IPhO 2016

Problem 1 Accelerated Particles (14+16 pts.) Particle accelerators are used in physics, but also in other fields, for a wide variety of purposes. Besides investigating the structure of matter, they also find application, for example, in medicine. In this problem, two types of particle accelerators are to be studied. 1.1 Cyclotron The idea of the cyclotron originated with the American physicist Ernest O. Lawrence, who was awarded the Nobel Prize in Physics for it in 1939. Until the 1950s, this type of particle accelerator was “the most powerful atom-smasher in the world”. A cyclotron consists of two hollow, semicircular electrodes in a homogeneous magnetic field of flux density $B$ , oriented perpendicular to the electrodes. Between the electrodes there is a very narrow gap across which a high-frequency voltage of the form $U (t) = U_{0} sin (ω t)$ is applied. Here $U_{0}$ denotes the amplitude and $ω$ the angular frequency of the voltage. Time is denoted by $t$ . Charged particles are introduced into the centre of the arrangement. The frequency of the voltage is set so that the particles are accelerated each time they cross the gap. As a result they move approximately along a spiral path outwards, until after many revolutions they reach the edge of the arrangement, where they leave the cyclotron (cf. Fig. 1). Magnet Magnet $B$ U Figure 1: Not-to-scale sketch of a cyclotron. The vacuum chamber enclosing the electrodes is not shown. In the following, consider a cyclotron such as the one developed by E.O. Lawrence in Berkeley at the end of the 1930s. The cyclotron’s electrodes had a radius of $R = 0, 76 m$ and the magnetic flux density, which can be assumed constant over the entire cyclotron cross-section, was $B = 0, 71 T$ . In the cyclotron, protons with a charge $e = 1, 602 \cdot 1 0^{- 19} As$ and a mass $m = 1, 673 \cdot 1 0^{- 27} kg$ were accelerated. The amplitude of the high-frequency voltage was $U_{0} = 87 kV$ . Neglect relativistic effects in your treatment. 1.a) Derive an expression for the angular frequency $ω$ needed to accelerate the protons and give the value of the angular frequency. (3 pts.) 1.b) Determine the kinetic energy as well as the speed of the protons as they leave the cyclotron and assess to what extent it is permissible to neglect relativistic effects. (3 pts.) 1.c) Calculate the minimum number of revolutions a proton makes in the cyclotron before it exits, and also the time it spends inside the cyclotron. (3 pts.) If, instead of protons, electrons with a mass of $m_{e} = 9, 109 \cdot 1 0^{- 31} kg$ are accelerated, one reaches regimes in which relativistic effects play a role much more quickly. 1.d) Consider electrons that have been accelerated to the kinetic energy determined in problem 1.b) and show that their speed is very close to the speed of light. (3 pts.) At these very high speeds the relativistic mass increase of the electrons must be taken into account, which leads to the speed of the electrons in the cyclotron no longer increasing by the required amount on each revolution to be accelerated again on the next revolution. One way to circumvent this is to let the magnetic field grow stronger towards the outside at a fixed high-voltage frequency. 1.e) Derive an expression for the dependence of the magnetic flux density needed for this on the distance $r$ from the centre of the cyclotron. (2 pts.) 1.2 Betatron The betatron is in a sense a further development of the cyclotron, in which the acceleration of the particles takes place not via an applied high voltage but through the temporal variation of the magnetic field. The motion in such a field is to be studied in the following. Consider an electron moving in a cylindrically symmetric magnetic field. The magnitude $B = ∣ B ∣$ of the magnetic flux density therefore depends only on the time $t$ , the $z$ -coordinate and the distance $r$ from the $z$ -axis, but not on the angle $θ$ (cf. Fig. 2). In the plane with $z = 0$ , let the magnetic field be oriented in the $z$ -direction. In this plane electrons therefore move on a circular orbit. The radius of the circular orbit for the electron under consideration is denoted by $R$ in the sketch. To accelerate the electron on this circular orbit, the magnetic flux density is varied. Let $Δ B$ denote the change of the magnetic flux density over a small time interval $Δ t$ . Through the temporal change of the magnetic field the electron can be accelerated on this circular orbit to very large energies. To simplify the calculations you may also work non-relativistically here. z R $θ$ $B$ Figure 2: Sketch of the course of the magnetic field lines in the betatron with the electron circular orbit at $z = 0$ . 1.f) Derive a relation between the change $Δ B$ of the magnitude of the magnetic flux density $B$ on the circular orbit, i.e. at $z = 0$ and $r = R$ , and the change $ΔΦ$ of the magnetic flux through the area enclosed by the circular orbit, which is necessary so that the electron stays on the circular orbit while being accelerated. (5 pts.) Assume that the $z$ -component of the magnetic field near the circular orbit can be written in the form $B_{z} (r) = B_{0} (\frac{R}{r})^{n}$ with a constant $n \in R$ . Here $B_{0}$ denotes the magnetic flux density on the circular orbit. The $z$ -component of the flux density in this case therefore depends only on the distance from the $z$ -axis. 1.g) Determine for which values of $n$ the circular orbit is stable under both small radial and small vertical perturbations. Assume that the speed of the electron does not change under the orbit perturbation and that the perturbations can be studied separately from one another. (11 pts.)

