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<image>A soap film ( $n=4 / 3$ ) of thickness $d$ is illuminated at normal incidence by light of wavelength 500 nm . Calculate the approximate intensities of interference maxima relative to the incident intensity as $d$ is varied, when viewed in reflected light.
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0.08
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<image>As shown in Fig. 3.26, the switch has been in position A for a long time. At $t=0$ it is suddenly moved to position B. Immediately after contact with B: What is the time rate of change of the current through $R$ ?
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$-10^{4} \mathrm{~A}/\mathrm{s}$
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<image>At low temperatures, a mixture of ${ }^{3} \text{He}$ and ${ }^{4} \text{He}$ atoms form a liquid which separates into two phases: a concentrated phase (nearly pure ${ }^{3} \text{He}$), and a dilute phase (roughly $6.5 \text{\textperthousand} { }^{3} \text{He}$ for $T \text{\textless}= 0.1 \text{ K}$). The lighter ${ }^{3} \text{He}$ floats on top of the dilute phase, and ${ }^{3} \text{He}$ atoms can cross the phase boundary (see Fig. 2.23). The superfluid ${ }^{4} \text{He}$ has negligible excitation, and the thermodynamics of the dilute phase can be represented as an ideal degenerate Fermi gas of particles with density $n_{d}$ and effective mass $m^{*}\text{(}m^{*}\text{ is larger than }m_{3}\text{, the mass of the bare }{ }^{3} \text{He atom, due to the presence of the liquid }{ }^{4} \text{He, actually }m^{*}=2.4 m_{3}\text{)}$. We can crudely represent the concentrated phase by an ideal degenerate Fermi gas of density $n_{c}$ and particle mass $m_{3}$. How much heat is required to warm each phase from $T=0 \text{ K}$ to $T$?
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$\frac{\alpha_{c} T^{2}}{2 T_{Fc}}, \frac{\alpha_{d} T^{2}}{2 T_{Fd}}$
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<image>The current-voltage characteristic of the output terminals A, B (Fig. 3.3) is the same as that of a battery of emf $\varepsilon_{0}$ and internal resistance $r$. Find the short-circuit current provided by the battery.
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$0.625 \mathrm{~A}$
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<image>A block of mass $M$ is rigidly connected to a massless circular track of radius $a$ on a frictionless horizontal table as shown in Fig. 2.7. A particle of mass $m$ is confined to move without friction on the circular track which is vertical. Set up the Lagrangian, using $\theta$ as one coordinate.
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$\frac{1}{2} M \dot{x}^{2}+\frac{1}{2} m\left[\dot{x}^{2}+a^{2} \dot{\theta}^{2}+2a\dot{x}\dot{\theta}\cos\theta\right]+mga\cos\theta$
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<image>A block of mass $M$ is rigidly connected to a massless circular track of radius $a$ on a frictionless horizontal table as shown in Fig. 2.7. A particle of mass $m$ is confined to move without friction on the circular track which is vertical. In the limit of small angles, solve that equations of motion for $\theta$ as a function of time.
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$\theta=A\sin(\omega t)+B\cos(\omega t)$, with $\omega=\sqrt{\frac{(M+m)g}{Ma}}$
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<image>A useful optical grating configuration is to diffract light back along itself. If there are $N$ grating lines per unit length, what wavelength(s) is (are) diffracted back at the incident angle $\theta$, where $\theta$ is the angle between the normal to the grating and the incident direction?
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$\lambda = 2d \sin\left( \frac{\theta}{m} \right)$
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<image>Static charges are distributed along the $x$-axis in the interval $-a \le x' \le a$. The charge density is $\rho(x')$ for $|x'| \le a$ and $0$ for $|x'| > a$. Derive a multipole expansion for the potential valid for $x>a$.
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$\Phi(x)=\frac{1}{4\pi\varepsilon_{0}}\left[\frac{1}{x}\int_{-a}^{a}\rho(x')\,dx'+\frac{1}{x^2}\int_{-a}^{a}x'\rho(x')\,dx'+\frac{1}{x^3}\int_{-a}^{a}x'^2\rho(x')\,dx'+\ldots\right]$
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<image>A horizontal ray of light passes through a prism of index 1.50 and apex angle $4^{\\circ}$ and then strikes a vertical mirror, as shown in the figure. Through what angle must the mirror be rotated if after reflection the ray is to be horisontal?
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$1^{\circ}$
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<image>A simple pendulum is attached to a support which is driven horizontally with time as shown in Fig. 2.25. Set up the Lagrangian for the system in terms of the generalized coordinates $\theta$ and $y$, where $\theta$ is the angular displacement from equilibrium and $y(t)$ is the horizontal position of the pendulum support. (Wisconsin)
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$L=\frac{m}{2}(\dot{y}_{s}^{2}+l^{2}\dot{\theta}^{2}+2l\dot{y}_{s}\dot{\theta}\cos\theta)+mgl\cos\theta$
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<image>As in Fig. $2.39$, what is the direction of the current in the resistor $r$ (from $A$ to $B$ or from $B$ to $A$) when the following operations are performed? In each case give a brief explanation of your reasoning? The switch $S$ is closed.
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from $B$ to $A$
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<image>The figure below shows an apparatus for the determination of $C_{p} / C_{v}$ for a gas, according to the method of Clement and Desormes. A bottle $G$, of reasonable capacity (say a few litres), is fitted with a tap $H$, and a manometer $M$. The difference in pressure between the inside and the outside can thus be determined by observation of the difference $h$ in heights of the two columns in the manometer. The bottle is filled with the gas to be investigated, at a very slight excess pressure over the outside atmospheric pressure. The bottle is left in peace (with the tap closed) until the temperature of the gas in the bottle is the same as the outside temperature in the room. Let the reading of the manometer be $h_{i}$. The tap $H$ is then opened for a very short time, just sufficient for the internal pressure to become equal to the atmospheric pressure (in which case the manometer reads $h=0$ ). With the tap closed the bottle is left in peace for a while, until the inside temperature has become equal to the outside temperature. Let the final reading of the manometer be $h$. From the values of $h_{i}$ and $h_{f}$ it is possible to find $C_{p} / C_{v}$. Suppose that the gas in question is oxygen. What is your theoretical prediction for $C_{p} / C_{v}$ at $20^{\text{C}}$, within the framework of statistical mechanics?
