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0 | [{"role": "user", "question": "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.", "answer": "0.08", "hint": "", "... | |
1 | [{"role": "user", "question": "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 $R$ ?", "answer": "$-10^{4} \\mathrm{~A}/\\mathrm{s}$", "hint": "", "options": {}... | |
10 | [{"role": "user", "question": "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.", "answer": "from $B$ to $A$", "hint": "", "opti... | |
100 | [{"role": "user", "question": "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 $R=R_{0}+R_{1}\\cos(\\omega t)$.", "answer": "all the electric multipole moments are zero.", "hint": "", "options"... | |
1000 | [{"role": "user", "question": "A wire has a conventional current $I$ directed to the right. At the instant shown in the figure, an electron has a velocity directed to the left. What is the direction of the magnetic force on the electron at this instant?", "answer": "Directed toward the bottom of the page.", "hint": "",... | |
1001 | [{"role": "user", "question": "What is the average acceleration of an object moving along a straight line during the time interval from $t=4.0 \\text{ s}$ to $t=9.0 \\text{ s}$, given the velocity vs. time graph?", "answer": "$-1.6 \\text{ m/s}^2$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": ... | |
1002 | [{"role": "user", "question": "For the inclined plane shown, a force of $150$ N applied parallel to the incline's surface is needed to pull the object of mass $M=10.0 \\text{ kg}$ along the entire length of the incline's surface at a constant speed. What is the actual mechanical advantage (AMA) for this scenario?", "an... | |
1003 | [{"role": "user", "question": "For the circuit shown, the four light bulbs have identical resistance, all wires have zero resistance, and the battery is assumed to be ideal with zero resistance added into the circuit. Which light bulb will be dimmer after the switch is closed?", "answer": "Only bulb 2", "hint": "", "op... | |
1004 | [{"role": "user", "question": "A girl twirls a small mass connected to the end of a string counterclockwise in a horizontal circle above her head. If the girl needs the mass to pass through the point labeled P in the figure, at which lettered point on the path should she let go of the string?", "answer": "Point B", "hi... | |
1005 | [{"role": "user", "question": "A mass moves according to the graph of position as a function of time shown below. For which time interval is the instantaneous velocity of the mass always negative?", "answer": "$2.5 \\text{s} < t < 7.5 \\text{s}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": ""... | |
1006 | [{"role": "user", "question": "An acceleration vs. time graph for an object moving along a line is shown. The object starts from rest at time $t=0 \\text{ s}$. At what time does the object attain a maximum displacement from its starting position?", "answer": "$t=10 \\text{ s}$", "hint": "", "options": {}, "choices": []... | |
