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<image>Metallic diffraction grating.\nPurcell (Phys. Rev. 92, 1069) has shown that electromagnetic radiation is emitted from a metallic diffraction grating when a beam of charged particles passes near and parallel to the surface of the grating.\n\nThe velocity of the charged particles is $v$ and the grating spacing is ...
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201
<image>A transmission line consists of two parallel conductors of arbitrary but constant cross‐sections. Current flows down one conductor and returns by way of the other. The conductors are immersed in an insulating medium of dielectric constant $\varepsilon$ and permeability $\mu$, as shown in Fig. 4.18. Under what co...
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202
<image>A camera (focal length 50 cm , apert ure diameter $D$ ), sensitive to visible light, is sharply focused on the stars; it is then used without refocusing for an object at a distance of 100 meters. Roughly, what aperture $D$ will give the best resolution for this object? Give $D$ in cm .
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203
<image>A sphere of mass $M$ and radius $R$ rolls without slipping down a triangular block of mass $m$ that is free to move on a frictionless horizontal surface, as shown in Fig. 2.18. Find the motion of the system by integrating Lagrange's equation, given that all objects are initially at rest and the sphere's center i...
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204
<image><image>A substance is found to have two phases, $N$ and $S$. In the normal state, the $N$ phase, the magnetization $M$ is negligible. At a fixed temperature $T<T_{\text{c}}$, as the external magnetic field $H$ is lowered below the critical field\n\[\nH_{\text{c}}(T)=H_{0}\left[1-\left(\frac{T}{T_{\text{c}}}\righ...
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205
<image>The decay $K \rightarrow \pi \gamma$ is absolutely forbidden by a certain conservation law, which is believed to hold exactly. Which conservation law is this?
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206
<image>It is believed that nucleons $(\mathrm{N})$ interact directly through the weak interaction and that the latter violates parity conservation. One way to study the nature of the $\mathrm{N}-\mathrm{N}$ weak interaction is by means of $\alpha$-decay, as typified by the decays of the $3^{+}, T=1$ and $3^{-}, T=0$ st...
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207
<image><image>A substance is found to have two phases, $N$ and $S$. In the normal state, the $N$ phase, the magnetization $M$ is negligible. At a fixed temperature $T<T_{\text{c}}$, as the external magnetic field $H$ is lowered below the critical field\n\[\nH_{\text{c}}(T)=H_{0}\left[1-\left(\frac{T}{T_{\text{c}}}\righ...
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208
<image>Find the reflection and transmission coefficients for the one-dimensional potential step shown in Fig. 1.27 if the particles are incident from the right.
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209
<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. Is ...
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210
<image>A curved mirror brings collimated light to focus at $x=20 \text{ cm}$. Then it is filled with water $n=\frac{4}{3}$ and illuminated through a pinhole in a white card (Fig. 1.25). A sharp image will be formed on the card at what distance, $X$ ?
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211
<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), solve it to lowest order in $(H/R)$ and verify that it gives a familiar result for the c...
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212
<image>A merry-go-round (MGR) has two orthogonal axes ($x, y$) painted on it, and is rotating on the earth (assume to be an inertial frame $x_{0}, y_{0}, z_{0}$) with constant angular velocity $\omega$ about a vertical axis. A bug of mass $m$ is crawling outward without slipping along the $x$-axis with constant velocit...
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<image>A particle of mass $m$ can slide without friction on the inside of a small tube bent in the form of a circle of radius $a$. The tube rotates about a vertical diameter at a constant rate of $\omega\,\mathrm{rad}/\mathrm{sec}$ as shown in Fig. 1.79. Write the differential equation of motion. (SUNY, Buffalo)
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214
<image>A heavy star of mass $M$ and radius $R$ moves with velocity $\mathbf{V}$ through a very dilute gas of mass density $\rho$. It pulls particles toward itself by its gravitational field and captures all of the atoms that strike its surface. Find the drag force on the star with the approximation that the thermal vel...
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215
<image>Two non-relativistic particles of equal energy and equal mass collide almost head-on. In a coordinate frame (the center of mass frame) moving with velocity $\mathbf{V}$, the particles appear to collide head-on. Find $\mathbf{V}$, the velocity of the center of mass frame.
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216
<image>Two non-relativistic particles of equal energy and equal mass collide almost head-on. In a coordinate frame (the center of mass frame) moving with velocity $\mathbf{V}$, the particles appear to collide head-on. Compare the total energy in the center of mass system to the original total energy.