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$\frac{7}{5}$
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<image>A beam of neutrons (mass $m$ ) traveling with nonrelativistic speed $v$ impinges on the system shown in Fig. 1.7, with beam-splitting mirrors at corners $B$ and $D$, mirrors at $A$ and $C$, and a neutron detector at $E$. The corners all make right angles, and neither the mirrors nor the beam-splitters affect the neutron spin. The beams separated at $B$ rejoin coherently at $D$, and the detector $E$ reports the neutron intensity $I$. In this part of the problem, assume the system to be in a vertical plane (so gravity points down parallel to AB and DC ). Given that detector intensity was $I_{0}$ with the system in a horizontal plane, derive an expression for the intensity $I_{g}$ for the vertical configuration.
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$I_{0} \cos ^{2}\left(\frac{m g H L}{2 \hbar v}\right)$
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<image>A long solenoid of radius $b$ and length $l$ is wound so that the axial magnetic field is $$\mathbf{B}=\begin{cases} B_{0}\,\mathbf{e}_{z}, & r<b,\\ 0, & r>b, \end{cases}$$ A particle of charge $q$ is emitted with velocity $v$ perpendicular to a central rod of radius $a$ (see Fig. 2.59). The electric force on the particle is given by $qE=f(r)\,\mathbf{e}_{r}$, where $\mathbf{e}_{r}\cdot\mathbf{e}_{z}=0$. We assume $\mathbf{v}$ is sufficiently large so that the particle passes out of the solenoid and does so without hitting the solenoid. If the electric field inside the solenoid is zero before the particle leaves the rod and after the particle has gone far away, it becomes $$\mathbf{E}=\begin{cases} -\frac{q}{2\pi\varepsilon_{0}lr}\,\mathbf{e}_{r}, & r>a,\\ 0, & r<a, \end{cases}$$ calculate the electromagnetic field angular momentum and discuss the final state of the solenoid if the solenoid can rotate freely about its axis. Neglect end effects.
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$\mathbf{J}_{EM}=\frac{qB_{0}(b^{2}-a^{2})}{2}\,\mathbf{e}_{z}$ and $\mathbf{J}_{S}=\frac{qB_{0}}{2}(b-a)^{2}\,\mathbf{e}_{z}$
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<image>Three point particles, two of mass $m$ and one of mass $M$, are constrained to lie on a horizontal circle of radius $r$. They are mutually connected by springs, each of constant $K$, that follow the arc of the circle and that are of equal length when the system is at rest as shown in Fig. 2.48. Assuming motion that stretches the springs only by a small amount from the equilibrium length $(2 \pi r / 3)$, find the frequency of each mode.
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$\omega_{1}=0,\quad \omega_{2}=\sqrt{\frac{3K}{m}},\quad \omega_{3}=\sqrt{\frac{(2m+M)K}{mM}}$
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<image>Four identical coherent monochromatic wave sources A, B, C, D, as shown in Fig. 4.2 produce waves of the same wavelength $\lambda$. Two receivers $R_{1}$ and $R_{2}$ are at great (but equal) distances from $B$. Which receiver, if any, picks up the greater signal if source $B$ is turned off?
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Both receivers pick up signals of the same intensity
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<image>Consider a solid cylinder of mass $m$ and radius $r$ sliding without rolling down the smooth inclined face of a wedge of mass $M$ that is free to move on a horizontal plane without friction (Fig. 1.174). Now suppose that the cylinder is free to roll down the wedge without slipping. How far does the wedge move in this case?
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$\frac{m h}{M+m}\cot\theta$
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<image>The one-dimensional quantum mechanical potential energy of a particle of mass $m$ is given by $$ V(x)=\begin{cases} V_{0}\delta(x), & -a<x<\infty \\ \infty, & x<-a \end{cases} $$ as shown in Fig. 1.19. At time $t=0$, the wave function of the particle is completely confined to the region $-a<x<0$. [Define the quantities $k=\sqrt{2 m E}/\hbar$ and $\alpha=2 m V_{0}/\hbar^{2}$] Find the (real) solutions for the energy eigenfunctions in the two regions (up to an overall constant) which satisfy the boundary conditions.
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$\psi_{k}(x)=\begin{cases} c_{k}\sin k(x+a), & (-a<x<0)\\ c_{k}\sin k(x+a)+A_{k}\sin kx, & (x\ge0)\\ 0, & (x<-a) \end{cases},\quad A_{k}=\frac{c_{k}\alpha}{k}\sin ka,\quad c_{k}=\left\{\frac{\pi}{2}\Bigl[1+\frac{\alpha\sin2ka}{k}+\Bigl(\frac{\alpha\sin ka}{k}\Bigr)^2\Bigr]\right\}^{-\frac{1}{2}}$
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<image>A long‐range rocket is fired from the surface of the earth (radius $R$) with velocity $\mathbf{v}=(v_r, v_\theta)$ (Fig. 1.29). Neglecting air friction and the rotation of the earth (but using the exact gravitational field), obtain an equation to determine the maximum height $H$ achieved by the trajectory.
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$H \approx \frac{v_{r}^{2}R}{2\left(\frac{GM}{R}-v_{\theta}^{2}\right)}$
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<image>1093 A block of mass $m$ slides without friction on an inclined plane of mass $M$ which in turn is free to slide without friction on a horizontal table (Fig. 1.65). Write sufficient equations to find the motion of the block and the inclined plane. You do not need to solve these equations. (Wisconsin)
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$M \ddot{X}=N \sin \alpha$, $m(\ddot{x}+\ddot{X}\cos \alpha)=-m g \sin \alpha$, $-m \ddot{X}\sin \alpha=N-m g\cos \alpha$
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<image>As in Fig. 4.17, two coaxial cylindrical conductors with $r_{1}$ and $r_{2}$ form a waveguide. The region between the conductors is vacuum for $z<0$ and is filled with a dielectric medium with dielectric constant $\varepsilon \neq 1$ for $z>0$. Describe the TEM mode for $z<0$ and $z>0$.
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For $z>0$: $\mathbf{E}^{\prime}(x,t)=\frac{C}{r}e^{i(k^{\prime} x-\omega t)}\mathbf{e}_{r}$ and $\mathbf{B}^{\prime}(x,t)=\frac{C\sqrt{\varepsilon}}{rc}e^{i(k^{\prime} x-\omega t)}\mathbf{e}_{\theta}$; for $z<0$: $\mathbf{E}(x,t)=\frac{A}{r}e^{i(kz-\omega t)}\mathbf{e}_{r}$ and $\mathbf{B}(x,t)=\frac{A}{rc}e^{i(kz-\omega t)}\mathbf{e}_{\theta}$
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<image>Figure 1.10 shows two capacitors in series, the rigid center section of length $b$ being movable vertically. The area of each plate is $A$. If the voltage difference between the outside plates is kept constant at $V_{0}$, what is the change in the energy stored in the capacitors if the center section is removed?
