline{S_1C_1}=+|R|=+5\;cm`$,
! $`n_{ini}=1`$ and $`n_{fin}=1.5`$.
! <br>
! * The second spherical refracting surface
! $DS2$ encountered by the light has the follwing characteristics :<br>
! $`\overline{S_2C_2}=-|R|=-5\;cm`$ ,
! $`n_{ini}=1.5`$ and $`n_{fin}=1`$
!
! * Algebraic distance between $DS1$ and $DS2$ is : $`\overline{S_1S_2}=+10\;cm`$
!
! </details>
! <details markdown=1>
! <summary>
! If you had to determine the characteristics of the image (position, size), how
! would you handle the problem?
! </summary>
!
! * $`DS1`$ gives an image $`B_1`$ of an object $`B`$. This image $`B_1`$ for $`DS1`$
! becomes the object for $`DS2`$. $`DS2`$ gives an image $`B'1`$ of the object $`B_1`$
!
! </details>
! </details>
! <!--FOR IMAGES 2 & 3-->
!
! <details markdown=1>
! <summary>
! Ready to answer M3P2 team questions for images 2 and 3?
! </summary>
!
! <details markdown=1>
! <summary>
! Where is the painting located?
! </summary>
!
! * The painting is located on the same side of the lens as you, behind you.
!
! </details>
! <details markdown=1>
! <summary>
!
! What are the two optical systems at the origin of the two images of the painting? And
! can you characterize each of the single optical elements (+ their relative distances)
! that make up each of these optical systems ?
! </summary>
!
! * A first optical system $`OS1`$ is composed of a simple convexe spherical mirror
! (the object is reflected on the front face of the ball lensball). Keaping the optical
! axis positively oriented in the direction of the incident light propagation on the lensball,
! the algebraic value of the mirror radius is : $`\overline{SC}=+5\;c`$.
!
! * The second optical system $`OS2`$ is composed of three simple optical elements :<br><br>
! 1) The light crosses a spherical refracting surface $`DS1`$ with characteristics :
! $`\overline{S_1C_1}=+|R|=+5\;cm`$ , $`n_{ini}=1`$ and $`n_{fin}=1.5`$.
!
! 2) Then the light is reflected at the surface of the last lensball interface that
! acts like a spherical mirror of characteristics : $`\overline{S_2C_2}=-|R|=-5\;cm`$,
! $`n=1.5`$.
!
! 3) Finally the light crosses back the first interface of the lensball that acts
! like a spherical refracting surface those characteristics are :
! $`\overline{S_3C_3}=+|R|=+5\;cm`$ , $`n_{ini}=1.5$ and $n_{fin}=1`$.
!
! Relative algebraic distances between the different elements of $`OS2`$ are :
!
! $`\overline{S_1S_2}=+10\;cm`$ and $`\overline{S_2S_3}=-10\;cm`$
!
! </details>
! <details markdown=1>
! <summary>
! Which image is associated with each of the optical systems?
! </summary>
!
! * It is difficult to be 100% sure before having made the calculations.
!
! </details>
! <details markdown=1>
! <summary>
! Why do we had to take the picture in the darkness, with only the painting
! illuminated behind the camera, to obtain images 2 and 3 ?
! </summary>
!
! * At a refracting interface, part of the light incident power is refracted,
! and part is reflected. For transparent material like glass and for visible light,
! the part of the reflected power is small. If the room had been homogeneously
! illuminated, the images 2 and 3 of the painting on the wall behind the camera would
! have been faintly visible compared to the image of the front wall through the lensball.
! </details>
! </details>
! </details>
! *TON DÉFI* : "Je regarde une cathédrale à travers une lentille boule. Puis-je prédire ce que j'obseve ?".
! <br>
! _Aptitude testée : Savoir poser un problème et conduire des calculs_<br>
! <br>
! <br>
! <br>
! *Durée de découverte : 2 heures*<br>
! *Durée de résolution : 30 minutes*
! <details markdown=1>
! <summary>
! Je choisis ce défi.
! </summary>
! La lentille-boule est une sphère de rayon $`R=5 cm`$, réalisée dans un verre
! d'indice de réfraction $`n_verre}=1.5`$. La cathédrale à une hauteur totale,
! depuis sa base jusqu'au sommets de ses tours, de 90 mètres. Je me tiens à
! 400 mètres de la cathédrale (cette distance est la distance cathédrale-oeil).
! Je regarde la cathédrale à travers la lentille-boule, située à 20 centimètres
! de mon oeil. Que-suis sensé voir ?
!
! * _La durée de résolution représente le temps qui serait typiquement alloué à la résolution_
! _de ce problème, s'il faisait partie d'un examen d'optique géométrique de_
! _première année universitaire._
!