1007 | [{"role": "user", "question": "A ray of monochromatic light enters a right-triangular glass with an index of refraction of 2.00, surrounded by air. Which lettered ray shows the path of the light after it exits the glass?", "answer": "Ray E", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig... | |
1008 | [{"role": "user", "question": "An object is being pushed at constant speed on an inclined plane. The free body diagram of the object is shown with the gravitational force represented by $W$, the friction force by $f$, the applied external push parallel to the incline by $F$, and the normal force with the surface by $n$... | |
1009 | [{"role": "user", "question": "An object starts at the origin and its velocity along a line vs. time is graphed. What is the proper interval(s) of time for which the object is moving away from the origin?", "answer": "$0 \\text{ s} < t < 7 \\text{ s}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasonin... | |
101 | [{"role": "user", "question": "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 ... | |
1010 | [{"role": "user", "question": "An electron moves at constant non-zero velocity directly between two long straight wires with currents in opposite directions. Ignoring gravity, what are the directions of the magnetic field at the location of the electron?", "answer": "To the right", "hint": "", "options": {}, "choices":... | |
1011 | [{"role": "user", "question": "A magnetic field directed into the plane of the page is decreasing in time. A constant $\\text{emf} \\text{ }\boldsymbol{\\text{ξ}}$ is produced for the square loop enclosing the field in the figure. The square loop has three identical light bulbs of resistance $R$ in it and an ideal volt... | |
1012 | [{"role": "user", "question": "The position vs. time graph of an object moving on a horizontal line is shown. At what time is the object moving with its greatest speed?", "answer": "$t=3.0 \\, \\mathrm{s}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "2", "level": 3, "subject"... | |
1013 | [{"role": "user", "question": "A ray of light is incident onto a thin glass lens as shown. Which arrow indicates the path of the refracted ray?", "answer": "Arrow E", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "", "level": 3, "subject": "OPT", "language_field": "English"}] | |
1014 | [{"role": "user", "question": "The circuit shown has 4 identical light bulbs connected to an ideal battery. If bulb #2 were removed leaving an open circuit there, what happens to the voltage from Q to X?", "answer": "Decrease", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "", "... | |
1015 | [{"role": "user", "question": "A string is connected to a mechanical oscillator on one end and to a cube-shaped mass, $M=8.0 \\text{ kg}$, at the other end as shown. The oscillator vibrates the string with a frequency of $f$ producing the standing wave in the figure on the left. When the mass is submerged completely in... | |
