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physics_en_no_proof_embedding

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884
A raindrop of mass $M=0.035 \mathrm{~g}$ is at height $H=2 \mathrm{~km}$ above a large lake. The raindrop then falls down (without initial velocity), mixing and coming to equilibrium with the lake. Assume that the raindrop, lake, air, and surrounding environment are at the same temperature $T=300 \mathrm{~K}$. Determin...
[ "$2.29 \\times 10^{-3}$" ]
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885
A rocket with mass of 563.17 (not including the mass of fuel) metric tons sits on the launchpad of the Kennedy Space Center (latitude $28^{\circ} 31^{\prime} 27^{\prime \prime} \mathrm{N}$, longitude $80^{\circ} 39^{\prime} 03^{\prime \prime} \mathrm{W}$ ), pointing directly upwards. Two solid fuel boosters, each with ...
[ "$18.44$" ]
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888
A spacecraft is orbiting in a very low circular orbit at a velocity $v_{0}$ over the equator of a perfectly spherical moon with uniform density. Relative to a stationary frame, the spacecraft completes a revolution of the moon every 90 minutes, while the moon revolves in the same direction once every 24 hours. The pilo...
[ "$3.06$" ]
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890
Consider two points $S$ and $S^{\prime}$ randomly placed inside a $D$-dimensional hyper-rectangular room with walls that are perfect-reflecting $(D-1)$-dimensional hyper-plane mirrors. How many different light-rays that start from $S$, reflect $N$ times on one of the walls and $N-1$ times on each of the rest, then go ...
[ "895" ]
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891
Two concentric isolated rings of radius $a=1 \mathrm{~m}$ and $b=2 \mathrm{~m}$ of mass $m_{a}=1 \mathrm{~kg}$ and $m_{b}=2 \mathrm{~kg}$ are kept in a gravity free region. A soap film of surface tension $\sigma=0.05 \mathrm{Nm}^{-1}$ with negligible mass is spread over the rings such that it occupies the region betwee...
[ "$2 \\pi \\sqrt{\\frac{10 \\ln (2)}{3 \\pi}}$" ]
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893
A table of unknown material has a mass $M=100 \mathrm{~kg}$, width $w=4 \mathrm{~m}$, length $\ell=3 \mathrm{~m}$, and 4 legs of length $L=0.5 \mathrm{~m}$ with a Young's modulus of $Y=1.02 \mathrm{MPa}$ at each of the corners. The cross-sectional area of a table leg is approximately $A=1 \mathrm{~cm}^{2}$. The surface...
[ "$18.71$" ]
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894
Dipole Conductor An (ideal) electric dipole of magnitude $p=1 \times 10^{-6} \mathrm{C} \cdot \mathrm{m}$ is placed at a distance $a=0.05 \mathrm{~m}$ away from the center of an uncharged, isolated spherical conductor of radius $R=0.02 \mathrm{~m}$. Suppose the angle formed by the dipole vector and the radial vector (t...
[ "$-25.22$" ]
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897
A uniform spherical metallic ball of mass $m$, resistivity $\rho$, and radius $R$ is kept on a smooth friction-less horizontal ground. A horizontal uniform, constant magnetic field $B$ exists in the space parallel to the surface of ground. The ball was suddenly given an impulse perpendicular to magnetic field such that...
[ "$0.019$" ]
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898
In quantum mechanics, when calculating the interaction between the electron with the proton in a hydrogen atom, it is necessary to compute the following volume integral (over all space): $$ \mathbf{I}=\int \mathbf{B}(\mathbf{r})|\Psi(\mathbf{r})|^{2} d V $$ where $\Psi(\mathbf{r})$ is the spatial wavefunction of the e...
[ "$0.0254$" ]
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899
Zed is trying to model the repulsive interaction between 2 objects, $A$ and $B$ (with masses $m_{A}$ and $m_{B}$, respectively), in a relativistic setting. He knows that in relativity, forces cannot act at a distance, so he models the repulsive force with a small particle of mass $m$ that bounces elastically between $A...
[ "378" ]
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900
Consider an optical system consisting of two thin lenses sharing the same optical axis. When a cuboid with a side parallel to the optical axis is placed to the left of the left lens, its final image formed by the optical system is also a cuboid but with 500 times the original volume. Assume the two lenses are $10 \mat...
[ "$2216$" ]
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901
Consider an infinite square grid of equal resistors where the nodes are exactly the lattice points in the 2D Cartesian plane. A current $I=2.7 \mathrm{~A}$ enters the grid at the origin $(0,0)$. Find the current in Amps through the resistor connecting the nodes $(N, 0)$ and $(N, 1)$, where $N=38$ can be assumed to be m...
[ "$1.488 \\times 10^{-4}$" ]
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902
Suppose we have a non-ideal gas, and in a certain volume range and temperature range, it is found to satisfy the state relation $$ p=A V^{\alpha} T^{\beta} $$ where $A$ is a constant, $\alpha=-\frac{4}{5}$ and $\beta=\frac{3}{2}$, and the other variables have their usual meanings. Throughout the problem, we will assum...