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$W-W'=\frac{A\varepsilon_{0}V_{0}^{2}}{2(a-b)}\frac{b}{a}$
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<image>The potential curves for the ground electronic state (A) and an excited electronic state (B) of a diatomic molecule are shown in Fig. 8.2. Each electronic state has a series of vibrational levels which are labelled by the quantum number $\nu$. Some molecules were initially at the lowest vibrational level of the electronic state B, followed by subsequent transitions to the various vibrational levels of the electronic state A through spontaneous emission of radiation. Which vibrational level of the electronic state A would be most favorably populated by these transitions?
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$\nu \approx 5$
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<image>Assume all surfaces to be frictionless and the inertia of pulley and cord negligible (Fig. $1.6$). Find the horizontal force necessary to prevent any relative motion of $m_{1}$, $m_{2}$ and $M$.
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$F=\frac{m_{2}\left(M+m_{1}+m_{2}\right)g}{m_{1}}$
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<image>Consider the circuit shown in Fig. 3.87. If one varies the frequency but not the amplitude of $V$, what is the maximum current that can flow? The minimum current? At what frequency will the minimum current be observed.
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$\left(I_{0}\right)_{\max}=\frac{V_{0}}{R},\;\left(I_{0}\right)_{\min}=0,\; \omega=\frac{1}{\sqrt{L_{1}C_{1}}}$
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<image>A $\pi$ meson with a momentum of $5 m_{\pi} c$ makes an elastic collision with a proton ($m_{p}=7 m_{\pi}$) which is initially at rest (Fig. 3.20). Find the momentum of the incident pion in the c.m. system.
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$3.18 m_{\pi} c$
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<image>A mass $M$ is constrained to slide without friction on the track $AB$ as shown in Fig. 2.24. A mass $m$ is connected to $M$ by a massless inextensible string. (Make small angle approximation.) Write a Lagrangian for this system.
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$L = \frac{1}{2} M \dot{x}^{2} + \frac{1}{2} m \left( \dot{x}^{2} + b^{2}\dot{\theta}^{2} + 2b\dot{x}\dot{\theta}\cos\theta \right) + m g b\cos\theta$
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<image>An ideal gas is contained in a large jar of volume $V_{0}$. Fitted to the jar is a glass tube of cross-sectional area $A$ in which a metal ball of mass $M$ fits snugly. The equilibrium pressure in the jar is slightly higher than atmospheric pressure $p_{0}$ because of the weight of the ball. If the ball is displaced slightly from equilibrium it will execute simple harmonic motion (neglecting friction). If the states of the gas represent a quasistatic adiabatic process and $\n\gamma$ is the ratio of specific heats, find a relation between the oscillation frequency $f$ and the variables of the problem.
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$f = \frac{1}{2 \pi} \sqrt{\frac{\gamma A^{2}\left(p_{0}+\frac{m g}{A}\right)}{V m}}$
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<image>Four identical coherent monochromatic wave sources A, B, C, D, as shown in Fig. 4.2 produce waves of the same wavelength $\lambda$. Two receivers $R_{1}$ and $R_{2}$ are at great (but equal) distances from $B$. Which receiver picks up the greater signal?
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$R_{2}$ picks up greater signal
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<image>Find the electric quadrupole moment for two point charges of charge $e$ located at the ends of a line of length $2l$ that rotates with a constant angular velocity $\omega/2$ about an axis perpendicular to the line and through its center.
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$Q_{11}=el^{2}[1+3\cos(\omega t')]$, $Q_{12}=Q_{21}=3el^{2}\sin(\omega t')$, $Q_{22}=el^{2}[1-3\cos(\omega t')]$.
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<image>A mirror is moving through vacuum with relativistic speed $v$ in the $x$ direction. A beam of light with frequency $\omega_{i}$ is normally incident (from $x=+\infty$) on the mirror, as shown in Fig. 3.9. What is the frequency of the reflected light expressed in terms of $\omega_{i}$, $c$ and $v$?
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$\left(\frac{c+v}{c-v}\right)\omega_{i}$
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<image>Two small spheres of mass $M$ are suspended between two rigid supports with three identical springs of spring constant $K$ (each of unstretched length $\frac{a}{2}$). Assuming small oscillations about the equilibrium configuration, find the frequencies for the two normal modes corresponding to vertical oscillations.
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$\omega_{3}=\sqrt{\frac{K}{2M}}, \quad \omega_{4}=\sqrt{\frac{3K}{2M}}$
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<image>$1254$ A rope is attached at one end to a wall and is wrapped around a capstan through an angle $\theta$. If someone pulls on the other end with a force $F$ as shown in Fig. 1.231(a), find the tension in the rope at a point between the wall and the capstan in terms of $F$, $\theta$ and $\mu_{s}$, the coefficient of friction between the rope and capstan. (Columbia)
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$T=F e^{-\mu_{s} \theta}$
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<image>A car is traveling in the $x$-direction and maintains constant horizontal speed $v$. The car goes over a bump whose shape is described by $y_{0}=A[1-\cos (\pi x / l)]$ for $0 \leq x \leq 2 l ; y_{0}=0$ otherwise. Determine the motion of the center of mass of the car while passing over the bump. Represent the car as a mass $m$ attached to a massless spring of relaxed length $l_{0}$ and spring constant $k$. Ignore friction and assume that the spring is vertical at all times.
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$y(t)=C_{1}\cos (\omega t)+B \cos \left(\frac{\pi v t}{l}\right)+A+l_{0}-\frac{m g}{k}$ with $\omega=\sqrt{\frac{k}{m}},\ C_{1}=-\frac{m\pi^{2}v^{2}A}{m\pi^{2}v^{2}-kl^{2}},\ B=\frac{kl^{2}A}{m\pi^{2}v^{2}-kl^{2}}$
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<image>Consider the ground state and \(n=2\) states of hydrogen atom. There are four corrections to the indicated level structure that must be considered to explain the various observed splitting of the levels. These corrections are: (a) Lamb shift, (b) fine structure, (c) hyperfine structure, (d) relativistic effects. Which of the above apply to the \(n=2, l=1\) state? Answer in the name of the corrections.