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217
<image>A chain with mass/length $=u$ hanging vertically from one end, where an upward force $F$ is applied to it, is lowered onto a table as shown in Fig. 1.103. Find the equation of motion for $h$, the height of the end above the table ( $h$ is the length of chain hanging freely).
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218
<image>A thin disk of radius $R$ and mass $M$ lying in the $xy$-plane has a point mass $m=5 M/4$ attached on its edge (as shown in Fig. 1.109). The moment of inertia of the disk about its center of mass is (the $z$-axis is out of the paper). The disk is constrained to rotate about the $y$-axis with angular velocity $\o...
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219
<image>Cylinder C (mass 10.0 kg and radius 0.070 m) rolls without slipping on hill H as shown in Fig. 1.121. The string does not stretch and is wrapped around the cylinder C. How far vertically upward does C move when the 2 kg mass moves down one meter?
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220
<image>Cylinder C (mass 10.0 kg and radius 0.070 m) rolls without slipping on hill H as shown in Fig. 1.121. The string does not stretch and is wrapped around the cylinder C. What are the magnitude and direction of the acceleration?
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<image>Cylinder C (mass 10.0 kg and radius 0.070 m) rolls without slipping on hill H as shown in Fig. 1.121. The string does not stretch and is wrapped around the cylinder C. What are the magnitude and direction of the force of static friction at the contact point P?
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<image>A thin flat rectangular plate, of mass $M$ and sides $a$ by $2a$, rotates with constant angular velocity $\omega$ about an axle through two diagonal corners, as shown in Fig. 1.138. The axle is supported at the corners of the plate by bearings which can exert forces only on the axle. Ignoring gravitational and f...
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223
<image>A homogeneous thin rod of mass $M$ and length $b$ is attached by means of an inextensible cord to a spring whose spring constant is $k$. The cord passes over a very small and smooth pulley fixed at $P$. The rod is free to rotate about $A$ without friction throughout the angular range $-\pi<\theta\leq\pi$. When $...
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224
<image>A uniform solid ball of radius $a$ rolling with velocity $v$ on a level surface collides inelastically with a step of height $h<a$, as shown in Fig. 1.172. Find, in terms of $h$ and $a$, the minimum velocity for which the ball will "trip" up over the step. Assume that no slipping occurs at the impact point, and ...
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225
<image>A rigid square massless frame contains 4 disks rotating as shown in Fig. 1.183. Each disk has mass $m$, moment of inertia $I_{0}$, and rotational velocity $\omega_{0}$. The frame is horizontal and pivots freely about a support at one corner. What is the precession rate? (MIT)
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<image><image>The velocity of sound in the atmosphere is $300 \mathrm{~m}/\mathrm{s}$. An airplane is traveling with velocity $600 \mathrm{~m}/\mathrm{s}$ at an altitude of $8000 \mathrm{~m}$ over an observer as shown in Fig. 1.226. How far past the observer will the plane be when he hears the sonic boom? (Wisconsin)
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<image>A rectangular plate of mass $M$, length $a$ and width $b$ is supported at each of its corners by a spring with spring constant $k$. The springs are confined so that they can move only in the vertical direction. For small amplitudes, find the normal‐modes of vibration and their frequencies. Describe each of the m...
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<image>A negligibly thin piece of metal of mass $m$ in the shape of a square hangs from two identical springs at two corners as shown in Fig. 2.58. The springs can move only in the vertical plane. Calculate the frequencies of vibration of the normal-modes of small amplitude oscillations. (UC, Berkeley)
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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 ma...
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<image>As seen by an inertial observer $O$, photons of frequency $\nu$ are incident, at an angle $\theta_{i}$ to the normal, on a plane mirror. These photons are reflected back at an angle $\theta_{r}$ to the normal and at a frequency $\nu^{\prime}$ as shown in Fig. 3.10. What is the result if the mirror were moving wi...
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<image>As in Fig. 2.15, you are given the not-so-parallel plate capacitor. Neglecting edge effects, when a voltage difference $V$ is placed across the two conductors, find the potential everywhere between the plates.
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<image>A sphere of radius $a$ has a bound charge $Q$ distributed uniformly over its surface. The sphere is surrounded by a uniform fluid dielectric medium with fixed dielectric constant $\varepsilon$ as in Fig. 1.52. The fluid also contains a free charge density given by $\rho(\mathbf{r})=-k V(\mathbf{r})$, where $k$ i...