[ "$\\frac{7}{4}$" ]
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903
The coin flip has long been recognized as a simple and unbiased method to randomly determine the outcome of an event. In the case of an ideal coin, it is well-established that each flip has an equal $50 \%$ chance of landing as either heads or tails. However, coin flips are not entirely random. They appear random to us...
[ "0" ]
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905
Suppose all cars on a (single-lane) highway are identical. Their length is $l=4 \mathrm{~m}$, their wheels have coefficients of friction $\mu=0.7$, and they all travel at speed $v_{0}$. Find the $v_{0}$ which maximizes the flow rate of cars (i.e. how many cars travel across an imaginary line per minute). Assume that th...
[ "$7.41$" ]
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907
In a resource-limited ecological system, a population of organisms cannot keep growing forever (such as lab bacteria growing inside culture tube). The effective growth rate $g$ (including contributions from births and deaths) depends on the instantaneous abundance of resource $R(t)$, which in this problem we will cons...
[ "2.1095" ]
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908
An incandescent lightbulb is connected to a circuit which delivers a maximum power of 10 Watts. The filament of the lightbulb is made of Tungsten and conducts electricity to produce light. The specific heat of Tungsten is $c=235 \mathrm{~J} /(\mathrm{K} \cdot \mathrm{kg})$. If the circuit is alternating such that the t...
[ "$1.68 \\times 10^{-3}$" ]
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909
In hyperdrive, Spaceship-0 is relativistically moving at the velocity $\frac{1}{3} c$ with respect to reference frame $R_{1}$, as measured by Spaceship-1. Spaceship-1 is moving at $\frac{1}{2} c$ with respect to reference frame $R_{2}$, as measured by Spaceship-2. Spaceship- $k$ is moving at speed $v_{k}=\frac{k+1}{k+3...
[ "19" ]
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910
The path of an asteroid that comes close to the Earth can be modeled as follows: neglect gravitational effects due to other bodies, and assume the asteroid comes in from far away with some speed $v$ and lever arm distance $r$ to Earth's center. On January 26, 2023, a small asteroid called 2023 BU came to a close distan...
[ "$3.74 \\times 10^{24}$" ]
[ 0.035400390625, -0.0159912109375, -0.006439208984375, 0.0299072265625, -0.0201416015625, 0.040771484375, 0.021240234375, 0.010498046875, -0.0262451171875, -0.00830078125, -0.036376953125, 0.0223388671875, -0.03173828125, -0.0157470703125, 0.019775390625, -0.011474609375, -0.041503906...
911
Pegasi and Betelgeuse are two star systems that can undergo a supernova. Betelgeuse is 548 light-years away from Earth and IK Pegasi is 154 light-years away from Earth. Assume that the two star systems are 500 light-years away from each other. Astronomers on Earth observe that the two star systems undergo a supernova e...
[ "400" ]
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912
A ball of mass $1 \mathrm{~kg}$ is thrown vertically upwards and it faces a quadratic drag with a terminal velocity of $20 \mathrm{~m} / \mathrm{s}$. It reaches a maximum height of $30 \mathrm{~m}$ and falls back to the ground. Calculate the energy dissipated until the point of impact (in J).
[ "515.83" ]
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915
Two parallel square plates of side length $1 \mathrm{~m}$ are placed a distance $30 \mathrm{~cm}$ apart whose centers are at $(-15 \mathrm{~cm}, 0,0)$ and $(15 \mathrm{~cm}, 0,0)$ have uniform charge densities $-10^{-6} \mathrm{C} / \mathrm{m}^{2}$ and $10^{-6} \mathrm{C} / \mathrm{m}^{2}$ respectively. Find the magnit...
[ "$11.9$" ]
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917
A space elevator consists of a heavy counterweight placed near geostationary orbit, a thread that connects it to the ground (assume this is massless), and elevators that run on the threads (also massless). The mass of the counterweight is $10^{7} \mathrm{~kg}$. Mass is continuously delivered to the counterweight at a r...
[ "$15.21$" ]
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918
Consider a spherical shell of thickness $\delta=0.5 \mathrm{~cm}$ and radius $R=5 \mathrm{~cm}$ made of an Ohmic material with resistivity $\rho=10^{-7} \Omega \mathrm{m}$. A spherical laser source with a tuned frequency of $f_{0}$ and intensity $I_{0}=10^{5} \mathrm{~W} / \mathrm{m}^{2}$ is placed at the center of the...
[ "$2.39078 \\times 10^{-15}$" ]
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919
A stable star of radius $R$ has a mass density profile $\rho(r)=\alpha(1-r / R)$. Here, "stable" means that the star doesn't collapse under its own gravity. If the internal pressure at the core is provided solely by the radiation of photons, calculate the temperature at the core. Assume the star is a perfect black body...
[ "$26718$" ]
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922
In this problem, we consider a simple model for a thermoacoustic device. The device uses heavily amplified sound to provide work for a pump that can then extract heat. Sound waves form standing waves in a tube of radius $0.25 \mathrm{~mm}$ that is closed on both sides, and a two-plate stack is inserted in the tube. A t...