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fine structure, hyperfine structure
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<image>Two uniform cylinders are spinning independently about their axes, which are parallel. One has radius $R_{1}$ and mass $M_{1}$, the other $R_{2}$ and $M_{2}$. Initially they rotate in the same sense with angular speeds $\Omega_{1}$ and $\Omega_{2}$ respectively as shown in Fig. 1.128. They are then displaced until they touch along a common tangent. After a steady state is reached, what is the final angular velocity of each cylinder? (CUSPEA)
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$\omega_{1}=\frac{M_{1} R_{1} \Omega_{1}-M_{2} R_{2} \Omega_{2}}{R_{1}(M_{1}+M_{2})}$, $\omega_{2}=\frac{M_{2} R_{2} \Omega_{2}-M_{1} R_{1} \Omega_{1}}{R_{2}(M_{1}+M_{2})}$
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<image>A thin ring of mass $M$ and radius $r$ lies flat on a frictionless table. It is constrained by two extended identical springs with relaxed length $l_{0}$ ($l_{0} \gg r$) and spring constant $k$ as shown in Fig. 1.55. What are the normal modes of small oscillations and their frequencies?
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$\omega_{x}=\sqrt{\frac{2k}{M}},\ \omega_{y}=\sqrt{\frac{k}{M}}$
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<image>An electric circuit consists of two resistors (resistances $R_{1}$ and $R_{2}$), a single condenser (capacitor $C$) and a variable voltage source $V$ joined together as shown in Fig. 3.83. When $V(t)$ is a very sharp pulse at $t=0$, we approximate $V(t)=A\delta(t)$. What is the time history of the potential drop across $R_{1}$ ?
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$V_{1}\propto \exp\left(-\frac{R_{1}+R_{2}}{CR_{1}R_{2}}t\right)$ for $t>0$ and $V_{1}=0$ for $t<0$
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<image>A very long conducting pipe has a square cross section of its inside surface, with side $D$ as in Fig. 1.47. Far from either end of the pipe is suspended a point charge located at the center of the square cross section. Determine the electric potential at all points inside the pipe, perhaps in the form of an infinite series.
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$\varphi(x, y, z)= \frac{2 Q}{\pi \varepsilon_{0} D} \sum_{m, m^{\prime}=0}^{\infty} \frac{\cos\frac{(2m+1)\pi x}{D}\cos\frac{(2m^{\prime}+1)\pi y}{D}}{\sqrt{(2m+1)^2+(2m^{\prime}+1)^2}}\; e^{-\frac{x \mid 21}{D}\sqrt{(2m+1)^2+(2m^{\prime}+1)^2}}$
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<image>In Fig. 5.10 a point charge $e$ moves with constant velocity $v$ in the $z$ direction so that at time $t$ it is at the point $Q$ with coordinates $x=0$, $y=0, z=v t$. Find at the time $t$ and at the point $P$ with coordinates $x=b, y=0, z=0$ the vector potential $\mathbf{A}$.
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$\mathbf{A}=\frac{e v}{4 \pi \varepsilon_{0} c^{2} \sqrt{\left(1-\beta^{2}\right) b^{2}+v^{2} t^{2}}} \mathbf{e}_{z}$
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<image>In the circuit shown in Fig. 3.23, the resistance of $L$ is negligible and initially the switch is open and the current is zero. Find the quantity of heat dissipated in the resistance $R_{2}$ when the switch is closed and remains closed for a long time. (Notice the circuit diagram and the list of values for $V, R_{1}, R_{2}$, and $L_{0}$).
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$45.5\;\mathrm{J}$
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<image>A perfectly uniform ball 20 cm in diameter and with a density of $5\,\mathrm{g}/\mathrm{cm}^{3}$ is rotating in free space at $1\,\mathrm{rev}/\mathrm{s}$. An intelligent flea of $10^{-3}\,\mathrm{g}$ resides in a small (massless) house fixed to the ball's surface at a rotational pole as shown in Fig. 1.197. The flea decides to move the equator to the house by walking quickly to a latitude of $45^{\circ}$ and waiting the proper length of time. How long should it wait? Indicate how you obtain this answer.
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$6 \times 10^{6}\,\mathrm{s}$
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<image>Consider a square loop of wire, of side length $l$, lying in the $xy$ plane as shown in Fig. 2.43. Suppose a particle of charge $q$ is moving with a constant velocity $v$, where $v \ll c$, in the $xz$-plane at a constant distance $z_{0}$ from the $xy$-plane. (Assume the particle is moving in the positive $x$ direction.) Suppose the particle crosses the $z$-axis at $t=0$. Give the induced emf in the loop as a function of time.
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$-\frac{\mu_{0}qv^{2}}{4\pi}\left\{\frac{1}{\sqrt{v^{2}t^{2}+(z-z_{0})^{2}}}-\frac{1}{\sqrt{(l-vt)^{2}+(z-z_{0})^{2}}}-\frac{1}{\sqrt{(l-vt)^{2}+l^{2}+(z-z_{0})^{2}}}+\frac{1}{\sqrt{v^{2}t^{2}+l^{2}+(z-z_{0})^{2}}}\right\}$
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<image>Consider an intrinsic semiconductor whose electronic density of states function $N(E)$ is depicted in Fig. Where is the Fermi level with respect to the valence and conduction bands?
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$\varepsilon_{\mathrm{F}}=\frac{\varepsilon_{\mathrm{C}}+\varepsilon_{\mathrm{V}}}{2}$
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<image>Four identical masses are connected by four identical springs and constrained to move on a frictionless circle of radius $b$ as shown in Fig. 2.30. What are the frequencies of small oscillations?
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$\sqrt{\frac{k}{m}}$ and $\sqrt{\frac{2 k}{m}}$
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<image>The betatron accelerates particles through the emf induced by an increasing magnetic field within the particle's orbit. Let $\bar{B}_{1}$ be the average field within the particle orbit of radius $R$, and let $B_{2}$ be the field at the orbit (see Fig. 2.80). What must be the relationship between $\bar{B}_{1}$ and $B_{2}$ if the particle is to remain in the orbit at radius $R$ independent of its energy?
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$B_{2}=\frac{\bar{B}_{1}}{2}$
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<image>The conductors of a coaxial cable are connected to a battery and resistor as shown in Fig. 2.15. Starting from first principles find, in the region between $r_{1}$ and $r_{2}$, the Poynting vector.
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$\mathbf{N}=\frac{V^{2}}{2 \pi r^{2} R \ln \frac{r_{2}}{r_{1}}}\mathbf{e}_{z}$
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<image>Two infinite parallel wires separated by a distance $d$ carry equal currents $I$ in opposite directions, with $I$ increasing at the rate $\frac{dI}{dt}$. A square loop of wire of length $d$ on a side lies in the plane of the wires at a distance $d$ from one of the parallel wires, as illustrated in Fig. 2.30. Find the emf induced in the square loop.