Topic: Magnetism, Electromagnetism, Special Relativity Metodi: Lorentz Force Analysis, Faraday’s Law of Induction, Differential Equations Competenze: Mathematical Modeling, Physical Reasoning Objects: Magnet, Electron Fonte: Testo (PDF) — p.2

Problema 1 Particelle accelerate 14 + 16 punti) Gli acceleratori di particelle sono utilizzati in fisica, ma anche in altri campi, per una vasta gamma di scopi. Inoltre indagando sulla struttura della materia, trovano anche applicazione, per esempio, in medicina. In questo problema, Due tipi di acceleratori di particelle sono da studiare. 1.1 Ciclotrone L’idea del ciclotrone è originata dal fisico americano Ernest O. Lawrence, che ha premiato il Premio Nobel di Fisica per questo nel 1939. Fino agli anni ‘50, questo tipo di acceleratore di particelle era “il più grande “potenti spaccatori di atomi nel mondo”. Un ciclotrone è composto da due buchi, elettrodi semicircolari in un campo magnetico omogeneo di densità di flusso $B$ , orientati perpendicolare agli elettrodi. Tra gli elettrodi c’è un gap molto stretto attraverso il quale a high-frequency voltage of the form $U (t) = U_{0} sin (ω t)$ è applicato. Qui $U_{0}$ denotes La amplitude and $ω$ the angular frequency of the voltage. Time is denoted by $t$ . Le particelle cariche vengono introdotte nel centro del

