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Given a nitrogen molecule in a room at room temperature and atmospheric pressure, and using the equipartition theorem with 3 translational degrees of freedom, what is the average speed of the molecule? | v ≈ √(3 k_B T / 28 m_p) ≈ 515 m/s |
Given a nitrogen molecule with radius r₀, in a room at temperature T and pressure P, what is the mean free path λ assuming ideal gas behavior? | λ ≈ k_B T / (4π r₀² P) ≈ 2.90 × 10⁻⁷ m |
A satellite of mass $m = \SI{500}{\kg}$ is in a circular orbit at an altitude $h = \SI{150}{\km}$ above the Earth's surface. As a result of air friction, the satellite's orbit degrades. Protected by a heat shield, the satellite eventually impacts with a velocity of $\SI{2}{\km\per\s}$. How much energy (in Joules) was r... | The lost energy $E_\mathrm{loss}$ is given by $E_\mathrm{loss} = \frac{GM_E m}{R_E(R_E + h)}(R_E - 2h) - \frac 12 m{v_f}²$, which evaluates to $\SI{2.816e10}{\J} = \SI{28.16}{\giga\J}$. |
What is the velocity of recoil of an ${}^{57}\mathrm{Fe}$ nucleus that emits a \SI{100}{\keV} photon? | The recoil velocity is $v_{Fe} = \frac{E_γ}{57m_p c} = \SI{561.4}{\m\per\s}$ or $v_{Fe} = \SI{1.87e-6}{c}$. |
A resistance $R$ and an inductance $L$ are connected in series, and an alternating voltage $V₀\cos ωt$ is impressed across the combination. The resulting steady state voltage across the resistance can be written as $V_R \cos(ωt + β)$. Find $V_R$ and $β$. | $V_R = V₀ \frac{R\sqrt{R² + ω²L²}}{R² + ω²L²}$ and $β = \arctan(-\frac{ωL}{R})$. |
Find the magnetic flux through a square loop of side $a$ due to current $I$ in a long straight wire, where the wire is coplanar with the loop and runs parallel to the loop's closest side, at a distance $b$ away. | The magnetic flux is given by $Φ = \frac{μ₀Ia}{2π}\ln(\frac{a+b}{a})$. |
By actually evaluating the integral, show that $\int_0^∞ \frac{\cos x}{1 + x²}\,dx = \frac{π}{2e}$. | The integral evaluates to $\frac{π}{2e}$. |
The drag force on a very high speed object of area $A$, passing through a gas of density $ρ$ at a velocity $v$ is expected to be of the form $\text{Force} \sim A^r ρ^s v^t$. Determine the value of the exponents $r$, $s$, and $t$. | The exponents are $r = 2$, $s = 1$, and $t = 2$. |
For waves in shallow water, the relation between frequence $ν$ and wavelength $λ$ is $ν = (\frac{2πT}{ρλ³})^{1/2}$, where $ρ$ and $T$ are the density and surface tension of water. What is the group velocity of these waves? | The group velocity is $v_g = \frac 32 \sqrt{\frac{kT}{ρ}} = \frac 32 \sqrt{\frac{2πT}{ρλ}}$. |
Find the eigenvalues and corresponding eigenvectors of the matrix $M = \begin{bmatrix} 1 & 0 & -i \ 0 & 2 & 0 \ i & 0 & -1 \end{bmatrix}$. | The eigenvalues are $λ = \{ -\sqrt 2, \sqrt 2, 2 \}$ with corresponding eigenvectors $v₁ = \begin{bmatrix} 1 \ 0 \ -i(1+\sqrt 2) \end{bmatrix}$, $v₂ = \begin{bmatrix} 1 \ 0 \ -i(1-\sqrt 2) \end{bmatrix}$, and $v₃ = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$. |
Suppose the electron were to have spin $\frac 32$ instead of spin $\frac 12$. What would then be the atomic numbers $Z$ of the three lowest-mass Nobel gases? | The atomic numbers of the three lowest-mass Nobel gases would be $Z = \{4, 20, 56 \}$. |
Given a cylindrical bucket filled with water to a height of \SI{150}{\cm}, at what height from the ground should a hole be punched in the side of the bucket to maximize the distance the water stream travels, and what is that maximum distance? | The height that maximizes the distance is \SI{75}{\cm}, and that distance is \SI{150}{\cm}. |