[ "$6.47$" ]
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923
The following information applies for the next two problems. For your mass spectroscopy practical you are using an apparatus consisting of a solenoid enclosed by a uniformly charged hollow cylinder of charge density $\sigma=50 \mu \mathrm{C} / \mathrm{m}^{2}$ and radius $r_{0}=7 \mathrm{~cm}$. There exists an infinites...
[ "$\\cot \\frac{\\pi}{8} \\sqrt{\\frac{2 \\sigma m}{q \\varepsilon_{0} r_{0}} \\ln \\frac{R}{r_{0}}}$" ]
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926
In the far future, the Earth received an enormous amount of charge as a result of Mad Scientist ecilA's nefarious experiments. Specifically, the total charge on Earth is $Q=1.0 \times 10^{11} \mathrm{C}$. (compare this with the current $5 \times 10^{5} \mathrm{C}$ ). Estimate the maximum height of a "mountain" on Earth...
[ "115" ]
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928
Follin is investigating the electrostatic pendulum. His apparatus consists of an insulating Styrofoam ball with a mass of $14 \mathrm{mg}$ and radius $r=0.5 \mathrm{~cm}$ suspended on a uniform electrically-insulating string of length $1 \mathrm{~m}$ and mass per unit length density of $1.1 \cdot 10^{-5} \mathrm{~kg} /...
[ "$0.0475$" ]
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929
Hoping to get a larger deflection, Follin replaces the insulating Styrofoam ball with a conducting pith ball of mass $250 \mathrm{mg}$ and $2 \mathrm{~cm}$ and daisy chains 4 additional $10 \mathrm{kV}$ High Voltage Power Supplies to increase the voltage drop across the plates to $50 \mathrm{kV}$. Leaving the plate sep...
[ "$4.48 \\times 10^{-10}$" ]
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936
(a) Where should a pin be placed on the optical axis such that its image is formed at the same place?
[ "15" ]
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939
(b) Equation 4 is a version of the damped harmonic oscillator, and can be solved by guessing a solution $\eta=\alpha e^{\lambda t}$. Plugging in this guess, what must $\lambda$ be?
[ "$\\lambda=i \\Omega$" ]
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940
(c) Using your answer to part (b), and defining $\alpha=A e^{i \phi}$ where $A$ and $\phi$ are real, find $\mathbf{x}(\mathbf{t})$ and $\mathbf{y}(\mathbf{t})$. This is the trajectory for a particle which is stationary with respect to the symmetry axis. While not required for this problem, an additional guess would re...
[ "$x(t)=A \\cos (\\Omega t+\\phi)$ , $y(t)=A \\sin (\\Omega t+\\phi)$" ]
[ 0.03466796875, 0.003143310546875, 0.0098876953125, 0.012939453125, -0.00836181640625, 0.029296875, -0.0159912109375, 0.0205078125, -0.0218505859375, -0.0263671875, -0.05078125, 0.015625, -0.028076171875, -0.028076171875, 0.0390625, -0.0272216796875, -0.004486083984375, 0.0159912109...
941
(d) The one-dimensional diffusion equation (also called the "heat equation") is given (for a free particle) by $$ \frac{\partial \psi}{\partial t}=a \frac{\partial^{2} \psi}{\partial x^{2}} \tag{5} $$ A spatial wave can be written as $\sim e^{i k x}$ (larger $k$ 's correspond to waves oscillating on smaller length sc...
[ "$\\omega=-i k^{2} a$" ]
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942
(e) The most important equation of non-relativistic quantum mechanics is the Schrödinger equation, which is given by $$ i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \psi}{\partial x^{2}} \tag{6} $$ Using your answer to part (d), what is the dispersion relation of the Schrödinger ...
[ "$\\omega=\\frac{\\hbar k^{2}}{2 m}$" ]
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944
(g) The theory of relativity instead posits that the energy of a particle is given by $E=\sqrt{p^{2} c^{2}+m^{2} c^{4}}$. In accordance with this, we can try to guess a relativistic version of the Schrödinger equation: $$ \frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}...
[ "$\\sqrt{k^{2} c^{2}+\\frac{m^{2} c^{4}}{\\hbar^{2}}}$, $-\\sqrt{k^{2} c^{2}+\\frac{m^{2} c^{4}}{\\hbar^{2}}}$" ]
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947
(c) The uniqueness theorem allows us to use "image" charges in certain settings to describe the system. Consider one such example: There is a point-like charge $q$ at a distance $L$ from a metallic sphere of radius $R$ attached to the ground. As you argued in part (a), sphere will be polarized to make sure the electric...
[ "$x=\\frac{R^{2}}{L}$ , $q_{0}=-q \\frac{R}{L}$" ]
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949
(e) Find the force of attraction between the charge and the sphere.
[ "$F=\\frac{1}{4 \\pi \\epsilon_{0}} \\frac{q^{2} \\frac{R}{L}}{\\left(L-\\frac{R^{2}}{L}\\right)^{2}}$" ]
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954
(a) What value of $\theta$ maximizes the range?
[ "$\\frac{\\pi}{4}$" ]
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955
(b) What value of $\theta$ maximizes the surface area under the trajectory curve?