! * _La durée de découverte est le temps estimé pour préparer ce défi, en absence_
! _d'entraînement. Cette durée est juste une indication. Prends autant de temps que_
! _nécessaire. Le temps de s'interroger sereinement sur la façon de traiter le problème,_
! _sur la méthode de résolution et sa validité, sur quelques approximations possibles_
! _si elles peuvent se justifier, et le temps qu'il faut pour vérifier les équations_
! _si vous ne les avez pas préalablement mémorisées et pour effectuer le calcul, sont importants._
!
! <details markdown=1>
! <summary>
! Prêt pour les questions ?
! </summary>
! <!--question 1-->
! <details markdown=1>
! <summary>
! (traduction en cours) What is the scientifical framework you choose to study this problem ?
! </summary>
! * All the characteristic sizes in this problem are much bigger than the wavelength of
! the visible radiation ($`\lambda\approx5\mu m`$), so I deal with this problem in the
! framework of geometrical optics, and in the paraxial approximation in order to
! characterize the image.
!
! * The cathedral sustains an angle of $`arctan\dfrac{90}{400}=13°`$ from the lensball.
! This value seems reasonable to justify at first order the use of the paraxial approximation
! (_we usually consider that angles of incidence would not exceed 10° on the various simple
! optical element encountered between the objet (here the cathedral) and the final image
! (retina of the eye or matrix sensor of a camera_).
! </details>
! <!--question 2-->
! <details markdown=1>
! <summary>
! Describe the optical system for this use of the lensball.
!</summary>
! * The lensball breaks down into two refracting spherical surfaces sharing the same centre of curvature C and of opposite radius (in algebraic values).
!</details>
!<!--question 3-->
!<detailsmarkdown=1>
!<summary>
! </summary>
!
! * The lensball breaks down into two refracting spherical surfaces sharing the same
! centre of curvature C and of opposite radius (in algebraic values).
!
! </details>
! <!--question 3-->
! <details markdown=1>
! <summary>
! What is your method of resolution ?
!</summary>
! * You don't use general equations 3a and 3b for a thick lens, they are too complicated to remind, and you don't have in m3p2 to "use" but to "build a reasoning". And you don't know at this step how to handle with centered optical systems.
! </summary>
! * You don't use general equations 3a and 3b for a thick lens, they are too complicated
! to remind, and you don't have in m3p2 to "use" but to "build a reasoning". And you don't
! know at this step how to handle with centered optical systems.
!
! * But this system is simple, so you will calculate the image of the cathedral by the first spherical refracting surface $DS_1$ encountered by the light from the cathedral $DS_1$. Then this image becomes the object for the second spherical refracting surface $DS_2$ and so I can determine position and size of the final image.
! * But this system is simple, so you will calculate the image of the cathedral by the
! first spherical refracting surface $`DS_1`$ encountered by the light from the cathedral $`DS_1`$.
! Then this image becomes the object for the second spherical refracting surface $`DS_2`$
! and so I can determine position and size of the final image.
!
! * For a spherical refracting surface, general equations are :<br><br>
! $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$ for the position.<br>
! $`\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ for the transverse magnification.
! * The optical axis is the straight line that joins the center C of the lens to my eye, positively oriented in the direction of light propagation light for that observation, so from the cathedral to my eye.
! * First spherical refrating surface $DS1$ : $S_1C_1=+5\:cm$, $n_{ini}=1$ (air) and $n_{fin}=1.5$ (glass).<br>
! Second spherical refrating surface $DS2$ : $S_1C_1=-5\:cm$, $n_{ini}=1.5$ (glass) and $n_{fin}=1$ (air)<br>
! Distance between $DS1$ and $DS2$ vertices : $S_1S_2=+10\:cm$<br>
! Object cathedral $AB$ : $AB=90\;m$ and $S_1A=-400\;m$<br>
! Let us write $A_1B_1$ the intermediate image (the image of the cathedral given by $DS1$.
!
! * Specific equations for $DS1$ are :<br><br>
! $`\dfrac{1.5}{\overline{S_1A_1}}-\dfrac{1}{\overline{S_1A}}=\dfrac{0.5}{\overline{S_1C_1}}`$ (équ. DS1a), and
! Do you see some approximation that can be done ?
!</summary>
! * In the visible range, refractive index values of transparent material are in the range [1 ; 2], then the focal lengthes of a spherical refractive surface (object as well as image) are expected to remain in the same order of magnitude than the radius of curvature, so a few centimeters in this case (we talk in absolute value here).