Ice on a pond is \SI{10}{\cm} thick and the water temperature just below the ice is \SI{0}{\celsius}. If the air temperature is \SI{-20}{\celsius}, by how much will the ice thickness increase in 1 hour? Assuming that the air temperature stays the same over a long period, how will the ice thickness increase with time? | The ice thickness will increase by \SI{0.0501}{\cm} in 1 hour. |
A specimen of wood containing \SI{3}{\g} of carbon has a measured count rate of \SI{12.8(1)}{\minute^{-1}} with a counter efficiency of \SI{18}{\percent}. Given that \SI{1}{\g} of living wood contains \SI{16.1}{\minute^{-1}} radioactive carbon-14 decays, what is the age of this specimen, and its uncertainty? | The age of the specimen is \SI{4250(160)}{\year}. |
A \SI{1000}{\kg} automobile has ground clearance of \SI{18}{\cm}, but when loaded with an extra \SI{500}{\kg} it only clears the ground by \SI{12}{\cm}. The car's shock absorbers are ineffective. At what speed (in miles per hour) will the car bounce in resonance when it travels along a smooth road containing transverse... | \SI{39.42}{mph} |
Suppose a particle of mass $m$ moves in a 1-dimensional square potential well of width $L$ and depth $V$. What is the minimum depth of the well such that the particle will have two bound states? | $V = \frac{π²ℏ²}{2mL²}$ |
The cross-section for collisions between helium atoms is about \SI{e-16}{\cm\squared}. Estimate the mean free path of helium atoms in helium gas at atmospheric pressure and temperature. | \SI{4.06}{\micro\m} |
A single closed loop of chain with mass $m$ and length $L$ rests horizontally on a smooth frictionless cone with half-angle $α$. What is the tension in the chain? | $T = \frac{Mg}{2π} \cot α$ |
A laser beam (photon energy \SI{1}{\eV}) collides head-on with a \SI{50}{\GeV} ultra-relativistic electron beam. What is the energy of the photons reflected backwards in the collision? | \SI{4.273}{\eV} |
In an LR circuit, $\mathcal E = \SI{100}{\V}$, $R₁ = \SI{5}{\ohm}$, $R₂ = \SI{10}{\ohm}$, $R₃ = \SI{15}{\ohm}$, and $L = \SI{1.0}{\henry}$. Find the values of the currents $I₁$ and $I₂$ immediately after the switch $S$ is closed. | $I₁(0) = \SI{6.66}{\A}$ and $I₂(0) = \SI{6.66}{\A}$ |
In an LR circuit, $\mathcal E = \SI{100}{\V}$, $R₁ = \SI{5}{\ohm}$, $R₂ = \SI{10}{\ohm}$, $R₃ = \SI{15}{\ohm}$, and $L = \SI{1.0}{\henry}$. Find the values of the currents $I₁$ and $I₂$ a long time later. | $I₁(∞) = \SI{9.09}{\A}$ and $I₂(∞) = \SI{5.54}{\A}$ |
In an LR circuit, $\mathcal E = \SI{100}{\V}$, $R₁ = \SI{5}{\ohm}$, $R₂ = \SI{10}{\ohm}$, $R₃ = \SI{15}{\ohm}$, and $L = \SI{1.0}{\henry}$. Find the values of the currents $I₁$ and $I₂$ immediately after switch $S$ is opened again. | $I₁(0) = 0$ and $I₂(0) = \SI{-3.64}{\A}$ |
In an LR circuit, $\mathcal E = \SI{100}{\V}$, $R₁ = \SI{5}{\ohm}$, $R₂ = \SI{10}{\ohm}$, $R₃ = \SI{15}{\ohm}$, and $L = \SI{1.0}{\henry}$. How long must you wait, after the switch is opened, before $I₂$ falls by a factor of $e$? | \SI{0.04}{\s} |
A particle is confined within a cubical box with sides of length $L$ and is initially in the ground state. If the length of one side of the box (along the $x$-direction) is abruptly increased to a length $2L$, what is the probability that the particle remains in the ground state? | $\mathscr P ≈ 0.60$ |