La situazione è diversa. La frequenza del voltage è impostato in modo che le particelle sono Accelerato ogni volta che attraversano il gap. Come risultato si muovono Approximately along a spiral path verso l’esterno, fino a dopo molte rivoluzioni che raggiungono l’orlo dell’accordo, dove
il ciclotrone (cfr. Fig. 1). Magnete Magnete $B$ U Figura 1: Sketch non a scala di un ciclotrone. La camera a vuoto che chiude gli elettrodi non è dimostrato. In seguito, considerate un ciclotron come quello sviluppato da E.O. Lawrence a Berkeley alla fine degli anni ‘30. Gli elettrodi del ciclotrone avevano un raggio di $R = 0, 76 m$ e la densità del flusso magnetico, che può essere assumita costante su tutta la sezione di cross-section del ciclotrone, che è $B = 0, 71 T$ . In questo ciclo, i protoni con una carica $e = 1, 602 \cdot 1 0^{- 19} As$ e una massa $m = 1, 673 \cdot 1 0^{- 27} kg$ sono state accelerate. L’ampiezza della tensione ad alta frequenza $U_{0} = 87 kV$ . Negli effetti relativistici nel trattamento. 1.a) Derivare un’espressione per la frequenza angolare $ω$ necessaria per accelerare i protoni e dare il valore della frequenza angolare. (3 punti) 1.b) Determinare l’energia cinetica e la velocità dei protoni quando lasciano il La Commissione ha inoltre adottato una decisione che prevede di limitare il numero di persone che possono essere soggette a un’accettazione di tali effetti. (3 punti) 1.c) Calcolare il numero minimo di rivoluzioni che un protone fa nel ciclotrone prima di E’ il tempo che passa all’interno del ciclotrone. (3 punti) Se, invece di protoni, gli elettroni con una massa di $m_{e} = 9, 109 \cdot 1 0^{- 31} kg$ sono accelerati, Uno raggiunge regimi in cui gli effetti relativistici svolgono un ruolo molto più rapidamente. 1.d) Considerare gli elettroni che sono stati accelerati all’energia cinetica determinata nel problema 1.b) e mostrano che la loro velocità è molto vicina alla velocità della luce. (3 punti) A queste velocità molto elevate l’aumento di massa relativistica degli elettroni deve essere preso in considerazione, che porta alla velocità degli elettroni nel cyclotron non più aumentando per la quantità necessaria su ogni rivoluzione da essere Accelerato di nuovo sulla prossima rivoluzione. Un modo per aggirare questo è lasciare che il campo magnetico cresca più forte verso l’esterno ad una frequenza di alta tensione fissa. 1.e) Derivare un’espressione per la dipendenza della densità del flusso magnetico necessaria per questo sulla distanza $r$ dal centro del ciclotrone. - 2 punti 1.2 Betatron Il betatron è in un senso un ulteriore sviluppo del cyclotron, in cui l’accelerazione è di particelle non si verifica a causa di un’elevata tensione applicata, ma attraverso la variazione temporale del campo magnetico. La mozione in un campo simile è da studiare nel seguente. Considerate un elettrone che si muove in un campo magnetico cilindricamente simmetrico. La magnitudo $B = ∣ B ∣$ del magnetic flux density therefore depends only on the time $t$ , the $z$ coordinate and the distance $r$ from the $z$ axis, but not in the angle $θ$ (cfr. Fig. 2). In the plane with $z = 0$ , let Il campo magnetico è orientato nella direzione $z$ . In questo piano gli elettroni si muovono quindi in orbita circolare. Il raggio di Circular orbit for the electron under consideration is denoted by $R$
In un sketch. Per accelerare l’elettrone in questa orbita circolare, il La densità del flusso magnetico è variabile. Let $Δ B$ denota il cambiamento della densità del flusso magnetico over a small time interval $Δ t$ . Attraverso il cambiamento temporale del campo magnetico L’elettrone può essere accelerato in questa orbita circolare a energie molto grandi. Per semplificare i calcoli Quindi, non lavorate in modo relativistico qui. z R $θ$ $B$ Figura 2: Sketch of the course di linee di campo magnetico nel betatron con l’orbita circolare dell’elettrone a $z = 0$ . 1.f) Derivare una relazione tra il cambiamento $Δ B$ della magnitudine del flusso magnetico densità $B$ sull’orbita circolare, cioè a $z = 0$ e $r = R$ , e il cambiamento $ΔΦ$ del magnetico Il flusso attraverso l’area chiusa dall’orbita circolare, che è necessario affinché il L’elettrone rimane in orbita circolare mentre viene accelerato. (cfr. Supponiamo che il $z$ componente del campo magnetico vicino all’orbita circolare possa essere scritto nel forma $B_{z} (r) = B_{0} (\frac{R}{r})^{n}$ con una costante $n \in R$ . Qui $B_{0}$ denota la densità del flusso magnetico sul Orbita circolare. Il componente $z$ della densità di flusso in questo caso dipende quindi solo dalla distanza dal flusso. L’asse di $z$ . 1.g) Determine for which values of $n$ the circular orbit is stable under both small radial and
Le perturbazioni verticali. Supponiamo che la velocità del L’elettrone non cambia sotto l’orbita di perturbazione e che le perturbazioni possono essere studiate separatamente gli uni dagli altri. (11 pag.)

Problem 1 Accelerated particles (including the following: Particle accelerators are used in physics, but also in other fields, for a wide variety of purposes. In addition They also find application, for example, in medicine. In this problem, Two types of particle accelerators are to be studied. 1.1 Cyclotron The idea of the cyclotron originated with the American physicist Ernest O. Lawrence, who was awarded the Nobel Prize in Physics for it in 1939. Until the 1950s, this type of particle accelerator was “the most “the most powerful atomic smasher in the world”. A cyclotron consists of two hollow, semicircular electrodes in a homogeneous magnetic field of flux density $B$ , oriented perpendicular to the electrodes. Between the electrodes there is a very narrow gap across which a high-frequency voltage of the form $U (t) = U_{0} sin (ω t)$ is applied. Here $U_{0}$ denotes the following: The amplitude and $ω$ the angular frequency of the