[ "$\\pi / 3$" ]
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956
(c) What is the answer for (a), if the point mass is thrown from an apartment of height $h$ ? Now assume that we have a frictional force that is proportional to the velocity vector, such that the equation of motion is as follows $$ \frac{d \vec{v}}{d t}=\vec{g}-\beta \vec{v} $$
[ "$\\arcsin \\left(\\frac{1}{\\sqrt{2}} \\frac{v_{0}}{\\sqrt{v_{0}^{2}+g h}}\\right)$" ]
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957
(d) Supposing that $\beta<<\frac{g}{v_{0}}$, find the duration of the motion $T$.
[ "$T=\\frac{2 v_{0} \\sin \\theta}{g} \\frac{1}{1+\\frac{\\beta v_{0} \\sin \\theta}{g}}$" ]
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972
(a) Compute the electric potential inside and outside the sphere.
[ "$\\Phi_{-}=\\frac{Q}{4 \\pi \\varepsilon_{0} R}$,$\\Phi_{+}=\\frac{Q}{4 \\pi \\varepsilon_{0} r}$" ]
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973
(a) Knowing that Legendre polynomials are orthogonal $\left(\left\langle P_{m}(x), P_{n}(x)\right\rangle=0\right.$ if $m \neq n)$ and $\operatorname{deg} P_{n}(x)=n$, obtain $P_{2}(x)$ and $P_{3}(x)$. For reaching the usual and most convenient form of these polynomials, divide your results by the norm: $\left\|P_{n}(x)...
[ "$P_{2}(x)=C_{2}\\left(x^{2}+\\lambda_{1} x+\\lambda_{0}\\right)$ , $P_{3}=\\frac{1}{2}\\left(5 x^{3}-3 x\\right)$" ]
[ 0.0203857421875, -0.007049560546875, -0.0128173828125, -0.0159912109375, 0.00007390975952148438, -0.00144195556640625, 0.0169677734375, -0.01348876953125, -0.0252685546875, 0.009765625, -0.0302734375, 0.003662109375, -0.0169677734375, -0.0059814453125, 0.0152587890625, 0.034423828125, ...
974
(b) Compute the electric potential both inside and outside the sphere.
[ "$\\Phi_{-}=\\frac{q}{4 \\pi \\varepsilon_{0} r}-\\frac{q}{4 \\pi \\varepsilon_{0} R}+\\frac{V_{0} \\cos \\theta}{R} r$,$\\Phi_{+}=\\frac{V_{0} R^{2}}{r^{2}} \\cos \\theta$" ]
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982
(b) Any real inductor has undesired, or parasitic, resistance. We can model the real inductor as an ideal inductor $L$ in series with a parasitic resistance $R$. Due to the thermal noise $\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R$ of its parasitic resistance, this (real) inductor will support a nonzero per-...
[ "$\\frac{4 k T R}{R^{2}+4 \\pi^{2} f^{2} L^{2}}$" ]
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984
(b) Find the period $T$ and angular frequency $\omega$ of the orbital motion.
[ "$\\omega=\\sqrt{\\frac{G\\left(M_{1}+M_{2}\\right)}{\\left|\\overrightarrow{r_{2}}-\\overrightarrow{r_{1}}\\right|^{3}}}$ , $T=2 \\pi \\sqrt{\\frac{\\left|\\overrightarrow{r_{2}}-\\overrightarrow{r_{1}}\\right|^{3}}{G\\left(M_{1}+M_{2}\\right)}}$" ]
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985
(a) Express $\alpha$ in terms of $M_{1}$ and $M_{2}$.
[ "$\\alpha=\\frac{M_{2}}{M_{1}+M_{2}}$" ]
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986
(b) Let $\rho_{1}(t)$ and $\rho_{2}(t)$ be the distances from $m$ to $M_{1}$ and $M_{2}$ respectively. Express $\rho_{1}(t)$ and $\rho_{2}(t)$ in terms of the coordinates and constants given.
[ "$\\rho_{1}(t)=\\sqrt{(x(t)+\\alpha R)^{2}+(y(t))^{2}}$ , $\\rho_{2}(t)=\\sqrt{(x(t)-(1-\\alpha) R)^{2}+(y(t))^{2}}$" ]
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987
(c) By considering the centrifugal acceleration $\omega^{2} \vec{r}$ and Coriolis acceleration $-2 \omega \times$ $\vec{v}$, find the acceleration $\frac{d^{2}}{d t^{2}} \vec{r}$ of the third mass in terms of the coordinates and constants given, including $\rho_{1}$ and $\rho_{2}$.
[ "$\\frac{d^{2}}{d t^{2}} \\vec{r}=-G M_{1} \\frac{\\vec{r}-\\vec{r_{1}}}{\\rho_{1}^{3}}-G M_{2} \\frac{\\vec{r}-\\vec{r_{2}}}{\\rho_{2}^{3}}+\\omega^{2} \\vec{r}-2 \\omega \\times \\vec{r}$" ]
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988
(d) Express $\frac{d^{2} x}{d t^{2}}$ and $\frac{d^{2} y}{d t^{2}}$ in terms of $U$, where $U=-\frac{G M_{1}}{\rho_{1}}-\frac{G M_{2}}{\rho_{2}}-\frac{\omega^{2}}{2}\left(x^{2}+y^{2}\right)$.