The frequency $f$ of a deep water gravity wave is given by $f =\sqrt{\frac{1}{2π}} ρ^a g^b λ^c$, where $ρ$, $g$, and $λ$ are the water density, gravitational acceleration, and wavelength of the wave, respectively. What are the values of the exponents $a$, $b$, and $c$? | $a = 0$, $b = \frac 12$, $c = \frac 12$ |
The frequency $f$ of a deep water gravity wave is given by $f =\sqrt{\frac{1}{2π}} ρ^a g^b λ^c$, where $ρ$, $g$, and $λ$ are the water density, gravitational acceleration, and wavelength of the wave, respectively. What is the ratio of the wave group velocity to phase velocity? | $\frac{v_g}{v_p} = \frac 12$ |
A zipper has $N$ links, each with a closed state (energy 0) and an open state (energy $ε$). The zipper can only unzip from the left, and link $s$ can only open if links $1, 2, ..., s-1$ are already open. Find an explicit expression for the partition function. | Z = \frac{1 - e^{-(N+1)ε/k_BT}}{1 - e^{-ε/k_BT}} |
An electric bulb is rated at \SI{100}{\W} when used with a DC voltage of \SI{110}{\V}. What is the total power dissipated if this voltage is applied to two such bulbs connected in series, assuming each bulb dissipates heat by radiation from its filament similar to a black body and that the resistance of the filament is... | The total power dissipated is $\frac{\SI{100}{\W}}{2^{4/5}} = \SI{57.435}{\W}$ |
If an impulse is delivered to the end of a uniform rod of length $ℓ$, lying on a frictionless plane, and the impulse is in the plane of the table and perpendicular to the rod, how far will the rod travel while making one revolution? | $\vec x = \frac{πℓ}{3} \hat J$, where $\hat J$ is the direction of the applied impulse. |
A time-independent magnetic field is given by $\vec B = 2bxy \,\hat ı + ay² \,\hat ȷ$. What is the relationship between the constants $a$ and $b$? | $b = -a$ |
A time-independent magnetic field is given by $\vec B = 2bxy \,\hat ı + ay² \,\hat ȷ$. Determine the steady current density $J$ that gives rise to this field. | $\vec J = \frac{2a}{μ₀} x \,\hat k$ |
A set of four point charges $q₁$, $q₂$, $q₃$, and $q₄$ are arranged collinearly along the $z$-axis at $z₁ = 0$, $z₂ = a$, $z₃ = 2a$, $z₄ = 4a$, respectively. Given $q₂ = +2$ and $q₃ = +4$, and the resulting electric field at a distant point $\vec r$ ($r ≫ a$) decays faster than $1/r³$, determine the values of $q₁$ and ... | $q₁ = -\frac 72$ and $q₄ = -\frac 52$ |
The Lyman-α transition in atomic hydrogen has a wavelength $λ = \SI{121.5}{\nm}$, and a transition rate of \SI{0.6e9}{\s^{-1}}. Estimate the minimum value of $Δλ/λ$. | $\frac{Δλ}{λ} ≈ \num{1.935e-8} ≈ \text{1 part in 50 million}$ |
A rock is found to contain \SI{4.20}{\mg} of ${}^{238}U$ and \SI{2.00}{\mg} of ${}^{206}Pb$. Assume the rock contained no lead at the time of its formation, and the half-life of ${}^{238}U$ is \SI{4.47e9}{\year}. Find the age of the rock. | $t = \SI{2.83e9}{\year}$ |
The applied AC voltage in the circuit is given by $V(t) = V₀ \sin ωt$, with a frequency fixed at $ω = 1/(LC)^{1/2}$. Determine the steady state amplitude and phase of the current through the resistor $R$. | The current's amplitude is $V₀\sqrt{C/L}$ and has a phase of $-π/2$ with respect to the voltage. |
Given an elevator with acceleration $a$ for a time equal to the time with acceleration $-a$, does the elevator operator work more than, exactly, or less than 8 hours, and why? | The elevator operator actually spends more than 8 hours in the elevator during his shift. |