What? Time is denoted by $t$ . Charged particles are introduced into the centre of the The arrangement. The frequency of the The voltage is set so that the particles are Accelerated every time they crossed the gap. As a result they move approximately along a spiral path Outwards, until after many revolutions They reach the edge of the arrangement, where They leave the cyclotron (cf. Fig. 1). Magnetic Magnetic $B$ U Figure 1: Not-to-scale sketch of a cyclotron. The vacuum chamber enclosing the electrodes is not shown. In the following, consider a cyclotron such as the one developed by E.O. Lawrence in Berkeley at the end of the 1930s. The cyclotron’s electrodes had a radius of $R = 0, 76 m$ and the magnetic flux density, which can be assumed constant over the entire cyclotron cross-section, which is $B = 0, 71 T$ . In the cyclotron, protons with a charge $e = 1, 602 \cdot 1 0^{- 19} As$ and a mass $m = 1, 673 \cdot 1 0^{- 27} kg$ were accelerated. The amplitude of the high-frequency voltage what $U_{0} = 87 kV$ . Neglect relativistic effects in your treatment. 1.a) Derive an expression for the angular frequency $ω$ needed to accelerate the protons And give the value of the angular frequency. (Page 3 of this report)

(b) Determine the kinetic energy as well as the speed of the protons as they leave the The Commission has also taken a number of measures to ensure that the Commission is able to take account of the impact of the new measures on the environment. (Page 3 of this report) 1.c) Calculate the minimum number of revolutions a proton makes in the cyclotron before It exits, and also the time it spends inside the cyclotron. (Page 3 of this report) If, instead of protons, electrons with a mass of $m_{e} = 9, 109 \cdot 1 0^{- 31} kg$ are accelerated, One reaches regimes in which relativistic effects play a role much more quickly.
(d) Consider electrons that have been accelerated to the kinetic energy determined in problem 1. (b) And show that their speed is very close to the speed of light. (Page 3 of this report) At these very high speeds the relativistic mass increase of the electrons must be taken into account, which leads to the speed of the electrons in the cyclotron no longer increasing by the required amount on each revolution to be Accelerated again on the next revolution. One way to circumvent this is to let the magnetic field grow stronger towards the outside at a fixed high-voltage frequency.
(e) Derive an expression for the dependence of the magnetic flux density needed for this on the distance $r$ from the centre of the cyclotron. (c) the number of persons who are not members of the 1.2 Betatron The betatron is in a sense a further development of the cyclotron, in which the acceleration The particles are not subject to applied high voltage but to the temporal variation of the magnetic field. The motion in such a field is to be studied in the following. Consider an electron moving in a cylindrically symmetrical magnetic field. The magnitude $B = ∣ B ∣$ of the magnetic flux density therefore depends only on the time $t$ , the $z$ coordinate and the distance $r$ from the $z$ axis, but not on the angle $θ$ (cf. Fig. 2). In the plane with $z = 0$ , let The magnetic field is oriented in the $z$ direction. In this plane electrons therefore move on a circular orbit. The radius of the circular orbit for the electron under consideration is denoted by $R$ In the sketch. To accelerate the electron on this circular orbit, the The magnetic flux density is varied. Let $Δ B$ denotes the change of the magnetic flux density over a small time interval $Δ t$ . Through the temporal change of the magnetic field the electron can be accelerated on this circular orbit to very large energies. To simplify the calculations you may So work non-relativistically here. z R $θ$ $B$ Figure 2: Sketch of the course of the magnetic field lines in the betatron with the electron circular orbit at $z = 0$ . 1.f) Derive a relation between the change $Δ B$ of the magnitude of the magnetic flux density $B$ on the circular orbit, i.e. at $z = 0$ and $r = R$ , and the change $ΔΦ$ of the magnetic The flow through the area enclosed by the circular orbit, which is necessary so that the electron stays on the circular orbit while being accelerated. (five points) Assume that the $z$ component of the magnetic field near the circular orbit can be written in the form $B_{z} (r) = B_{0} (\frac{R}{r})^{n}$ with a constant $n \in R$ . Here $B_{0}$ denotes the magnetic flux density on the The orbit is circular. The $z$ component of the flux density in this case therefore depends only on the distance from the $z$ -axis.
(g) Determine for which values of $n$ the circular orbit is stable under both small radial and small radial The following is a list of the types of electrical equipment used in the manufacture of electrical equipment: Assume that the speed of the The electron does not change under the orbit perturbation and that the perturbations can be studied separately from each other. (Page 11)