[ "$\\ddot{x} =2 \\omega \\dot{y}-\\frac{\\partial U}{\\partial x}$ , $\\ddot{y} =-2 \\omega \\dot{x}-\\frac{\\partial U}{\\partial y}$" ]
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989
(e) Hence, write down an expression of the motion of $m$ which is a constant.
[ "$-2 U-v^{2}$" ]
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1,006
a - Calculate the flattening factor, given that the gravitational constant is $6.67 \times 10^{-11}$ N.m $\cdot \mathrm{kg}^{-2}$.
[ "$3.7 \\times 10^{-4}$" ]
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1,012
2.1 Find the ratio $\frac{\rho_{i} T_{i}}{\rho_{a} T_{a}}$ in terms of $\gamma, P_{a}$ and $R_{0}$.
[ "$1+\\frac{4 \\gamma}{R_{0} P_{a}}$" ]
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1,013
2.2 Find the numerical value of $\frac{\rho_{i} T_{i}}{\rho_{a} T_{a}}-1$ using $\gamma=0.0250 \mathrm{Nm}^{-1}, R_{0}=1.00 \mathrm{~cm}$, and $P_{a}=1.013 \times 10^{5} \mathrm{Nm}^{-2}$.
[ "1.0001" ]
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1,014
2.3 The bubble is initially formed with warmer air inside. Find the minimum numerical value of $T_{i}$ such that the bubble can float in still air. Use $T_{a}=300 \mathrm{~K}, \rho_{s}=1000 \mathrm{kgm}^{-3}$, $\rho_{a}=1.30 \mathrm{kgm}^{-3}, t=100 \mathrm{~nm}$ and $g=9.80 \mathrm{~ms}^{-2}$.
[ "307.1" ]
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1,015
2.4 Find the minimum velocity $u$ of an updraught (air flowing upwards) that will keep the bubble from falling at thermal equilibrium. Give your answer in terms of $\rho_{s}, R_{0}, g, t$ and the air's coefficient of viscosity $\eta$. You may assume that the velocity is small such that Stokes's law applies, and ignore ...
[ "$\\frac{4 R_{0} \\rho_{s} t g}{6 \\eta}+\\frac{\\frac{4}{3} R_{0}^{2} \\rho_{a} g\\left(\\frac{4 \\gamma}{R_{0} P_{a}}\\right)}{6 \\eta}$" ]
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1,016
2.5 Calculate the numerical value for $u$ using $\eta=1.8 \times 10^{-5} \mathrm{kgm}^{-1} \mathrm{~s}^{-1}$.
[ "$0.36$" ]
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1,017
2.6 If this spherical bubble is now electrified uniformly with a total charge $q$, find an equation describing the new radius $R_{1}$ in terms of $R_{0}, P_{a}, q$ and the permittivity of free space $\varepsilon_{0}$.
[ "$\\left(\\frac{R_{1}}{R_{0}}\\right)^{4}-\\left(\\frac{R_{1}}{R_{0}}\\right)-\\frac{q^{2}}{32 \\pi^{2} \\varepsilon_{0} R_{0}^{4} P_{a}}=0$" ]
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1,018
2.7 Assume that the total charge is not too large (i.e. $\frac{q^{2}}{\varepsilon_{0} R_{0}^{4}}<<P_{a}$ ) and the bubble only experiences a small increase in its radius, find $\Delta R$ where $R_{1}=R_{0}+\Delta R$. Given that $(1+x)^{n} \approx 1+n x$ where $x \ll 1$.
[ "$\\frac{q^{2}}{96 \\pi^{2} \\varepsilon_{0} R_{0}^{3} P_{a}}$" ]
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1,050
A.1 Determine the minimal possible energy $E_{\min }$ of the quantum particle in the well. Express your answer in terms of $m, L$, and the Planck's constant $h$.
[ "$E_{\\min }=\\frac{h^{2}}{8 m L^{2}}$" ]
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1,051
A.2 Find the general expression for the energy $E_{n}$ (here $n=1,2,3, \ldots$ ).
[ "$E_{n}=\\frac{h^{2} n^{2}}{8 m L^{2}}$" ]
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1,052
A.3 Particle can undergo instantaneous transition from one state to another only by emitting or absorbing a photon of the corresponding energy difference. Find the wavelength $\lambda_{21}$ of the photon emitted during the transition of the particle from the first excited state $\left(E_{2}\right)$ to the ground state ...
[ "$\\lambda_{21}=\\frac{8 m c L^{2}}{3 h}$" ]
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1,053
C.1 Given a non-interacting gas of ${ }^{87} \mathrm{Rb}$ atoms in thermal equilibrium, write the expressions for their typical linear momentum $p$ and the typical de Broglie wavelength $\lambda_{\mathrm{dB}}$ as a function of atom's mass $m$, temperature $T$ and physical constants.
[ "$p=\\sqrt{3 m k_{\\mathrm{B}} T}$ , $\\lambda_{\\mathrm{dB}}=\\frac{h}{\\sqrt{3 m k_{\\mathrm{B}} T}}$" ]
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1,054
C.2 Calculate the typical distance between the particles in a gas, $\ell$, as a function of particle density $n$. Hence deduce the critical temperature $T_{c}$ as a function of atom's mass, their density and physical constants.