For a classical particle subject to an attractive central force proportional to $r^α$, what is required of $α$ in order for the particle to have a stable circular orbit? | For a stable circular orbit, $α > -3$. |
A neutral conductor A with cavities B, C, and D contains positive charges $q_B$ and $q_C$ introduced at the centers of B and C, respectively. What is the amount and distribution of the induced charges on the surfaces of A, B, C, and D? | Cavity B has induced charge $-q_B$, cavity C has induced charge $-q_C$, cavity D has no induced charge, and surface A has total charge $q_B + q_C$. |
If another positive charge $q_E$ is introduced at a distance $r > R$ from the center of A, how does the distribution of induced charges on the surfaces of A, B, C, and D change? | The surfaces B, C, and D remain unaffected, while the distribution on surface A shifts with negative charge concentration greatest on the side nearest to $q_E$ and positive concentration on the opposite side. |
What is the amount of the induced charges on the surfaces of A, B, C, and D when another positive charge $q_E$ is introduced at a distance $r > R$ from the center of A? | The total charges on surfaces A, B, C, and D remain unchanged. |
The dielectric strength of air is \SI{3e6}{\V\per\m}. What is the maximum intensity in \si{\W\per\m^2} for a monochromatic laser that can be used in the laboratory? | The maximum power of a laser usable in the lab is \SI{1.19e10}{\W\per\m\squared}. |
What is the minimum energy of the projectile proton required to induce the reaction $p + p \rightarrow p + p + p + \bar p$ if the target proton is at rest? | The minimum energy of the projectile proton is approximately \SI{6.567}{\GeV\per c\squared}. |
Assuming hydrostatic equilibrium and adiabatic conditions, find an expression for the pressure $P$ of the atmosphere as a function of the height $z$. | $P(z) = P₀ e^{z/ξ}$, where $ξ = \frac{k_B T}{mg}$. |
Given a system with mass $m₁$ moving along a horizontal rod and a massless string of length $ℓ$ connecting $m₁$ to mass $m₂$ undergoing pendulum motion, find the Lagrangian of the system. | $\sL = \frac{1}{2} (m₁ + m₂)\dot x² + \frac{1}{2} m₂ \left( ℓ²\dot φ² + 2ℓ\dot φ\dot x\cos φ \right) + m₂gℓ\cos φ$ and approximately $\sL ≈ \frac{1}{2} (m₁ + m₂)\dot x² + \frac{1}{2} m₂ \left( ℓ²\dot φ² + 2ℓ\dot φ\dot x \right) + m₂gℓ - \frac{1}{2}m₂gℓφ²$ |
Given the Lagrangian of the system, derive the equations of motion and the corresponding conservation laws. | The equations of motion are: $\ddot x + \frac{m₂}{m₁+m₂} ℓ \ddot φ = 0$ and $\ddot φ + \frac{1}{ℓ}\ddot x + \frac{g}{ℓ}φ = 0$ |
Assuming initial conditions $x(0)=x₀$, $\dot x(0)=0$, $φ(0)=φ₀$ ($|φ₀| ≪ 1$), and $\dot φ(0)=0$, find $x(t)$ and $φ(t)$ for $t > 0$. | $φ(t) = φ₀\cos(ωt)$ where $ω² = \frac{g}{ℓ}\frac{m₁+m₂}{m₁}$ and $x(t) = x₀ - \frac{g}{ω²}\frac{m₂}{m₁}\cos(ωt)$ |
A particle in a one-dimensional infinite square well is initially in a state Ψ(x,0). What is the time T after which the particle will always return to that state, where a is the width of the well? | T = 4ma²/πħ |
A photon collides with a stationary electron. If the photon scatters at an angle θ, what is the resulting wavelength λ' in terms of the original wavelength λ and the electron mass m? | λ' = λ + (h/mc) (1 - cos θ) |