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$\varepsilon=-\frac{\mu_{0}d}{2\pi}\ln\left(\frac{4}{3}\right)\frac{dI}{dt}$
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<image>Two uniform discs in a vertical plane of masses $M_{1}$ and $M_{2}$ with radii $R_{1}$ and $R_{2}$ respectively have a thread wound about their circumferences, and are thus connected as shown in Fig. 1.158. The first disc has fixed frictionless horizontal axis of rotation through its center. Set up the equations to determine the acceleration of the center of mass of the second disc if it falls freely.
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$M_{2}\ \ddot{x}=M_{2}g-F$, $I_{1}\ \ddot{\theta}_{1}=FR_{1}$, $I_{2}\ \ddot{\theta}_{2}=FR_{2}$, $\ddot{x}=R_{1}\ \ddot{\theta}_{1}+R_{2}\ \ddot{\theta}_{2}$, with $I_{1}=\frac{m_{1}R_{1}^{2}}{2}$ and $I_{2}=\frac{m_{2}R_{2}^{2}}{2}$
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<image>In Fig. 3.37 the capacitor is originally charged to a potential difference $V$. The transformer is ideal: no winding resistance, no losses. At $t=0$ the switch is closed. Assume that the inductive impedances of the windings are very large compared with $R_{p}$ and $R_{s}$. Calculate: The initial secondary current.
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$i_{s}(0)=\frac{N_{p} N_{s} V}{N_{s}^{2} R_{p}+N_{p}^{2} R_{s}$
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<image>A rectangular coil of dimensions $a$ and $b$ and resistance $R$ moves with constant velocity $v$ into a magnetic field $\mathbf{B}$ as shown in Fig. 2.36. Derive an expression for the vector force on the coil in terms of the given parameters.
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$-\frac{v b^{2} B^{2}}{R}$
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<image>A parallel-plate capacitor is made of circular plates as shown in Fig. 2.10. The voltage across the plates (supplied by long resistanceless lead wires) has the time dependence $V=V_{0} \cos \omega t$. Assume $d \ll a \ll c / \omega$, so that fringing of the electric field and retardation may be ignored. What current flows in the lead wires and what is the current density in the plates as a function of time?
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$I=-\frac{\pi a^{2}\varepsilon_{0}V_{0}\omega}{d}\sin\omega t,\quad j_{r}(r)=\frac{(a^{2}-r^{2})\varepsilon_{0}V_{0}\omega}{2dr}\sin\omega t\,e_{r}$
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<image>A ray of light enters a spherical drop of water of index $n$ as shown (Fig. 1.21). Find the angle $\phi$ which produces minimum deflection.
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$\cos ^{2} \phi = \frac{n^{2}-1}{3}$
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<image>Consider a rectangular waveguide, infinitely long in the $x$-direction, with a width ($y$-direction) 2 cm and a height ($z$-direction) 1 cm. The walls are perfect conductor, as in Fig. 4.13. For the lowest mode that can propagate, find the phase velocity and the group velocity.
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Phase velocity: $v=\frac{\omega}{\sqrt{\frac{\omega^{2}}{c^{2}}-\frac{\pi^{2}}{4}}}>c$, Group velocity: $v_{g}=\frac{c^{2}}{v}$.
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<image>Let us apply a shearing force on a rectangular solid block as shown in Fig. 2.77. Find the relation between the displacement $u$ and the applied force within elastic limits.
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$u=l\varphi=\frac{lF}{An}$
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<image>A plane monochromatic wave (wavelength $\lambda$ ) is incident on a set of 5 slits spaced at a distance $d$ (Fig. 2.55). You may assume that the width of the individual slits is much less than $d$. For the resulting interference pattern, which is focused on a screen, compute either analytically or approximately the following. Angular width of central image (angle between principal maximum and first minimum.)
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$\frac{\lambda}{5 d}$
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<image>Consider the ground state and \(n=2\) states of hydrogen atom. Indicate in the diagram (Fig. 1.14) the complete spectroscopic notation for all four states. There are four corrections to the indicated level structure that must be considered to explain the various observed splitting of the levels. These corrections are: (a) Lamb shift, (b) fine structure, (c) hyperfine structure, (d) relativistic effects. Which of the above apply to the \(n=1\) state?
|
hyperfine structure
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<image>Consider the following energy level structure (Fig. 2.30): The ground states form an isotriplet as do the excited states (all states have a spin-parity of $0^{+}$). The ground state of $_{21}^{42}\text{Sc}$ can $\beta$-decay to the ground state of $_{20}^{42}\text{Ca}$ with a kinetic end-point energy of 5.4 MeV (transition II in Fig. 2.30). Using the results of parts (a) and (c), ignoring $H_{2}$, and given that $\frac{\text{Γ}_{I}}{\text{Γ}_{II}}=6 \times 10^{-5}$, calculate the value of the reduced matrix element which mixes the ground and excited states of $_{20}^{42}\text{Ca}$.
|
$\left|\left\langle \alpha, 1 \left\| P_{1} \right\| \alpha^{\prime}, 1 \right\rangle\right|^{2} = 1.48 \times 10^{-3} \mathrm{MeV}^{2},\ \left|\left\langle \alpha, 1 \left\| P_{1} \right\| \alpha^{\prime}, 1 \right\rangle\right| = 38 \mathrm{keV}$
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<image>Find the expression for the speed of the transverse elastic wave.
|
$v=\sqrt{\frac{n}{\rho}}$
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<image>According to the Weinberg-Salam model, the Higgs boson $\phi$ couples to every elementary fermion $f$ ( $f$ may be a quark or lepton) in the form
$$
\frac{e m_{f}}{m_{W}} \phi \bar{f} f,
$$
where $m_{f}$ is the mass of the fermion $f, e$ is the charge of the electron, and $m_{W}$ is the mass of the $W$ boson. Some theorists believe that the Higgs boson weighs approximately 10 GeV . If so do you believe it would be observed (in practice) as a resonance in $e^{+} e^{-}$ annihilation (Fig. 3.22)? Roughly how large would the signal to background ratio be at resonance?
|
$1.5 x 10^{-4}$
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<image>Consider a two-dimensional classical system with Hamiltonian\n\nH=\frac{1}{2 m}\left(P_{1}^{2}+P_{2}^{2}\right)+\frac{1}{2} \mu^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-\frac{1}{4} \lambda\left(x_{1}^{2}+x_{2}^{2}\right)^{2} .\n\nA system of $N$ particles of mass $m$ each is in thermal equilibrium at temperature $T$ within the potential well that appears in the Hamiltonian. $T$ is small enough so that an overwhelming majority of the particles reside within the quadratic part of the well. However, some particles will always possess enough thermal energy to escape from the well by passing over the "top" of the well; in the one-dimensional slice of $V(\mathbf{x})$ shown in Fig. 2.14, this occurs at $x_{1}=b$, where $b$ can be determined from the above equation.\n\nCalculate the escape rate for particles to leave the well by passing over the top.
|
$\frac{N \mu^{3}}{\sqrt{2 \pi m \lambda k T}} \cdot e^{-\frac{\mu^{4}}{6 k T \lambda}}$
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<image>One mole of the paramagnetic substance whose $T S$ diagram is shown below is to be used as the working substance in a Carnot refrigerator operating between a sample at 0.2 K and a reservoir at 1 K: How much work will be performed on the paramagnetic substance per cycle?