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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 is ...
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234
<image>In Fig. 2.31 two conductors of infinite length carry a current $I$. They are parallel and separated by a distance $2 a$. A circular conducting ring of radius $a$ in the plane of the parallel wires lies between the two straight conductors and is insulated from them. Find the coefficient of mutual inductance betwe...
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235
<image>Suppose the input voltages $V_{1}$, $V_{2}$, and $V_{3}$ in the circuit of Fig. 3.1 can assume values of either 0 or 1 (0 means ground). There are thus 8 possible combinations of input voltage. Compute $V_{\text{out}}$ for each of these possibilities.
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236
<image>In the circuit shown in Fig. 3.21, the capacitors are initially charged to a voltage $V_{0}$. At $t=0$ the switch is closed. Derive an expression for the voltage at point $A$ at a later time $t$.
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237
<image>The switch S in Fig. 3.24 has been opened for a long time. At time $t=0$, S is closed. Calculate the current $I_{L}$ through the inductor as a function of the time.
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238
<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. Find the instantaneous current $i(t)$ the oscillator must deliver as a function of its frequency.
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239
<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 1: $V_{\mathrm{A}}=4.0$ volts, give $V_{\mathrm{B}}, V_{\mathrm{C}}, V_{\mathrm{E}}$.
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240
<image>A semi-infinite electrical network is formed from condensers $C$ and inductances $L$, as shown in Fig. 3.84. The network starts from the left at the terminals $A$ and $B$; it continues infinitely to the right. An alternating voltage $V_{0}\cos\omega t$ is applied across the terminals $A$ and $B$ and this causes ...
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241
<image>Consider the circuit shown in Fig. 3.87. Find the impedance to a voltage $V$ of frequency $\omega$ applied to the terminals.
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242
<image>What type of radiation is emitted by this system? What is the frequency?
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243
<image>Two large parallel plates (non-conducting), separated by a distance $d$ and oriented as shown in Fig. 5.5, move together along $x$-axis with velocity $v$, not necessarily small compared with $c$. The upper and lower plates have uniform surface charge densities $+\sigma$ and $-\sigma$ respectively in the rest fra...
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244
<image>A particle of mass $m$ and momentum $p$ is incident from the left on the potential step shown in Fig. 1.26. Calculate the probability that the particle is scattered backward by the potential if $p^{2}/2m>V_{0}$.
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245
<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...
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246
<image>A point source $Q$ emits coherent light isotropically at two frequencies $w$ and $w+ \Delta w$ with equal power I joules/sec at each frequency. Two detectors $A$ and $B$ each with a (small) sensitive area $s$, capable of responding to individual photons are located at distances $l_{A}$ and $l_{B}$ from $Q$ as sh...
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247
<image>A cylinder contains a perfect gas in thermodynamic equilibrium at $p, V, T, U$ (internal energy) and $S$ (entropy). The cylinder is surrounded by a very large heat reservoir at the same temperature $T$. The cylinder walls and piston can be either perfect thermal conductors or perfect thermal insulators. The pist...
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248
<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...
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249
<image>The single-particle energies for neutrons and protons in the vicinity of ${ }_{82}^{208} \text{Pb}_{126}$ are given in Fig. 2.13. Using this figure as a guide, estimate or evaluate the following. The spins and parities of the ground state and the first two excited states of ${ }^{207} \text{Pb}$.
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250
<image>It is believed that nucleons $(\mathrm{N})$ interact directly through the weak interaction and that the latter violates parity conservation. One way to study the nature of the $\mathrm{N}-\mathrm{N}$ weak interaction is by means of $\alpha$-decay, as typified by the decays of the $3^{+}, T=1$ and $3^{-}, T=0$ st...
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251
<image>The $\gamma$-ray total nuclear cross section $\sigma_{total}$(excluding $e^+e^-$ pair pro-duction) on neodymium 142 is given in the Figure. Using a simple model of the nucleus as A particles bound in an harmonic oscillator potential, estimate the resonance energy as a function of A. Does this agree with the obse...
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252
<image><image>An object is placed 10 cm in front of a convering lens of focal length 10 cm . A diverging lens of focal length -15 cm is placed 5 cm behind the converging lens (Fig. 1.27). Find the position of the final image.