[ "$\\ell=n^{-1 / 3}$ , $T_{c}=\\frac{h^{2} n^{2 / 3}}{3 m k_{\\mathrm{B}}}$" ]
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1,060
A.1 Assume that the Sun radiates like a perfect blackbody. Use this fact to calculate the temperature, $T_{\mathrm{s}}$, of the solar surface.
[ "$5.76 \\times 10^{3} $" ]
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A2 Using the Wien approximation, express the total radiated solar power, $P_{\mathrm{in}}$, incident on the surface of the <br> solar cell, in terms of $A, R_{\odot}, d_{\odot}, T_{\mathrm{S}}$ and the fundamental constants $c, h, k_{\mathrm{B}}$
[ "$P_{\\mathrm{in}}=\\frac{12 \\pi k_{\\mathrm{B}}^{4}}{c^{2} h^{3}} T_{\\mathrm{s}}^{4} A \\frac{R_{\\odot}^{2}}{d_{\\odot}^{2}}$" ]
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1,062
A3 Express the number of photons, $n_{\gamma}(v)$, per unit time per unit frequency interval incident on the surface of <br> the solar cell in terms of $A, R_{\odot}, d_{\odot}, T_{\mathrm{s}}, v$ and the fundamental constants $c, h, k_{\mathrm{B}}$.
[ "$n_{\\gamma}(\\nu)=A \\frac{R_{\\odot}^{2}}{d_{\\odot}^{2}} \\frac{2 \\pi}{c^{2}} \\nu^{2} e^ {\\frac{-h \\nu }{k_{\\mathrm{B}} T_{\\mathrm{s}}}}$" ]
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A4 Define $x_{\mathrm{g}}=h v_{\mathrm{g}} / k_{\mathrm{B}} T_{\mathrm{s}}$ where $E_{\mathrm{g}}=h v_{\mathrm{g}}$. Express the useful output power of the cell, $P_{\text {out }}$, in terms of $x_{\mathrm{g}}, A$, <br> $R_{\odot}, d_{\odot}, T_{\mathrm{s}}$ and the fundamental constants $c, h, k_{\mathrm{B}}$.
[ "$P_{\\text {out }}=\\frac{2 \\pi k_{\\mathrm{B}}^{4}}{c^{2} h^{3}} T_{\\mathrm{s}}^{4} A \\frac{R_{\\odot}^{2}}{d_{\\odot}^{2}} x_{\\mathrm{g}}(x_{\\mathrm{g}}^{2}+2 x_{\\mathrm{g}}+2) e^{-x_{\\mathrm{g}}}$" ]
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1,064
A5 Express the efficiency, $\eta$, of this solar cell in terms of $x_{\mathrm{g}}$.
[ "$\\eta=\\frac{x_{\\mathrm{g}}}{6}\\left(x_{\\mathrm{g}}^{2}+2 x_{\\mathrm{g}}+2\\right) e^{-x_{\\mathrm{g}}}$" ]
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1,065
A6 Make a qualitative sketch of $\eta$ versus $x_{\mathrm{g}}$. The values at $x_{\mathrm{g}}=0$ and $x_{\mathrm{g}} \rightarrow \infty$ should be clearly shown. What <br> is the slope of $\eta\left(x_{\mathrm{g}}\right)$ at $x_{\mathrm{g}}=0$ and $x_{\mathrm{g}} \rightarrow \infty$ ?
[ "$\\frac{1}{3}$ , 0" ]
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A7 Let $x_{0}$ be the value of $x_{\mathrm{g}}$ for which $\eta$ is maximum. Obtain the cubic equation that gives $x_{0}$. Estimate the <br> value of $x_{0}$ within an accuracy of \pm 0.25. Hence calculate $\eta\left(x_{0}\right)$.
[ "$x_0=2.27$ , $\\eta(2.27)=0.457$" ]
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1,067
A8 The band gap of pure silicon is $E_{\mathrm{g}}=1.11 \mathrm{eV}$. Calculate the efficiency, $\eta_{\mathrm{Si}}$, of a silicon solar cell using this <br> value.
[ "0.457" ]
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A9 Let us assume that the density of matter is uniform inside the Sun. Find the total gravitational potential energy, $\Omega$, of the Sun at present, in terms of $G, M_{\odot}$ and $R_{\odot}$.
[ "$\\Omega=-\\frac{3}{5} \\frac{G M_{\\odot}^{2}}{R_{\\odot}}$" ]
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A.10 Estimate the maximum possible time, $\tau_{\mathrm{KH}}$ (in years), for which the Sun could have been shining, according to the KH hypothesis. Assume that the luminosity of the Sun has been constant throughout this period.
[ "$1.88 \\times 10^{7}$" ]
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B1 Calculate the flux density, $\Phi_{v}$, of the number of neutrinos arriving at the Earth, in units of $\mathrm{m}^{-2} \mathrm{~s}^{-1}$. The energy released in the above reaction is $\Delta E=4.0 \times 10^{-12} \mathrm{~J}$. Assume that the energy radiated by the Sun is entirely due to this reaction.