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$6.6 \times 10^{7} \mathrm{ergs} / \mathrm{mol}$
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<image>A simple molecular beam apparatus is shown in Fig. 2.40. The oven contains $\mathrm{H}_{2}$ molecules at 300 K and at a pressure of 1 mm of mercury. The hole on the oven has a diameter of $100 \mu \mathrm{~m}$ which is much smaller than the molecular mean free path. After the collimating slits, the beam has a divergence angle of 1 mrad. Find: the mean speed of molecules in the beam;
|
$\frac{3}{2} \sqrt{\frac{\pi k T}{2 m}}$
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<image>In Fig. 2.71, the cylindrical cavity is symmetric about its long axis. For the purposes of this problem, it can be approximated as a coaxial cable (which has inductance and capacitance) shorted at one end and connected to a parallel plate disk capacitor at the other. Find the direction and radial-dependence of the Poynting vector $N$ in the regions near points $A$ and $B$.
|
At point $A$, $\mathbf{N} \sim \frac{1}{r^{2}}\mathbf{e}_{z}$; at point $B$, $\mathbf{N} \sim r\mathbf{e}_{r}$
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<image>Refer to Fig. 3.67. When this monostable circuit is triggered how long will $Q_{2}$ be off?
|
7 \mu \mathrm{~s}
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<image>A dipole of fixed length $2 R$ has mass $m$ on each end, charge $+Q_{2}$ on one end and $-Q_{2}$ on the other. It is in orbit around a fixed point charge $+Q_{1}$. (The ends of the dipole are constrained to remain in the orbital plane.) Figure 1.57 shows the definitions of the coordinates $r, \theta, lpha$. Figure 1.58 gives the radial distances of the dipole ends from $+Q_{1}$. The dipole is in a circular orbit about $Q_{1}$ with $\dot{r} pprox \ddot{r} pprox \ddot{\theta} pprox 0$ and $lpha \ll 1$. Find the period of small oscillations in the $lpha$ coordinate.
|
$T = 2 \pi \sqrt{\frac{4 \pi \varepsilon_{0}}{Q_{1} Q_{2}} \cdot m R r^{2}}$
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<image>A pinhole camera consists of a box in which an image is formed on the film plane which is a distance $P$ from a pinhole of diameter $d$. The object is at a distance $L$ from the pinhole, and light of wavelength $\nabla$ is used (Fig. 2.66). Approximately what diameter $d$ of the pinhole will give the best image resolution?
|
$\sqrt{\frac{\lambda * L * P}{L + P}}$
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<image>Figure 1.24 is an energy level diagram for the ground state and first four excited states of a helium atom. Given electrons of sufficient energy, which levels could be populated as the result of electrons colliding with ground state atoms?
|
$(1 s 2 s)^{1} S_{0}, (1 s 2 s)^{3} S_{1}$
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<image>A very long conducting pipe has a square cross section of its inside surface, with side $D$ as in Fig. 1.47. Far from either end of the pipe is suspended a point charge located at the center of the square cross section. Give the asymptotic expression for this potential for points far from the point charge.
|
$\varphi=\frac{\sqrt{2} Q}{\varepsilon_{0} \pi D}\cos\frac{\pi x}{D}\cos\frac{\pi y}{D}e^{-\frac{\sqrt{3} x}{\partial}|z|}$
|
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<image>The space between two long thin metal cylinders is filled with a material with dielectric constant $\varepsilon$. The cylinders have radii $a$ and $b$, as shown in Fig. 1.19. What is the electric field between the cylinders?
|
$\mathbf{E}=-\frac{V}{r \ln \left(\frac{a}{b}\right)} \mathbf{e}_{r}$
|
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<image>A particle of mass m is released at $t=0$ in the one-dimensional double square well shown in the figure in such a way that its wave function at $t=0$ is just one sinusoidal loop (half a sine wave with nodes just at the edges of the left half of the potential as shown). Find the average value of the energy at $t=0$ (in terms of symbols defined above).
|
$\frac{\hbar^{2}\pi^{2}}{2ma} - V_{0}$
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<image>Inelastic neutrino scattering in the quark model. Consider the scattering of neutrinos on free, massless quarks. We will simplify things and discuss only strangeness‐conserving reactions, i.e. transitions only between the $u$ and $d$ quarks. Assume that inelastic $u$ (or $\bar{u}$)-nucleon cross sections are given by the sum of the cross sections for the four processes that have been listed above. Derive the quark model prediction for the ratio of the total cross section for antineutrino‐nucleon scattering compared with neutrino‐nucleon scattering, $\sigma^{\bar{u} N} / \sigma^{u N}$.
|
3
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<image>Three identical objects, each of mass $m$, are connected by springs of spring constant $K$, as shown in Fig. 1.95. The motion is confined to one dimension. At $t=0$, the masses are at rest at their equilibrium positions. Mass $A$ is then subjected to an external time‐dependent driving force $F(t)=f \cos(\omega t)$, $t>0$. Calculate the motion of mass $C$.
|
$x_{C}=2a+\frac{f}{3m\omega^{2}}[1-\cos(\omega t)]+\frac{f}{2m(\omega_{2}^{2}-\omega^{2})}[\cos(\omega t)-\cos(\omega_{2}t)] + \frac{f}{6m(\omega_{3}^{2}-\omega^{2})}[\cos(\omega t)-\cos(\omega_{3}t)]$, where $\omega_{2}=\sqrt{\frac{K}{m}}$ and $\omega_{3}=\sqrt{\frac{3K}{m}}$
|
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<image>A simple molecular beam apparatus is shown in Fig. 2.40. The oven contains $\mathrm{H}_{2}$ molecules at 300 K and at a pressure of 1 mm of mercury. The hole on the oven has a diameter of $100 \mu \mathrm{~m}$ which is much smaller than the molecular mean free path. After the collimating slits, the beam has a divergence angle of 1 mrad. Find: the speed distribution of molecules in the beam;
|
$n v \left(\frac{m}{2 \pi k T}\right)^{3/2} e^{-m v^2/(2 k T)} v^2 dv d\Omega$
|
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<image>Figure 2.19 gives the low-lying states of ${}^{18}\text{O}$ with their spin-parity assignments and energies (in MeV) relative to the $0^{+}$ ground state. What $J^{p}$ values are expected for the low-lying states of ${}^{19}\text{O}$?