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253
<image><image>An object is placed 10 cm in front of a convering lens of focal length 10 cm . A diverging lens of focal length -15 cm is placed 5 cm behind the converging lens (Fig. 1.27). Find the size of the final image.
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254
<image>Point charge in the field of a magnetic monopole. The equation of motion of a point electric charge $e$, of mass $m$, in the field of a magnetic monopole of strength $g$ at the origin is $m \ddot{\mathbf{r}} = -g e \frac{\dot{\mathbf{r}} \times \mathbf{r}}{r^{3}}$. The monopole may be taken as infinitely heavy. ...
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255
<image>A perfectly reflecting sphere of radius $r$ and density $\rho=1$ is attracted to the sun by gravity, and repelled by the sunlight reflecting off its surface. Calculate the value of $r$ for which these effects cancel. The luminosity of the sun is $I_{s}=4 \times 10^{33}$ erg/sec and its mass is $M_{s}=2 \times 10...
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256
<image>A spaceship is in a circular orbit of radius $r_{0}$ around a star of mass $M$. The spaceship's rocket engine may be fired briefly to alter its velocity (instantaneously) by an amount $\Delta \mathbf{v}$. The direction of firing is specified by the angle $\theta$ between the ship's velocity $\mathbf{v}$ and the ...
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257
<image>A merry-go-round (carousel) starts from rest and accelerates with a constant angular acceleration of $0.02$ revolution per second per second. A woman sitting on a chair $6$ meters from the axis of revolution holds a $2$ kg ball (see Fig. 1.66). Calculate the magnitude and direction of the force she must exert to...
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258
<image>Take a very long chain of beads connected by identical springs of spring constant $K$ and equilibrium length $a$, as shown in Fig. 1.93. Each bead is free to oscillate along the $x$ direction. All beads have mass $m$ except for one which has mass $m_{0}<m$. The mass of the spring is negligibly small. For a wave ...
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259
<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 $...
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260
<image>A thin disk of radius $R$ and mass $M$ lying in the $xy$-plane has a point mass $m=5 M/4$ attached on its edge (as shown in Fig. 1.109). The moment of inertia of the disk about its center of mass is (the $z$-axis is out of the paper). Find the principal moments and the principal axes about point $A$.
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261
<image>A structure is made of equal‐length beams, 1 to 11, as shown in Fig. 1.112, hinged at the joints $A, B, \ldots, G$. Point $A$ is supported rigidly while $G$ is only supported vertically. Neglect the beam weights. A weight $w$ is placed at $E$. Each member is under pure tension $T$ or compression $C$. Find the te...
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262
<image>A uniform thin rigid rod of mass $M$ is supported by two rapidly rotating rollers, whose axes are separated by a fixed distance $a$. The rod is initially placed at rest asymmetrically, as shown in Fig. 1.114. Now consider the case in which the directions of rotation of the rollers are reversed, as shown in Fig. ...
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263
<image>A coin with its plane vertical and spinning with angular velocity $\omega$ in its plane as shown in Fig. 1.120 is set down on a flat surface. What is the final angular velocity of the coin? (Assume the coin stays vertical; neglect rolling friction.) (Wisconsin)
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264
<image>Find the ratio of the periods of the two torsion pendula shown in Fig. $1.130$. The two differ only by the addition of cylindrical masses as shown in the figure. The radius of each additional mass is $1 / 4$ the radius of the disc. Each cylinder and disc have equal mass. (Wisconsin)
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265
<image>Two equal point masses $M$ are connected by a massless rigid rod of length $2A$ (a dumbbell) which is constrained to rotate about an axle fixed to the center of the rod at an angle $\theta$. The center of the rod is at the origin of coordinates, the axle along the $z$-axis, and the dumbbell lies in the $xz$-plan...
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266
<image>A cone of height $h$ and base radius $R$ is constrained to rotate about its vertical axis, as shown in Fig. 1.141. A thin, straight groove is cut in the surface of the cone from apex to base as shown. The cone is set rotating with initial angular velocity $\omega_{0}$ around its axis and a small (point-like) bea...