Problem 2 Floating Disc (30 pts.) (Idea: Fabian Bühler) Most people have probably seen at some point an ornamental fountain in which a stone sphere rests on a thin film of water. Thanks to the water cushion, the usually tonne-heavy sphere in such a fountain can even be turned by hand. In this problem you are to use a somewhat simpler arrangement to investigate how it comes about that a stone body slides on a film of water. To this end, consider, as sketched in Figure 4, a cylindrical granite disc with a radius $R$ and a thickness $D$ above a stone base. From below, water flows at a rate $Q$ into the gap between the disc and the base, so that the disc rests on a film of water. Restrict yourself to the consideration of a non-rotating disc. Figure 3: Sphere fountain in Breisach. The base covers the lateral surface in the angular range $- ϑ_{max} \leq ϑ \leq ϑ_{max}$ and is shaped so that the gap has the same width everywhere. In the gap the water flows with varying flow velocity $u$ . Along a cross-section running in the radial direction, the flow velocity depends on the distance $x$ from the lateral surface of the disc (cf. the right part of Fig. 4). Directly at the disc ( $x = 0$ ) and at the base ( $x = h$ ) the flow velocity is in each case equal to zero. The gap is very narrow, so the gap width $h$ is much smaller than $R$ . R $ϑ_{max}$ D Q h x u(x) Figure 4: Sketch of the water-borne disc on the base (left) and an enlarged detail of the water-filled gap (right). The gap width is exaggerated for clarity. For the following considerations, assume that the water in the gap moves only tangentially to the lateral surface, i.e., as indicated in Figure 4, neither in the radial direction nor along the cylinder axis. The water pressure $p$ in the gap depends on the angle $ϑ$ . You may assume, however, that the water pressure in the gap is constant in the radial direction, i.e. at a fixed angle $ϑ$ , and that the hydrostatic pressure of the water can be neglected. 2.a) Determine an expression for the velocity profile $u (x)$ in the gap as a function of the gap width $h$ , the rate $Q$ at which water flows into the gap, and the thickness $D$ of the disc. Sketch the shape of the velocity profile. (14 pts.) 2.b) Derive an expression for the water pressure $p (ϑ)$ in the gap as a function of the given quantities, the dynamic viscosity $η$ of the water and the ambient pressure $p_{0}$ . (4 pts.) Now consider in the following a granite disc with a radius $R = 50 cm$ and a thickness $D = 30 cm$ , which is covered by the water gap up to an angle $ϑ_{max} = 3 5^{\circ}$ . The density of granite is $ρ_{Granit} = 2, 75 \cdot 1 0^{3} kg m^{- 3}$ and for the viscosity of water you may take the value $η = 1, 0 \cdot 1 0^{- 3} Pa s$ . The rate at which water flows into the gap is $Q = 3, 0 \cdot 1 0^{- 4} m^{3} s^{- 1}$ . 2.c) Using the given values, calculate the gap width $h$ as well as the overpressure $p (0) - p_{0}$ at the water inlet point. (7 pts.) 2.d) Using your results, justify that the pressure difference in the radial direction and the gravitational pressure of the water are in this case indeed very small compared with the overpressure at the water inlet point, and that neglecting these contributions was therefore permissible. (5 pts.)

Topic: Fluid Mechanics Metodi: Differential Equations, Hydrostatic Equilibrium, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Disk, Cylinder Fonte: Testo (PDF) — p.4

Problema 2 Disco galleggiante (punto 30) (idea: Fabian Bühler) La maggior parte delle persone probabilmente hanno visto a un certo punto una fontana ornamentale in cui una sfera di pietra riposa su un sottile film di acqua. Grazie al cuscino d’acqua, il solitamente una sfera pesante in una fonte può anche essere trasformato a mano. In questo problema si deve usare un po ‘più semplice “Arrangement to Investigate How It Comes About That A Stone Body Slides on a Film of Water”. Per questo motivo, considerate, come Scatto in figura 4, un disco di granito cilindrico con un raggio $R$ e una spessura $D$ above a stone base. Da sotto, l’acqua scorre a un ritmo $Q$ in il gap tra il disco e la base, in modo che Il disco è un film di acqua. Restrict se stessi alla considerazione di un disco non rotante. Figura 3: Sphere fountain a Breisach. The base covers the lateral surface in the angular range $- ϑ_{max} \leq ϑ \leq ϑ_{max}$ and is shaped so that Il gap ha la stessa larghezza ovunque. In the gap the water flows with varying flow velocity $u$ . Along a cross-section running in the radial direction, the flow velocity depends on the distance $x$ from the lateral surface of the disc (cf. la parte destra di Fig. 4). Direttamente a disc ( $x = 0$ ) e alla base ( $x = h$ ) la velocità di flusso è in ogni caso pari a