[ "$6.8 \\times 10^{14}$" ]
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B2 In terms of $N_{1}$ and $N_{2}$, calculate what fraction, $f$, of $v_{\mathrm{e}}$ is converted to $v_{\mathrm{x}}$.
[ "$f=\\frac{6}{5}(1-\\frac{N_{2}}{N_{1}})$" ]
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B.3 Assume that an electron knocked out by a neutrino loses energy at a constant rate of $\alpha$ per unit time, while it travels through water. If this electron emits Cherenkov radiation for a time, $\Delta t$, determine the energy imparted to this electron ( $E_{\text {imparted }}$ ) by the neutrino, in terms of $\al...
[ "$E_{\\text {imparted }}=\\alpha \\Delta t+(\\frac{n}{\\sqrt{n^{2}-1}}-1) m_{\\mathrm{e}} c^{2}$" ]
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B.4 If $\Delta E_{r m s}=5.54 \times 10^{-17} \mathrm{~J}$, calculate the rms speed of the Be nuclei, $\mathrm{V}_{\mathrm{Be}}$, and hence estimate $T_{\mathrm{c}}$. (Hint: $\Delta E_{r m s}$ depends on the rms value of the component of velocity along the line of sight).
[ "$V_{\\mathrm{Be}}=2.01 \\times 10^{5}$ , $T_{\\mathrm{c}}=1.13 \\times 10^{7}$" ]
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A small puck of mass $m$ is carefully placed onto the inner surface of the thin hollow thin cylinder of mass $M$ and of radius $R$. Initially, the cylinder rests on the horizontal plane and the puck is located at the height $R$ above the plane as shown in the figure on the left. Find the interaction force $F$ between t...
[ "$F=3 m g(1+\\frac{m}{3 M})$" ]
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1 Find formula for the molar heat capacity of the gas in the bubble for such a process when the gas is heated so slowly that the bubble remains in a mechanical equilibrium and evaluate it; Hint: Laplace showed that there is pressure difference between inside and outside of a curved surface, caused by surface tension ...
[ "$33.2 $" ]
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2 Find formula for the frequency $\omega$ of the small radial oscillations of the bubble and evaluate it under the assumption that the heat capacity of the soap film is much greater than the heat capacity of the gas in the bubble. Assume that the thermal equilibrium inside the bubble is reached much faster than the per...
[ "$108 $" ]
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A1 Estimate $b$ and express it in terms of the diameter of the molecules $d$.
[ "$b=N_{A} d^{3}$" ]
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A1 Find $_{0}, a, b$ and express them in terms of $Z_{\text {ext }}$ and $r$.
[ "$n_{0}=0$ , $a=\\sqrt{\\frac{Z_{e x t}}{r}}$ , $b=\\sqrt{r Z_{e x t}}$" ]
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A2 Find the electron number density $n_{e}$ at equilibrium when both external ionizers are switched on simultaneously.
[ "$20.0 \\cdot 10^{10}$" ]
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A3 Express the electric current $I$ in the tube in terms of $U, \beta, L, S, Z_{\mathrm{ext}}, r$ and $e$ which is the elementary charge.
[ "$I=\\frac{e \\beta^{2} U^{2} S}{r L^{3}}(\\sqrt{1+\\frac{4 r Z_{e x t} L^{4}}{\\beta^{2} U^{2}}}-1)$" ]
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A4 Find the resistivity $\rho_{gas}$ of the gas at sufficiently small values of the voltage applied and express it in terms of $\beta,L,Z_{ext},r$ and $e$
[ "$\\frac{1}{2 e \\beta} \\sqrt{\\frac{r}{Z_{e x t}}}$" ]
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i. By adjusting the launching angle for a ball thrown with a fixed initial speed $v_{0}$ from the origin, targets can be hit within the region given by $$ z \leq z_{0}-k x^{2} $$ You can use this fact without proving it. Find the constants $z_{0}$ and $k$.
[ "$\\frac{v_{0}^{2}}{ 2 g}$ , $\\frac{g}{2 v_{0}^{2}}$" ]
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i. During much of the collapse, the gas is so transparent that any heat generated is immediately radiated away, i.e. the ball stays in thermodynamic equilibrium with its surroundings. What is the number of times, $n$, by which the pressure increases when the radius is halved to $r_{1}=0.5 r_{0}$ ? Assume that the gas d...
[ "$n=8$" ]
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ii. Estimate the time $t_{2}$ needed for the radius to shrink from $r_{0}$ to $r_{2}=0.95 r_{0}$. Neglect the change of the gravity field at the position of a falling gas particle.
[ "$\\sqrt{\\frac{0.1 r_{0}^{3}}{G m}}$" ]
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iii. Assuming that the pressure remains negligible, find the time $t_{r \rightarrow 0}$ needed for the ball to collapse from $r_{0}$ down to a much smaller radius, using Kepler's Laws.
[ "$t_{r \\rightarrow 0}=\\pi \\sqrt{\\frac{r_{0}^{3}}{8 G m}}$" ]
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iv. At some radius $r_{3} \ll r_{0}$, the gas becomes dense enough to be opaque to the heat radiation. Calculate the amount of heat $Q$ radiated away during the collapse from the radius $r_{0}$ down to $r_{3}$.