For a many particle system of weakly interacting particles, will quantum effects be more important for (a) high densities or low densities and (b) high temperatures or low temperatures? Explain your answers in terms of the de Broglie wavelength $\lambda$ defined as $\lambda^2 \equiv \frac{h^2}{3mk_B T}$ where $m$ is th... | High density — The de Broglie wavelength gives a 'size' of the particle, and in the high density limit, the wavefunctions overlap significantly so quantum effects and interactions are critical to the behavior of the system. Low temperature — Since $\lambda^2 \propto T^{-1}$, as $T \to 0$, $\lambda$ increases so that ag... |
The ground state energy of Helium is -79 eV. What is its ionization energy, which is the energy required to remove just one electron? | The ground state energy of the singly ionized Helium atom (He⁺) is approximately -24.6 eV (using a Hydrogen-like model with $Z=2$). The ionization energy is the energy difference between the neutral Helium atom and the singly ionized Helium atom. Therefore, the ionization energy is $-79\,\text{eV} - (-24.6\,\text{eV}) ... |
The force per unit area ($F/A$) between two neutral conducting plates due to polarization fluctuations of the vacuum (namely, the Casimir force) is a function of $h$ (Planck's constant), $c$ (speed of light), and $z$ (distance between the plates) only. Using only dimensional analysis, obtain $F/A$ as a function of $h$,... | Dimensional analysis gives $F/A \sim \frac{hc}{z^4}$. The correct full expression is $F/A \approx -\frac{\hbar c}{240\pi z^4}$, but dimensional analysis cannot determine the numerical prefactor. |
In the circuit diagram, initially the two identical capacitors with capacitance $C$ are uncharged. The battery is an ideal EMF and supplies a voltage $V$. (a) At first Switch A is closed and Switch B is kept open. What is the final stored energy on capacitor $C_a$? (b) Switch A is opened and afterwards Switch B is clos... | (a) Stored energy on $C_a$ is $\frac{1}{2}CV^2$. (b) After redistribution, each capacitor stores $\frac{1}{8}CV^2$, so total energy is $\frac{1}{4}CV^2$. |
A planet of mass $m$ moves around the sun, mass $M$, in an elliptical orbit with minimum and maximum distances of $r_1$ and $r_2$, respectively. Find the angular momentum of the planet relative to the center of the sun in terms of these quantities and the gravitational constant $G$. | $L = \sqrt{\frac{2GMm^2 r_1 r_2}{r_1 + r_2}}$ |
A particle moves in a circular orbit under the influence of a central force that varies as the $n$-th power of the distance. Show that this motion is unstable if $n < -3$. | The orbit is unstable if $n < -3$ because the second derivative of the potential energy becomes negative, indicating no restoring force. |
A classical, ideal, monatomic gas of $N$ particles is reversibly compressed isentropically, i.e.~with the entropy kept constant, from an initial temperature $T_0$ and pressure $P$ to a pressure $2P$. Find (a) the work done on the system, and (b) the net change in entropy of the system and its surroundings. | (a) $W = \frac{3}{2}Nk_B T_0 (2^{2/5} - 1)$ using $\gamma = 5/3$. (b) $\Delta S = 0$ since the process is isentropic and reversible. |
For an ideal Fermi gas of $N$ neutral spin-1/2 particles in a volume $V$ at $T = 0$, calculate the following: (1) The chemical potential (2) The average energy per particle (3) The pressure | (1) $\mu = \frac{\hbar^2}{2m} (3\pi^2 N/V)^{2/3}$ (2) $\langle \varepsilon \rangle = \frac{3}{5}\mu$ (3) $P = \frac{2}{5}(N/V)\mu$ |