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267
<image>A sphere of mass $m$, radius $a$, and moment of inertia $\frac{2}{5} m a^{2}$ rolls without slipping from its initial position at rest atop a fixed cylinder of radius $b$ (see Fig. 1.162). What are the components of the velocity of the sphere's center at the instant it leaves the cylinder? (Wisconsin)
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268
<image>A small homogeneous sphere of mass $m$ and radius $r$ rolls without sliding on the outer surface of a larger stationary sphere of radius $R$ as shown in Fig. 1.165. Let $\theta$ be the polar angle of the small sphere with respect to a coordinate system with origin at the center of the large sphere and $z$-axis v...
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269
<image>A heavy symmetrical top with one point fixed is precessing at a steady angular velocity $\Omega$ about the vertical axis $z$. What is the minimum spin $\omega^{\prime}$ about its symmetrical axis $z^{\prime}$ (where $z^{\prime}$ is inclined at an angle $\theta$ with respect to the $z$-axis)? The top has mass $m$...
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270
<image>A long wave packet with amplitude $A$ composed predominantly of frequencies very near $\omega_{0}$ propagates on an infinitely long string of linear mass density $\mu$ stretched with a tension $T$ as shown in Fig. 1.218. The packet encounters a bead of mass $m$ attached to the string as shown in the sketch. What...
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271
<image>A device consisting of a thin vertical tube and wide horizontal tube joined together in the way shown in Fig. 1.240 is immersed in a fluid of density $\rho_{f}$. The density and pressure of the external atmosphere are $\rho_{a}$ and $p_{a}$ respectively. The end of the horizontal tube is then sealed, and subsequ...
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272
<image>Disc 1 obtains all the kinetic energy when the system, consisting of two identical discs of mass $M$ and radius $R$ supported by three identical torsion bars with restoring torque $\tau=-k \theta$, with displacements $\theta_{1}, \theta_{2}$ from the equilibrium position and initial conditions $\theta_{1}(0)=0$,...
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273
<image>A spherical pendulum consists of a point mass $m$ tied by a string of length $l$ to a fixed point, so that it is constrained to move on a spherical surface as shown in Fig. 2.14. The mass in the circular orbit as in part (a) above receives an impulse perpendicular to its velocity, resulting in an orbit which has...
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274
<image>A simple pendulum is attached to a support which is driven horizontally with time as shown in Fig. 2.25. For small angular displacements and a sinusoidal motion of the support $y=y_{0}\cos(\omega t)$, find the steady-state solution to the equation of motion. (Wisconsin)
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275
<image>A particle moves without friction on the inside wall of an axially symmetric vessel given by $z:=\frac{1}{2} b\left(x^{2}+y^{2}\right)$ where $b$ is a constant and $z$ is in the vertical direction, as shown in Fig. 2.38. The particle in the horizontal circular orbit is poked downwards slightly. Obtain the freque...
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276
<image>Two masses, $2 m$ and $m$, are suspended from a fixed frame by elastic springs as shown in Fig. 2.52. The elastic constant (force/unit length) of each spring is $k$. Consider only vertical motion. The upper mass $2 m$ is slowly displaced downwards from the equilibrium position by a distance $l$ and then let go, ...
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277
<image>A flywheel of moment of inertia $I$ rotates about its center in a horizontal plane. A mass $m$ can slide freely along one of the spokes and is attached to the center of the wheel by a spring of natural length $l$ and force constant $k$ as shown in Fig. 2.64. Suppose the flywheel initially has a constant angular ...
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278
<image>Three point-like masses (two of them equal) and the massless springs (constant $K$) connecting them are constrained to move in a frictionless tube of radius $R$. This system is in gravitational field ($g$) as shown in Fig. 2.65. The springs are of zero length at equilibrium and the masses may move through one an...
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279
<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 ma...
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280
<image>A particle is confined inside a box and can move only along the $x$-axis. The ends of the box move toward the center with a speed small compared with the particle's speed (Fig. 2.75). When the walls are a distance $x$ apart what average external force must be applied to each wall in order to move it at constant ...
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281
<image>Consider the system of particles $m_{1}=m_{2}$ connected by a rope of length $l$ with $m_{2}$ constrained to stay on the surface of an upright cone of half‐angle $\alpha$ and $m_{1}$ hanging freely inside the cone, the rope passing through a hole at the top of the cone. Neglect friction. Give an appropriate gene...
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282
<image>Consider the system of particles $m_{1}=m_{2}$ connected by a rope of length $l$ with $m_{2}$ constrained to stay on the surface of an upright cone of half‐angle $\alpha$ and $m_{1}$ hanging freely inside the cone, the rope passing through a hole at the top of the cone. Neglect friction. Write the Lagrangian of ...