1016 | [{"role": "user", "question": "A mass moves along the $x$-axis following the position vs. time graph provided. During what time intervals does the mass have an acceleration of $0 \frac{\\text{m}}{\\text{s}^{2}}$?", "answer": "From $0 \\text{s} < t < 3 \\text{s}$ and $7 \\text{s} < t < 10 \\text{s}$", "hint": "", "optio... | |
1017 | [{"role": "user", "question": "A metal bar is moving to the left across a set of frictionless conducting rails with a uniform magnetic field directed into the plane of the page. The resistors labeled $X$ and $Y$ are identical. how do the magnitudes of the currents compare?", "answer": "$I_{X} > I_{Y}$.", "hint": "", "o... | |
1018 | [{"role": "user", "question": "A semi-circle has radius $R$, total charge $Q$, and a charge per unit length given as $\\\\lambda=\\\\lambda_{0} \\\\cos \\\\theta$ with $\\\\theta$ defined in the figure and $\\\\lambda_{0}$ a positive constant. What is the electric field strength at the point labeled $P$?", "answer": "$... | |
1019 | [{"role": "user", "question": "Once the circuit shown reaches equilibrium, what is the magnitude of the potential difference across the capacitor?", "answer": "$\\frac{5}{9} \\xi$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "", "level": 3, "subject": "EM", "language_field": "... | |
102 | [{"role": "user", "question": "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 momen... | |
1020 | [{"role": "user", "question": "The capacitor is initially uncharged. What is the current at Point X immediately after the switch is closed?", "answer": "$24$ mA", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "", "level": 3, "subject": "EM", "language_field": "English"}] | |
1021 | [{"role": "user", "question": "A $15.0-\\text{mW}$ laser emits a beam that is 2.00 mm in diameter. What is the intensity of this beam?", "answer": "$75 \\text{μC}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "2", "level": 3, "subject": "EM", "language_field": "English"}] | |
1022 | [{"role": "user", "question": "Shown here is accelerometer data for an elevator ride in a skyscraper in Chicago. The spikes at roughly 10 s and 72 s are caused by the handling of the accelerometer which rested on the floor of the elevator and occurred before the elevator started moving up, and after it reached the top,... | |
1023 | [{"role": "user", "question": "Two balls, A and B, are launched simultaneously from the origin (O) on a level surface at the respective angles of $60^{\\circ}$ and $30^{\\circ}$. Ball A has a launch speed $v$ while Ball B has a launch speed $2v$. Ball A lands $T$ seconds after launch at a horizontal distance $D$ from t... | |
1024 | [{"role": "user", "question": "An amusement park ride called 'The Rotor' has people standing in a vertical cylinder, in this case 6 m in diameter, which rotates and presses them against the wall with sufficient force that the floor can then drop without them sliding down the wall. Assuming the high plateau averages $27... | |
1025 | [{"role": "user", "question": "A triangular glass prism $(\\text{n}=1.6)$ is immersed in a liquid $(\\text{n}=1.1)$ as shown. A laser light is incident on face $AB$ making an angle of $20^{\\circ}$ with the normal $(N)$. What is the angle that the ray emerging from $AC$ makes with the ground when it leaves $AC$ and str... | |