[ "$Q=\\frac{3 m R T_{0}}{\\mu} \\ln \\frac{r_{0}}{r_{3}}$" ]
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v. For radii smaller than $r_{3}$ you may neglect heat loss due to radiation. Determine how the temperature $T$ of the ball depends on its radius for $r<r_{3}$.
[ "$T(r)=T_{0}\\left(\\frac{r_{3}}{r}\\right)^{3 \\gamma-3}$" ]
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vi. Eventually we cannot neglect the effect of the pressure on the dynamics of the gas and the collapse stops at $r=r_{4}$ (with $r_{4} \ll r_{3}$ ). However, the radiation loss can still be neglected and the temperature is not yet high enough to ignite nuclear fusion. The pressure of such a protostar is not uniform an...
[ "$r_{3}\\left(\\frac{R T_{0} r_{3}}{\\mu m G}\\right)^{\\frac{1}{3 \\gamma-4}}$,$T_{0}\\left(\\frac{R T_{0} r_{3}}{\\mu m G}\\right)^{\\frac{3 \\gamma-3}{4-3 \\gamma}}$" ]
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A.1 Find the exact expression for the final velocity $v$ of the protons as a function of the accelerating voltage $V$, and physical constants.
[ "$v=c \\cdot \\sqrt{1-(\\frac{m_{p} \\cdot c^{2}}{m_{p} \\cdot c^{2}+V \\cdot e})^{2}}$" ]
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A.2 For particles with high energy and low mass the relative deviation $\Delta=(c-v) / c$ of the final velocity $v$ from the speed of light is very small. Find a first order approximation for $\Delta$ and calculate $\Delta$ for electrons with an energy of $60.0 \mathrm{GeVusing}$ the accelerating voltage $V$ and physic...
[ "$\\frac{1}{2}\\left(\\frac{m_{e} \\cdot c^{2}}{m_{e} \\cdot c^{2}+V \\cdot e}\\right)^{2}$, $\\Delta=3.63 \\cdot 10^{-11}$", "$\\frac{1}{2}\\left(\\frac{m_{e} \\cdot c^{2}}{V \\cdot e}\\right)^{2}$, $\\Delta=3.63 \\cdot 10^{-11}$" ]
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A.3 Derive an expression for the uniform magnetic flux density $B$ necessary to keep the proton beam on a circular track. The expression should only contain the energy of the protons $E$, the circumference $L$, fundamental constants and numbers. You may use suitable approximations if their effect is smaller than precis...
[ "$B=\\frac{2 \\pi \\cdot E}{e \\cdot c \\cdot L}$ , $B=5.50$" ]
[ 0.0208740234375, -0.0103759765625, 0.02294921875, 0.0186767578125, -0.0035400390625, 0.0076904296875, 0.001922607421875, -0.017333984375, 0.007049560546875, 0.00665283203125, -0.0341796875, 0.02734375, -0.0250244140625, -0.03369140625, 0.01507568359375, -0.00762939453125, -0.02709960...
1,216
A.4 Use dimensional analysis to find an expression for the radiated power $P_{\text {rad }}$.
[ "$P_{\\text {rad }}=a^{\\alpha} \\cdot q^{\\beta} \\cdot c^{\\gamma} \\cdot \\epsilon_{0}^{\\delta}$" ]
[ -0.01080322265625, -0.0029296875, -0.00726318359375, 0.00885009765625, 0.00872802734375, 0.0167236328125, -0.008056640625, -0.0184326171875, -0.0027008056640625, -0.0005645751953125, -0.051025390625, 0.01171875, -0.0194091796875, -0.01080322265625, 0.01409912109375, -0.0040283203125, ...
1,217
A.5 Calculate $P_{\text {tot }}$, the total radiated power of the LHC, for a proton energy of E= 7.00 $\mathrm{TeV}$ (Note table 1). You may use suitable approximations.
[ "5.13" ]
[ 0.016845703125, 0.01141357421875, 0.00872802734375, 0.01336669921875, 0.027587890625, 0.01263427734375, -0.01611328125, -0.0027923583984375, -0.0089111328125, 0.015625, -0.026123046875, 0.0263671875, -0.0166015625, -0.00732421875, 0.00323486328125, -0.005157470703125, -0.033203125, ...
1,223
3.3 Calculate the energy of the ion $\mathrm{A}^{(\mathrm{Z}-1)+}$ using the value of the lowest energy. The calculation should be approximated based on the following principles: 3.3.A The potential energy of the ion should be expressed in terms of the average value of $\frac{1}{r}$. (ie. $\frac{1}{\mathrm{r}_{\mat...
[ "$-E_{R} \\cdot Z^{2}$" ]
[ 0.005157470703125, 0.0045166015625, 0.0264892578125, 0.0019989013671875, 0.00848388671875, 0.03515625, 0.0238037109375, -0.0024261474609375, 0.00030517578125, 0.021728515625, -0.060546875, -0.0009765625, -0.021484375, -0.025390625, 0.00665283203125, 0.01483154296875, -0.0019683837890...