A piece of p-doped silicon has a carrier density $n = 10^{15}\,\text{cm}^{-3}$ and dimensions of $\Delta x = 10\,\text{mm}$, $\Delta y = 2\,\text{mm}$, and $\Delta z = 1\,\text{mm}$. A magnetic field of $B_z = 1\,\text{T}$ is applied in the $z$-direction and a current $I_x = 1\,\text{A}$ flows in the $x$-direction, and... | (1) $j_x = ne v_x$ (2) $q v_x B_z = q V_y / \Delta y \Rightarrow V_y = v_x B_z \Delta y$ (3) $V_y = 12.48\,\text{V}$ |
Consider $N$ non-interacting particles with magnetic moment $\vec μ$ in a magnetic field $\vec B$. Each particle can be oriented only parallel or anti-parallel to the field. What is the partition function $Z$ for $N$ particles? | Z = 2^N cosh^N (μB/kT) |
Consider $N$ non-interacting particles with magnetic moment $\vec μ$ in a magnetic field $\vec B$. Each particle can be oriented only parallel or anti-parallel to the field. What is the total energy $U$? | U = -NμB tanh (μB/kT) |
Consider $N$ non-interacting particles with magnetic moment $\vec μ$ in a magnetic field $\vec B$. Each particle can be oriented only parallel or anti-parallel to the field. What is the magnetization $\expect{M}$? | $\expect{M} = -Nμ \tanh(\frac{μB}{kT})$ |
Consider $N$ non-interacting particles with magnetic moment $\vec μ$ in a magnetic field $\vec B$. The magnetic moment can rotate freely. What is the partition function $Z$ for $N$ particles? | Z = (2kT/μB)^N sinh^N(μB/kT) |
Consider $N$ non-interacting particles with magnetic moment $\vec μ$ in a magnetic field $\vec B$. The magnetic moment can rotate freely. What is the total energy $U$? | U = NkT - NμB coth (μB/kT) |
Consider $N$ non-interacting particles with magnetic moment $\vec μ$ in a magnetic field $\vec B$. The magnetic moment can rotate freely. What is the magnetization $\expect{M}$? | $\expect{M} = \frac{NkT}{B}$ |
For both cases (parallel/anti-parallel and free rotation), show that the total magnetization $\vec M$ can be written as a derivative of the partition function. | $\expect{M} = \frac{∂\ln Z}{∂B}$ |
Assuming the electron to be a classical particle, a sphere of radius \SI{e-15}{\m} and of a uniform mass density with an intrinsic angular momentum of order $ℏ$, compute the speed of rotation at the electron's equator. | v = \SI{2.89e11}{\m\per\s} = \SI{965}{c} |
Two electrons can be considered distinguishable if they are well separated in space from each other, that is, their single particle wavefunctions are non-overlapping. In that case, for every possible $x₁$ value, either $ψ_α(x₁)$ and $ψ_β(x₁)$ is zero. Show that for non-overlapping wavefunctions as defined above, the pr... | ψ^*ψ = \frac 12 \Big( |ψ_α(x₁)|² |ψ_β(x₂)|² + |ψ_α(x₂)|² |ψ_β(x₁)|² \Big) |
A particle of mass $m$ moves in a circular orbit of radius $r$ in a hypothetical atom where the force on the particle is in the form of a generalized Hooke's law: $F = -Cr$ directed towards the center of the atom, where $C$ is the `spring constant'. Assuming that Bohr's postulates for the atom apply in this case, in pa... | r = (\frac{ℏ²}{Cm})^{1/4} \sqrt{n} |
A particle of mass $m$ moves in a circular orbit of radius $r$ in a hypothetical atom where the force on the particle is in the form of a generalized Hooke's law: $F = -Cr$ directed towards the center of the atom, where $C$ is the `spring constant'. Assuming that Bohr's postulates for the atom apply in this case, in pa... | E_n = ℏn\sqrt{\frac{C}{m}} |
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