Non c’è niente. Il gap è molto stretto, quindi il gap width $h$ è molto più piccolo di $R$ . R $ϑ_{max}$ D Q h x u(x) Figura 4: Sketch of the water-borne disc on the base (left) and an enlarged detail di quella spazzatura di acqua (a destra). La larghezza del gap è esagerata per la chiarezza. Per le seguenti considerazioni, supponiamo che l’acqua nel gap si muova solo tangentialmente alla superficie laterale, cioè come indicato in figura 4, né nella direzione radiale né lungo il L’asse cilindrico. La pressione dell’acqua $p$ nel gap dipende dall’angolo $ϑ$ . Si può presumere, tuttavia, che la pressione dell’acqua nel gap è costante nella direzione radial, cioè a un angolo fisso $ϑ$ , e che il la pressione idrostatica dell’acqua può essere trascurata. 2.a) Determina un’espressione per il profilo di velocità $u (x)$ nel gap come funzione di la larghezza del gap $h$ , il tasso $Q$ al quale l’acqua scorre nel gap, e lo spessore $D$ del disco. Sketta la forma del profilo di velocità. (14 pag.) 2.b) Derivare un’espressione per la pressione di acqua $p (ϑ)$ nel gap come funzione del dato Quantità, la viscosità dinamica $η$ dell’acqua e la pressione ambientale $p_{0}$ . - 4 punti Ora considerate nel seguente un disco di granito con un raggio $R = 50 cm$ e uno spessore $D = 30 cm$ , che è coperto dal gap di acqua fino ad un angolo $ϑ_{max} = 3 5^{\circ}$ . La densità di granite is $ρ_{Granit} = 2, 75 \cdot 1 0^{3} kg m^{- 3}$ and for the viscosity of water you may take the value $η = 1, 0 \cdot 1 0^{- 3} Pa s$ . Il tasso al quale l’acqua scorre nel gap è $Q = 3, 0 \cdot 1 0^{- 4} m^{3} s^{- 1}$ . 2.c) Calcolare, utilizzando i dati forniti, la larghezza del gap $h$ e la sovrapposizione $p (0) - p_{0}$ al punto di ingresso idrico. (7 punti) 2.d) Usando i risultati, giustificare che la differenza di pressione nella direzione radial e la La pressione gravitazionale dell’acqua sono in questo caso indeed very small compared with the overpressure at the Il problema è che la Commissione ha deciso di non dare il massimo di informazioni. (cfr.

Problem 2 Floating disc (Page 30 of the report) (Ideas: Fabian Bühler) Most people have probably seen at some point an ornamental fountain in which a stone sphere rests on a thin The film of water. Thanks to the water cushion, the Usually a tonne-heavy sphere in such a fountain Can even be turned by hand. In this problem you are to use a somewhat simpler arrangement to investigate how it comes about that a stone body slides on a film of water. To this end, consider, as sketched in Figure 4, a cylindrical granite disc with a radius $R$ and a thickness $D$ above a stone Base. From below, water flows at a rate $Q$ into the gap between the disc and the base, so that The disc rests on a film of water. Restrict You’re not going to be able to get yourself to consider a non-rotating disc. Figure 3: Sphere fountain in Breisach. The base covers the lateral surface in the angular range $- ϑ_{max} \leq ϑ \leq ϑ_{max}$ and is shaped so that The gap has the same width everywhere. In the gap the water flows with varying flow velocity $u$ . Along a cross-section running in the radial direction, the flow velocity depends on the distance $x$ from the lateral surface of the disc (cf. the right part of Fig. 4). Directly At the disc ( $x = 0$ ) and at the base ( $x = h$ ) the flow velocity is in each case equal to

It’s zero. The gap is very narrow, so the gap width $h$ is much smaller than $R$ . R $ϑ_{max}$ D Q h x u(x) Figure 4: Sketch of the water-borne disc on the base (left) and an enlarged detail of the water-filled gap (right). The gap width is exaggerated for clarity. For the following considerations, assume that the water in the gap moves only tangentially to the lateral surface, i.e. as indicated in Figure 4, neither in the radial direction nor along the cylinder axis. The water pressure $p$ in the gap depends on the angle $ϑ$ . You may assume, however, that The water pressure in the gap is constant in the radial direction, i.e. at a fixed angle $ϑ$ , and that the The hydrostatic pressure of the water can be neglected.