! </summary>
! * In the visible range, refractive index values of transparent material are in the range [1 ; 2],
! then the focal lengthes of a spherical refractive surface (object as well as image) are
! expected to remain in the same order of magnitude than the radius of curvature,
! so a few centimeters in this case (we talk in absolute value here).
!
! * We can if we want just check this fact for $`DS1`$ ($`|S_1C_1|=5\;cm`$) using équation DS1 :<br>
! \- considering $`\overline{S_1A_1}\longrightarrow\infty`$ to obtain the object focal length $`\overline{S_1F_1}`$} we get :<br>
! * The distance of the cathedral from the lensball $`|\overline{S_1A}|=90\;m`$ is huge compared to the object focal length $`|\overline{S_1F_1}|=10\;cm`$, we can consider that the cathedral is at infinity from the lensball and so the image $`\overline{A_1B_1}`$ of the cathedral stands quasi in the image focal plane of $`DS1`$ : $`\overline {S_1A_1}=\overline {S_1F'_1}=+15cm`$. So we can directly use equation DS2a with :<br>
! With more than 2 significant figures, your calculator would tell you $`0.150037`$, which nearly exactly the value of $`\overline{S_1F'_1}=+0.15\;m`$, so the approximation $`\overline{S_1A_1}=\overline{S_1F'_1}`$ you could have done is fully justified.
! With more than 2 significant figures, your calculator would tell you $`0.150037`$,
! which nearly exactly the value of $`\overline{S_1F'_1}=+0.15\;m`$, so the approximation
! $`\overline{S_1A_1}=\overline{S_1F'_1}`$ you could have done is fully justified.
! * The final image is real, and stands 2.5 cm in front of the lensball in the side of your eye. Do not bring your eye or camera too close of the lensball \!
! * The final image is real, and stands 2.5 cm in front of the lensball in the side
! of your eye. Do not bring your eye or camera too close of the lensball \!
!
! * The size of an image (transversally to the optical axis) is given by the transverse magnification $`M_T`$. By Definition $`M_T`$ is the ratio of the algebraic size of the final image $`\overline{A'B'}`$ to the algebraic size of the initial object $`\overline{AB}`$. With an intermediate image, it can be break down :<br><br>
! * As calculated previously, standing 400 metres from the cathedral, the 90 m heigh cathedral sustends the apparent angles of $`\alpha=arctan\left(\dfrac{90}{400}\right)=0.221\;rad=12.7°`$ at your eye.
! * The image of the cathedral is 1.7 cm heigth and is located between the lens (from its vertex $`S2`$) and your eyes and at 2.5cm from the lens. If your eye is 20cm away from the lens, so the distance eye-image is 17.5 cm (we use no algebraic values). Thus the image of the catedral subtends the apparent angle $`\alpha'=arctan\left(\dfrac{1.7}{17.5}\right)=0.097\;rad=5.6°`$ at your eye.
!
! * The apparent magnification $`M_A`$ of the cathedral throught the lensball for my eye in that position is<br>
! * As calculated previously, standing 400 metres from the cathedral, the 90 m heigh
! cathedral sustends the apparent angles of $`\alpha=arctan\left(\dfrac{90}{400}\right)=0.221\;rad=12.7°`$
! at your eye.<br>
! <br>
! * The image of the cathedral is 1.7 cm heigth and is located between the lens
! (from its vertex $`S2`$) and your eyes and at 2.5cm from the lens. If your eye is
! 20cm away from the lens, so the distance eye-image is 17.5 cm (we use no algebraic values).
! Thus the image of the catedral subtends the apparent angle
! $`\alpha'=arctan\left(\dfrac{1.7}{17.5}\right)=0.097\;rad=5.6°`$ at your eye.<br>
! <br>
! * The apparent magnification $`M_A`$ of the cathedral throught the lensball for my
! Taking into account that the image is reversed, the algebraic value of the apparent magnification is $`\overline{M_A}=-0.44`$.
!
! * You could obtained directly this algebraic value of $`M_A`$ by considering algebraic lengthes and angles values in the calculations :<br><br> $`\overline{M_A}=\dfrac{\overline{\alpha'}}{\overline{\alpha}}`$ $`=\dfrac {arctan\left(\frac{-0.017}{-0.175}\right)} {arctan\left(\frac{90}{-400}\right)}`$ $`=\dfrac{0.097}{-0.221}=-0.44`$
! Taking into account that the image is reversed, the algebraic value of the apparent
! magnification is $`\overline{M_A}=-0.44`$.<br>
! <br>
! * You could obtained directly this algebraic value of $`M_A`$ by considering algebraic
! lengthes and angles values in the calculations :<br><br>