1026 | [{"role": "user", "question": "What is the average speed range of the object for the twenty-eight second interval?", "answer": "Between $6.5$ and $7.0 \\text{ m/s}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "2", "level": 3, "subject": "CM", "language_field": "English"}] | |
1027 | [{"role": "user", "question": "An electrical circuit is constructed with a battery and four resistors as shown in the diagram at the right. What is the current measured by the ammeter shown in the diagram?", "answer": "$2$ A", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "", "sig_figs": "", "le... | |
1028 | [{"role": "user", "question": "如力图1.7.1所示，球1和球2均从同一点水平抛出，起抛点离水平地面的高度为$H$。两球的水平初速分别为$v_{1}$和$v_{2}$（$v_{1}>v_{2}$）。球1抛出后刚好能越过位于$x_{P}$处的坚直杆的顶端，并落于地面上的$R$点，$R$点与$O$点的距离为$R$。球2抛出后落于地面，与地面做弹性碰撞，反弹后也刚好越过杆顶，并落在同一点$R$。试求：杆的位置$x_{P}$；", "answer": "$\\frac{R}{2}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reaso... | |
1029 | [{"role": "user", "question": "如力图1.7.1所示，球1和球2均从同一点水平抛出，起抛点离水平地面的高度为$H$。两球的水平初速分别为$v_{1}$和$v_{2}$（$v_{1}>v_{2}$）。球1抛出后刚好能越过位于$x_{P}$处的坚直杆的顶端，并落于地面上的$R$点，$R$点与$O$点的距离为$R$。球2抛出后落于地面，与地面做弹性碰撞，反弹后也刚好越过杆顶，并落在同一点$R$。试求：杆的高度$h$。", "answer": "$\\frac{3}{4}H$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasonin... | |
103 | [{"role": "user", "question": "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 $L$ ?", "answer": "$I_{R_{1}}=0.25\\,\\mathrm{mA}$ (rightward), $I_{R_{2}}=0.75\\,\\mathrm{mA}$ (upward), $I_{\\varepsilon_{2}}=0.25\\,... | |
1030 | [{"role": "user", "question": "如热图3.19.1所示，在温度为$1000^{\\circ}\\text{C}$的真空加热室中蒸发产生的钨原子（原子量为184）气体经小孔逸出，透过准直窄缝形成原子束后，进入长度为1m的真空大容器中，容器的右壁与原子束垂直。钨原子每行进1m所需的平均时间$\\bar{\\tau}$是多少？", "answer": "$4.9 \\times 10^{-4} \\mathrm{~s}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "$$\n\\bar{\\tau} = \\f... | |
1031 | [{"role": "user", "question": "如热图 3.19.1 所示，在温度为 $1000{ }^{\\text{°} \\text{C}}$ 的真空加热室中蒸发产生的钨原子 （原子量为 9）气体经小孔逸出，透过准直窄缝形成原子束后，进入长度为 1 m 的真空大容器中，容器的右壁与原子束垂直。设钨原子束刚进入真空容器时的粒子数密度为 $n_{0}=10^{10} \\text{~cm}^{-3}$ ，设钨原子与容器壁做完全非弹性碰撞，试估计原钨原子束与容器壁碰撞对后者所施加的压强。", "answer": "$2.6 \\times 10^{-4} \\mathrm{~Pa}$", "hint": "", "op... | |
1032 | [{"role": "user", "question": "如热图 3．26．1 所示，半径为 $r$ 的圆柱形保险丝与电路中的两个电阻块 A 和 B连接。设 A 和 B 与周围环境具有相同的恒定温度 $T_{\\text{c}}$ 。设保险丝中通过稳恒电流 $I$ ，保险丝已达到热稳定状态，设保险丝中任一正截面上的温度都相同，记为 $T_{\\text{c}}+T$ ，不同正截面上的温度不同，若某正截面与 A 相距为 $x$ ，它的温度表示为 $T(x)$ 。设保险丝的电阻率为 $\\rho$ ，设保险丝沿其长度方向 （即 $x$ 方向）的热传导率为 $\\lambda$ ，设保险丝单位时间通过其侧面单位面积向外界散发的热量为 ... | |
1033 | [{"role": "user", "question": "如热图 3.2.1 所示，一块平而薄的长方形钨质玻璃板，用两根质量可略的等长细线悬挂起来，玻璃板两个表面的半个面分别对称地涂了一层化学性质活泼的金属，整个装置放在容器之中，容器内有压强为 p 的氢气。\\n\\n设每一个氢气分子遇到金属分子后发生化学反应的概率为 q(q<1) ，且 q 为恒量，设玻璃板两边的氢气密度相等，设在化学反应过程中，氢气压强的减小可忽略不计，设玻璃板的质量为 m ，有关几何量如热图 3.2.1 所示。\\n\\n观察到玻璃板绕它的坚直轴转过一个很小的角度 α 后，处于平衡状态，试求 α 的大小，", "answer": "$\\frac{q b^... | |