(a) Determine an expression for the velocity profile $u (x)$ in the gap as a function of the gap width $h$ , the rate $Q$ at which water flows into the gap, and the thickness $D$ of the disc. Sketch the shape of the velocity profile. (Page 14) 2.b) Derive an expression for the water pressure $p (ϑ)$ in the gap as a function of the given The water pressure of the water and the water pressure of the water are measured in terms of quantities, the dynamic viscosity $η$ of the water and the ambient pressure $p_{0}$ . The Commission has also adopted a proposal for a directive on the protection of workers’ rights. Now consider in the following a granite disc with a radius $R = 50 cm$ and a thickness $D = 30 cm$ , which is covered by the water gap up to an angle $ϑ_{max} = 3 5^{\circ}$ . The density of granite is $ρ_{Granit} = 2, 75 \cdot 1 0^{3} kg m^{- 3}$ and for the viscosity of water you may take the value $η = 1, 0 \cdot 1 0^{- 3} Pa s$ . The rate at which water flows into the gap is $Q = 3, 0 \cdot 1 0^{- 4} m^{3} s^{- 1}$ . 2.c) Using the given values, calculate the gap width $h$ as well as the overpressure $p (0) - p_{0}$ at the water inlet point. (Page 77)
(d) Using your results, justify that the pressure difference in the radial direction and the pressure difference in the radial direction The gravitational pressure of the water are indeed very small compared to the overpressure at the The Commission has already made a number of proposals to the Council. (five points)

Problem 3 Experimental Problem - Interference Experiments (21+19 pts.) In this problem everything revolves around the interference of light waves. You are first to determine the wavelength of the light emitted by a laser pointer and subsequently the thickness of hairs. Besides the IPhO ruler sent out with these problems, you may use the following materials for experimenting: a laser pointer, ruler(s), a tape measure and other typical household items. If you do not own a laser pointer, you can surely borrow one at school. Caution! Under no circumstances look into the laser beam and do not point it at other people either! 3.1 Determining the wavelength of the laser pointer Let the light of the laser pointer fall at a shallow angle onto the millimetre scale of the IPhO ruler. If you set up the arrangement skilfully, you can observe on a wall onto which the light falls an interference pattern with alternating bright and dark regions. 3.a) Explain how this interference pattern comes about and derive an expression that relates the wavelength $λ$ of the light emitted by the laser to the diffraction orders observable in the interference pattern as well as to other measurable or known quantities. (7 pts.) 3.b) Using the ruler and the derived relation, experimentally determine the wavelength of the laser light used. Compare the determined wavelength either with the wavelength specified by the manufacturer or with the value to be expected from the colour of the light. (14 pts.) You may assume that the spacing of the mm markings on the ruler is $1, 00 mm$ . 3.2 Determining the thickness of hairs Persuade friends and acquaintances to provide you with a few of their head hairs for the following experiment1. Let the light of the laser pointer fall onto a slightly taut hair. On a screen or wall standing some distance away you can now again observe an interference pattern. This time, however, it looks a little different. Since you now know the wavelength of the laser light, you can deduce the thickness of the hair used from the interference pattern. 3.c) Explain how the interference pattern comes about in this case and now derive an expression that relates the wavelength $λ$ of the light emitted by the laser to the diffraction orders observable in the interference pattern, the thickness of the hair, as well as to other measurable quantities. (5 pts.) 3.d) Using the derived relation, experimentally determine the thickness of at least three different human hairs. (14 pts.) General notes on the experimental problem • Describe and document your procedure in enough detail that every step is easy to follow. In particular, sketch your experimental setups. • Carry out all your experiments so that the results are as accurate as possible. • In addition, estimate the errors of all results sensibly. 1It may help if you point out that it is for a scientific purpose.

Topic: Wave Optics, Oscillations & Waves Metodi: Interference & Diffraction Analysis, Experimental Data Analysis, Error Propagation Competenze: Experimental Data Analysis, Error Propagation, Measurement & Instrumentation Objects: Slit, Screen Fonte: Testo (PDF) — p.6

Problema 3 Problema sperimentale - Sperimenti di interferenza (cfr. In questo problema tutto gira attorno all’interferenza delle onde luminose. Tu sei il primo a determinare la lunghezza d’onda della luce emessa da un laser pointer e successivamente lo spessore dei capelli. Oltre al ruler IPhO inviato con questi problemi, è possibile utilizzare i seguenti materiali per sperimentazione: un laser pointer, ruler(s), un tape measure and other typical household

Istituzioni. Se non possiedi un laser pointer, puoi sicuramente prenderne uno a scuola. Attenzione! In nessun caso, guardate il fascio laser. E non puntarlo nemmeno su altre persone! 3.1 Determinare la lunghezza d’onda del laser pointer Lascia che la luce del laser pointer cadano a un angolo poco profondo sulla scala di millimetri dell’IPhO
Non è vero. Se impostate l’arrangimento abile, potete osservare su un muro su cui la luce si verifica un pattern di interferenza con regioni alternate luminose e scure.