|
$\frac{5}{2}^+$, $\frac{1}{2}^+$, $\frac{3}{2}^+$, $\frac{7^{+}}{2}$, $\frac{9^{+}}{2}$
|
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<image>Consider a $2-\text{cm}$ thick plastic scintillator directly coupled to the surface of a photomultiplier with a gain of $10^{6}$. A $10-\text{GeV}$ particle beam is incident on the scintillator as shown in Fig. 4.8(a). Suppose one could detect a signal on the anode of as little as $10^{-12}$ coulomb. If the beam particle is a neutron, estimate what is the smallest laboratory angle that it could scatter elastically from a proton in the scintillator and still be detected?
|
$5.0 \times 10^{-4} \mathrm{rad}$
|
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<image>In an experiment a beam of silver atoms emerges from an oven, which contains silver vapor at $T=1200 \, \mathrm{K}$. The beam is collimated by being passed through a small circular aperture. If the screen is at $L=1$ meter from the aperture, estimate numerically the smallest $D$ that can be obtained by varying $a$. (You may assume for simplicity that all atoms have the same momentum along the direction of the beam and have a mass of $\left.M_{\mathrm{Ag}}=1.8 \times 10^{-22} \, \mathrm{g}\right)$.
|
$8.0 \times 10^{-6} \mathrm{~m}$
|
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<image>The pions that are produced when protons strike the target at Fermilab are not all moving parallel to the initial proton beam. A focusing device, called a "horn", (actually two of them are used as a pair) is used to deflect the pions so as to cause them to move more closely towards the proton beam direction. This device consists of an inner cylindrical conductor along which a current flows in one direction and an outer cylindrical conductor along which the current returns. Between these two surfaces there is produced a toroidal magnetic field that deflects the mesons that pass through this region. By what angle would a $100\,\mathrm{GeV}/c$ meson be deflected if it traversed 2 meters of one horn's magnetic field at very nearly this radius of 15 cm ?
|
$\theta \approx 0.0013\,\mathrm{rad}$
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<image>In Fig. 3.70 the circuit is a "typical" TTL totom pole output circuit. You should assume that all the solid state devices are silicon unless you specifically state otherwise. Give the voltages requested within 0.1 volt. Case 2: $V_{\mathrm{A}}=0.2$ volts, give $V_{\mathrm{B}}, V_{\mathrm{C}}, V_{\mathrm{D}}, V_{\mathrm{E}}$.
|
$V_{\mathrm{B}}=0\,\mathrm{V},\; V_{\mathrm{C}}=5\,\mathrm{V},\; V_{\mathrm{D}}=4.3\,\mathrm{V},\; V_{\mathrm{E}}=3.6\,\mathrm{V}$
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<image>A thin spherical shell of radius $R$ has a fixed charge $+q$ distributed uniformly over its surface. A small circular section (radius $r \ll R$) of charge is removed from the surface. The cut section is replaced and the sphere is set in motion rotating with constant angular velocity $\omega=\omega_{0}$ about the $z$-axis. Now the sphere's angular velocity increases linearly with time: \(\omega=\omega_{0}+kt\). Calculate the line integral of the electric field around the circular path $P$ (located at the center of the sphere with normal along the $+z$ axis and radius $r_{P} \ll R$).
|
$\frac{\mu_{0} k r_{P}^{2} q}{6 R}$
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<image>A circular aperture of radius $a$ is uniformly illuminated by plane waves of wavelength $\lambda$. Let the $\boldsymbol{z}$-axis coincide with the aperture axis, with $z=0$ at the aperture, and with the incident flux travelling from negative values of $z$ toward $z=0$ (Fig.~2.30). Find the values of $z$ at which the intensity of illumination on the axis is zero (Fresnel diffraction). You may assume $z \gg a$.
|
$z = a^2 / (\lambda N) \text{for} N = 2, 4, 6, ...$
|
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<image>The Stark effect. The energy levels of the $n=2$ states of atomic hydrogen are illustrated in Fig. 5.14. The $S_{1/2}$ and $P_{1/2}$ levels are degenerate at an energy $\varepsilon_{0}$ and the $P_{3/2}$ level is degenerate at an energy $\varepsilon_{0}+A$. A uniform static electric field $E$ applied to the atom shifts the states to energies $\varepsilon_{1}$, $\varepsilon_{2}$ and $\varepsilon_{3}$. Assuming that all states other than these three are far enough away to be neglected, determine the energy of the $P_{3/2}$ level to second order in the electric field $E$, using the given perturbation Hamiltonian matrix elements.
|
\(\varepsilon_{0}+\Delta+\frac{|b|^{2}}{A}\)
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<image>A flyball governor consists of two masses $m$ connected to arms of length $l$ and a mass $M$ as shown in Fig. 2.68. The assembly is constrained to rotate around a shaft on which the mass $M$ can slide up and down without friction. Neglect the mass of the arms, air friction, and assume that the diameter of the mass $M$ is small. Suppose the shaft is now allowed to rotate freely. Does the frequency of small oscillation change? If so, calculate the new value.
|
$f=\frac{1}{2\pi}\sqrt{\frac{(m+M)g(1+3\cos^{2}\theta_{0})}{(m+2M\sin^{2}\theta_{0})l\cos\theta_{0}}}$
|
|
<image>A ray of light enters a spherical drop of water of index $n$ as shown (Fig. 1.21). Find an expression for the angle of deflection $\delta$.
|
$\pi - 4 \alpha + 2 \phi$
|
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<image>Consider a $2-\text{cm}$ thick plastic scintillator directly coupled to the surface of a photomultiplier with a gain of $10^{6}$. A $10-\text{GeV}$ particle beam is incident on the scintillator as shown in Fig. 4.8(a). Same as Part (b), but it scatters elastically from a carbon nucleus.
|
$1.73 \times 10^{-3} \mathrm{rad}$
|
|
<image>For the combination of one prism and 2 lenses shown (Fig. 1.42), find the location and size of the final image when the object, length 1 cm, is located as shown in the figure.
|
$10 \mathrm{cm}$ to the left of the second lens; $0.5 \mathrm{cm}$
|
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<image>Looking through a small hole is a well-known method to improve sight. If your eyes are near-sighted and can focus an object 20 cm away without using any glasses, estimate the required diameter of the hole through which you would have good sight for objects far away.