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283
<image>Consider the system of particles $m_{1}=m_{2}$ connected by a rope of length $l$ with $m_{2}$ constrained to stay on the surface of an upright cone of half‐angle $\alpha$ and $m_{1}$ hanging freely inside the cone, the rope passing through a hole at the top of the cone. Neglect friction. Write the Hamiltonian fo...
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284
<image>Consider the system of particles $m_{1}=m_{2}$ connected by a rope of length $l$ with $m_{2}$ constrained to stay on the surface of an upright cone of half‐angle $\alpha$ and $m_{1}$ hanging freely inside the cone, the rope passing through a hole at the top of the cone. Neglect friction. Express the angular freq...
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285
<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. The average energy flux of the incident beam is $P_{i}$ (watts$/\mathrm{m}^{2}$). What is the average reflec...
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286
<image>As seen by an inertial observer $O$, photons of frequency $\nu$ are incident, at an angle $\theta_{i}$ to the normal, on a plane mirror. These photons are reflected back at an angle $\theta_{r}$ to the normal and at a frequency $\nu^{\prime}$ as shown in Fig. 3.10. Find $\theta_{r}$ and $\nu^{\prime}$ in terms o...
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287
<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 charge per unit length on the cylinders when the potential between them is $V$ with the outer cylinder at the higher potential...
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288
<image>A parallel plate capacitor (having perfectly conducting plates) with plate separation $d$ is filled with two layers of material (1) and (2). The first layer has dielectric constant $\varepsilon_{1}$, conductivity $\sigma_{1}$, the second, $\varepsilon_{2}$, $\sigma_{2}$, and their thicknesses are $d_{1}$ and $d_...
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289
<image>In Fig. 1.22, a parallel-plate air-spaced condenser of capacitance $C$ and a resistor of resistance $R$ are in series with an ac source of frequency $\omega$. The voltage-drop across $R$ is $V_{R}$. Half the condenser is now filled with a material of dielectric constant $\varepsilon$ but the remainder of the cir...
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290
<image>As in Fig. 2.15, you are given the not-so-parallel plate capacitor. When this wedge is filled with a medium of dielectric constant $\varepsilon$, calculate the capacitance of the system in terms of the constants given.
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291
<image>As can be seen in Fig. 1.32, the inner conducting sphere of radius $a$ carries charge $Q$, and the outer sphere of radius $b$ is grounded. The distance between their centers is $c$, which is a small quantity. If the potential between the two spheres contains only $l=0$ and $l=1$ angular components, determine it ...
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292
<image>Two large flat conducting plates separated by a distance $D$ are connected by a wire. A point charge $Q$ is placed midway between the two plates, as in Fig. 1.40. Find an expression for the surface charge density induced on the lower plate as a function of $D, Q$ and $x$ (the distance from the center of the plat...
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293
<image>An electric dipole with dipole moment $\mathbf{P}_{1}=P_{1}\mathbf{e}_{z}$ is located at the origin of the coordinate system. A second dipole of dipole moment $\mathbf{P}_{2}=P_{2}\mathbf{e}_{z}$ is located on the $+z$ axis a distance $r$ from the origin. Show that the force between the two dipoles is attractive...
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294
<image>A sphere of radius $a$ has a bound charge $Q$ distributed uniformly over its surface. The sphere is surrounded by a uniform fluid dielectric medium with fixed dielectric constant $\varepsilon$ as in Fig. 1.52. The fluid also contains a free charge density given by $\rho(\mathbf{r})=-k V(\mathbf{r})$, where $k$ i...
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295
<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...
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296
<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...
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297
<image>As in Fig. 1.56, the central plate, bearing total charge $Q$, can move as indicated but makes a gastight seal where it slides on the walls. The air on both sides of the movable plate is initially at the same pressure $p_{0}$. Find value(s) of $x$ where the plate can be in stable equilibrium.
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298
<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 $r, \theta, lpha$. Figu...
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299
<image>An electromagnet is made by wrapping a current carrying coil $N$ times around a C-shaped piece of iron $(\mu \gg \mu_{0})$ as shown in Fig. 2.19. If the cross sectional area of the iron is $A$, the current is $i$, the width of the gap is $d$, and the length of each side of the "C" is $l$, find the $B$-field in t...
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