(a) Spiegare come questo pattern di interferenza viene prodotto e derivare un’espressione che Relates the wavelength $λ$ of the light emitted by the laser to the diffraction orders observable in the interference pattern come pure ad altre quantità misurabili o note. (7 punti) 3.b) Usando il ruler e la relazione derivata, determinare sperimentalmente il lunghezza d’onda della luce laser utilizzata. Compare the determined wavelength either con la lunghezza d’onda specificata dal produttore o con il valore da essere L’aspettato dal colore della luce. (14 pag.) Si può supporre che lo spaziamento dei segni mm sul ruotatore sia $1, 00 mm$ . 3.2 Determinare lo spessore dei capelli Persuadere amici e conoscenti a fornire alcuni dei loro capelli per il seguente esperimento1. Lascia che la luce del laser di punta cadano su un capello leggermente stronzo. On Un schermo o un muro che si trova a qualche distanza, puoi ora osservare di nuovo un modello di interferenza. Questa volta, però, sembra un po’ diverso. Dal momento che ora conosci la lunghezza d’onda di luce laser, si può dedurre lo spessore dei capelli utilizzati dal modello di interferenza. 3.c) Spiegare come il pattern di interferenza viene prodotto in questo caso e ora derivare da espressione che relati la lunghezza d’onda $λ$ della luce emessa dal laser agli ordini di diffrazione Le dimensioni di questo tipo di interferenza sono notevoli, come la spessura dei capelli, e anche altre quantità misurabili. (cfr. 3.d) Usando la relazione derivata, determinare sperimentalmente lo spessore di almeno tre diversi capelli umani. (14 pag.) Nota generale sul problema sperimentale • Descrivere e documentare la procedura in sufficiente dettaglio per rendere ogni passo facile da seguire. In particolare, schizziare le tue configurazioni sperimentali. • eseguire tutti i tuoi esperimenti in modo che i risultati siano il più accurati possibile. • Inoltre, stimare sensibilmente gli errori di tutti i risultati. 1Questo può aiutare se si fa notare che è per un scopo scientifico.

Problem 3 Experimental problem - interference experiments (c) the number of persons who are not members of the In this problem everything revolves around the interference of light waves. You are first to determine the wavelength of the light emitted by a laser pointer and subsequently the thickness of hairs. In addition to the IPhO ruler sent out with these problems, you may use the following materials for experimenting: a laser pointer, ruler(s), a tape measure and other typical household the items. If you don’t own a laser pointer, you can surely borrow one at school.

What? Under no circumstances look into the laser beam And don’t point it at other people either! 3.1 Determining the wavelength of the laser pointer Let the light of the laser pointer fall at a shallow angle onto the millimetre scale of the IPhO I’m not going to be a ruler. If you set up the arrangement skillfully, you can observe on a wall onto which the light The following is a list of the types of interference patterns with alternating bright and dark regions.

(a) Explain how this interference pattern comes about and derive an expression that relates the wavelength $λ$ of the light emitted by the laser to the diffraction orders observable in the interference pattern as well as to other measurable or known quantities. (Page 77) 3.b) Using the ruler and the derived relation, experimentally determine the wavelength of the laser light used. Compare the determined wavelength either with the wavelength specified by the manufacturer or with the value to be expected from the color of the light. (Page 14) You may assume that the spacing of the mm markings on the ruler is $1, 00 mm$ . 3.2 Determining the thickness of hairs Persuade friends and acquaintances to provide you with a few of their head hairs for the following experiment1. Let the light of the laser pointer fall on a slightly taut hair. On A screen or wall standing some distance away you can now again observe an interference pattern. This time, however, it looks a little different. Since you now know the wavelength of the laser light, you can deduce the thickness of the hair used from the interference pattern. 3.c) Explain how the interference pattern comes about in this case and now derive an expression that relates the wavelength $λ$ of the light emitted by the laser to the diffraction orders The measurement of the interference pattern, the thickness of the hair, as well as other measurable quantities. (five points) 3.d) Using the derived relation, experimentally determine the thickness of at least three different human hairs. (Page 14) General notes on the experimental problem • Describe and document your procedure in sufficient detail that every step is easy to follow. In particular, sketch your experimental setups. • Perform all your experiments so that the results are as accurate as possible. • In addition, estimate the errors of all results sensibly. 1It may help if you point out that it is for a scientific purpose.

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