|
$0.12 \mathrm{mm}$
|
|
<image>In Fig. 2.71, the cylindrical cavity is symmetric about its long axis. For the purposes of this problem, it can be approximated as a coaxial cable (which has inductance and capacitance) shorted at one end and connected to a parallel plate disk capacitor at the other. Derive an expression for the lowest resonant frequency of the cavity. Neglect end and edge effects ( $h \gg r_{2}, d \ll r_{1}$ ).
|
$\omega_{0}=\frac{2 d c^{2}}{h\left(2 d h+r_{1}^{2}\ln \frac{r_{2}}{r_{1}}\right)}$
|
|
<image>For the circuit shown in Fig. 3.35, the coupling coefficient of mutual inductance for the two coils $L_{1}$ and $L_{2}$ is unity. What is the average power supplied by the oscillator as a function of frequency?
|
$P=\frac{RV_{0}^{2}/2}{R^{2}+\left(\frac{\omega L}{1-\omega^{2}LC}\right)^{2}}$
|
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<image>A sound field is created by an arrangement of identical line sources grouped into two identical arrays of $N$ sources each as shown (Fig. 2.62) below.\nAll of the radiators lie in a plane perpendicular to the page, and produce waves of wavelength $\theta, N, c$ and $d$, the distance between the centers of the arrays.
|
$I = \frac{I_{m}}{N^{2}}\left(\frac{\sin \frac{N \delta}{2}}{\sin \frac{\delta}{2}}\right)^{2} \cos ^{2}\left(\frac{\pi d \sin \theta}{\lambda}\right)$
|
|
<image>Name the lowest electric multipole in the radiation field emitted by the following time-varying charge distributions. A uniform charged spherical shell whose radius varies as $R=R_{0}+R_{1}\cos(\omega t)$.
|
all the electric multipole moments are zero.
|
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<image>Three bodies of equal mass $m$ and indicated by $i=1,2,3$ are constrained to perform small oscillations along different coplanar axes forming $120^{\circ}$ angles at their common intersection, as shown in Fig. 1.94. Identical coupling springs hold these bodies near equilibrium positions which are at a distance $l$ from the intersection on each axis, that is, the equilibrium length of each spring is $\sqrt{3} l$. Show that the remaining normal modes are degenerate and determine their frequency.
|
$\omega'=\sqrt{\frac{K}{m}}$
|
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<image>Figure 2.67 shows a simplified electron lens consisting of a circular loop (of radius $a$) of wire carrying current $I$. For $\rho \ll a$ the vector potential is approximately given by $A_{\theta}=rac{\pi I a^{2} \rho}{\left(a^{2}+z^{2}\right)^{3/2}}$. Show that the canonical momentum $p_{\theta}$ vanishes for the orbit shown and find an expression for $\dot{\theta}$.
|
$P_{\theta}=0, \quad \dot{\theta}=-\frac{I \pi a^{2}q}{mc(a^{2}+z^{2})^{3/2}}$
|
|
<image>Refer to Fig. 3.25. The switch is suddenly moved to position $B$. Just after the switching, what are currents in $\varepsilon_{2}$, $R_{1}$, $R_{2}$ and $L$ ?
|
$I_{R_{1}}=0.25\,\mathrm{mA}$ (rightward), $I_{R_{2}}=0.75\,\mathrm{mA}$ (upward), $I_{\varepsilon_{2}}=0.25\,\mathrm{mA}$ (downward), $I_{L}=0.5\,\mathrm{mA}$ (downward)
|
|
<image>Suppose the radiation is observed far from the charges at an angle $\theta$ relative to the axis of rotation. What is the polarization for $\theta=0^{\circ}$, $90^{\circ}$, and $0<\theta<90^{\circ}$?
|
For $\theta=0^{\circ}$, no radiation is emitted; for $\theta=90^{\circ}$, the radiation is circularly polarized; for $0^{\circ}<\theta<90^{\circ}$, the radiation is not polarized.
|
|
<image>A marble bounces down stairs in a regular manner, hitting each step at the same place and bouncing the same height above each step (Fig. 1.5). The stair height equals its depth (tread=rise) and the coefficient of restitution $e$ is given. Find the necessary horizontal velocity and bounce height (the coefficient of restitution is defined as $e=-v_{f} / v_{i}$, where $v_{f}$ and $v_{i}$ are the vertical velocities just after and before the bounce respectively).
|
$v_{h} = \sqrt{\frac{g l}{2}\frac{1-e}{1+e}}$
|
|
<image>A coin spinning about its axis of symmetry with angular frequency $\omega$ is set down on a horizontal surface (Fig. 1.148). After it stops slipping, with what velocity does it roll away?
|
$\dot{x}_{c}=-\frac{1}{3}\omega R$
|
|
<image>A plane wave of light from a laser has a wavelength of $6000 \AA$. The light is incident on a double slit. After passing through the double slit the light falls on a screen 100 cm beyond the double slit. The intensity distribution of the interference pattern on the screen is shown (Fig. 2.48). What is the width of each slit and what is the separation of the two slits?
|
$6 \times 10^{-3} \mathrm{cm}$; $1. 5\times 10^{-3} \mathrm{cm}$
|
|
<image>The figure 2.21 shows the cross section of an infinitely long circular cylinder of radius $3a$ with an infinitely long cylindrical hole of radius $a$ displaced so that its center is at a distance $a$ from the center of the big cylinder. The solid part of the cylinder carries a current $I$, distributed uniformly over the cross section, and out from the plane of the paper. Find the magnetic field at all points on the plane $P$ containing the axes of the cylinders.
|
Inside the hole ($0<x<2a$): $H_y=\frac{I}{16\pi a}$; Inside the solid part ($2a\le x\le3a$ or $-3a\le x\le0$): $H_y=\frac{I(x^2-ax-a^2)}{16\pi a^2(x-a)}$; Outside the cylinder ($|x|>3a$): $H_y=\frac{(8x-9a)I}{16\pi x(x-a)}$
|
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<image>Flat metallic plates $P, P^{\prime}$, and $P^{\prime \prime}$ (see Fig. 1.53) are vertical and the plate $P$, of mass $M$, is free to move vertically between $P^{\prime}$ and $P^{\prime \prime}$. The three plates form a double parallel-plate capacitor. Let the charge on this capacitor be $q$. Ignore all fringing‐field effects. Assume that this capacitor is discharging through an external load resistor $R$, and neglect the small internal resistance. Assume that the discharge is slow enough so that the system is in static equilibrium at all times. Does the output voltage increase, decrease, or stay the same as this capacitor discharges?
|
$V_{0}=\sqrt{\frac{M g d}{\varepsilon_{0} a}}$
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