1034 | [{"role": "user", "question": "如热图 3.3.1 所示，在真空中有一个质量为 $M$ 的试管，它被质量为 $m$ 的隔板等分为两部分。隔板封闭的那部分有质量为 $n \\mathrm{~mol}$ ，温度为 $T$ ，摩尔质量为 $\\mu$ 的单原子分子理想气体。放开隔板后，隔板无摩擦地直立向上移动，在隔板离开试管顶端后气体方始从试管逸出，设隔板开始运动时，试管静止。试求试管的最终速度。", "answer": "$\\left[\\frac{3\\left(2^{2 / 3}-1\\right) m n R T}{2^{2 / 3}(n \\mu+M)(m+n \\mu+M)}\\right]^{... | |
1035 | [{"role": "user", "question": "如热图 4.6.1 所示，有一总长度为 $L$ ，粗细均匀的 $U$ 形毛细管，将它的两个开口端坚直向下稍稍浸入两个容器的等高液面之中，因毛细作用，液体 1 和液体 2 分别在毛细管中上升了 $h_{1}$ 和 $h_{2}$ 的高度，设液体 1 和 2 的密度分别为 $\\rho_{1}$ 和 $\\rho_{2}$ ，它们与毛细管壁的接触角均为 $0^{\\text{°}}$ ，设液体在毛细管中上升时管内气体无外漏，大气压强为 $p_{0}$ 。试求液体 1 与液体 2 的表面张力系数的比值", "answer": "$\\frac{\\left(h_{1}+h_{2... | |
1036 | [{"role": "user", "question": "如电图1．14．1 所示，在半径为 $R$ ，质量为 $m$ 且均匀分布的细圆环上，均匀地分布着总电量为 $q$ 的正电荷。在通过圆环中心 $O$ 点且垂直于圆环平面的 $x$ 轴上有两个固定的点电荷，它们分别在圆环的两侧，与环心的距离均为 $L$ ，各带有电量为 $Q$ 的正电荷。设圆环只能沿 $x$ 轴平动，显然，在上述条件下，圆环处于平衡位置。若该位置是稳定平衡位置，试求圆环在其附近做微小振动的周期。", "answer": "$2 \\pi \\sqrt{\\frac{2 \\pi \\varepsilon_{0}\\left(L^{2}+R^{2}\\right... | |
1037 | [{"role": "user", "question": "如电图1.19.1 所示，两个固定的均匀带电球面 A 和 B 分别带电 $4 Q$ 和 $Q(Q$ $>0)$ 。两球心之间的距离 $d$ 远大于两球的半径，两球心的连线 $M N$ 与两球面的相交处都开有足够小的孔，因小孔而损失的电量可以忽略不计。一个带负电的质点静止地放置在 A 球左侧某处 $P$ 点，且在 $M N$ 直线上。设质点从 $P$ 点释放后刚好能穿越三个小孔，并通过 B 球球心。试求质点开始时所在的 $P$ 点与 A 球球心的距离 $x$ 应为多少？", "answer": "$\\frac{2}{9}(\\sqrt{10}-1) d$", "hint"... | |
1038 | [{"role": "user", "question": "如电图 1.29.1 所示，三根等长的带电绝缘细棒首尾相接构成三角形，其中电荷的分布如同绝缘棒都换成等长导体棒且已达到静电平衡时的电荷分布，测得电图1.29.1 中中 $A$ 点和 $B$ 点的电势分别为 $U_{A}$ 和 $U_{B}$ 。假设若将 $a b$ 棒取走，并不影响 $a c$ 及 $b c$ 两棒的电荷分布，试求此时 $A$ 点和 $B$ 点的电势，", "answer": "$\\frac{2}{3} U_A; \\frac{1}{6} U_A + \\frac{1}{2} U_B$", "hint": "", "options": {}, "choi... | |
1039 | [{"role": "user", "question": "如电图1.2.1 所示，在边长为 $a$ 的正方形的四个顶点分别有电量均为 $Q$ 的固定的点电荷。在正方形对角线交点上放置一个质量为 $m$ ，电量为 $q$（ $q$ 与 $Q$ 同号）的自由点电荷。今将 $q$ 沿某一对角线移动一个很小的距离，试问：$q$ 是否将做周期性振动？若是，试求出振动周期。", "answer": "$\\pi\\sqrt{\\frac{2\\sqrt{2}\\pi\\varepsilon_0 m a^3}{Qq}}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "r... | |
104 | [{"role": "user", "question": "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}$?", "answer": "For $\\theta=0^{\\circ}$, no radiation is emitted; for $\\theta=90^{\\... | |
1040 | [{"role": "user", "question": "如电图 1.30 .1 所示，恒温的矩形箱内装有理想气体，当隔板将箱等分为二时，两侧气体压强均为 $P_{0}$ 。当隔板平行移动时，无摩擦，不漏气，两侧气体经历准静态过程。隔板是面积为 $A$ 的金属板，带电量为 $Q$ ，矩形箱上与它平行的两块板也是金属板，面积也为 $A$ ，相距为 $2 L$ ，固定，并接地。隔板两侧的电场均匀。箱的其余部分是不导电的绝缘板。试求隔板的平衡位置。", "answer": "$x=0，x = \\pm L\\cdot\\sqrt{1 - \\frac{4\\varepsilon_0 p_0 A^2}{Q^2}}$", "hint": ... | |
1041 | [{"role": "user", "question": "如电图 1.31.1 所示，在半径为 $R$ 的接地金属圆柱面的中央放有一根半径为 $r_{0}$ 的同轴细长导线，导线处于正的高电势 $U_{0}$ 。于是导线外侧附近介质原子被电离成自由电子与正离子，其中自由电子即被导线吸附，正离子则离开导线径向地运动。设正离子的径向迁移率（径向速度与电场强度的比值）为常数 $w$ ，且迁移过程中正离子始终围绕导线形成均匀的圆柱形薄层。 2.设全部正离子电量为 $Q$ ，为了使导线电势保持原来的 $U_{0}$不变，需给导线补充电量 $Q^{*}$ 。试导出 $Q^{*}$ 与时间 $t$ 的关系。", "answer": "$\\f... | |
1042 | [{"role": "user", "question": "如电图1．33．1 所示，两块足够大的接地导体平板 A 和 B 平行坚直放置，相距 $2 d, d=10 \\text{~cm}$ 。在两板之间的中间位置，用长 $l=1 \\text{~m}$ 的绝缘细线悬挂一个质量 $m=0.1 \\text{~g}$ ，电量 $q=5 \\times 10^{-9} \\text{C}$ 的小摆球。让小摆球稍稍偏离平衡位置后释放，使之小角度摆动。忽略各种电磁阻尼和空气阻力。试求小球的摆动周期 $T$ 。", "answer": "$2.3 \\mathrm{~s}$", "hint": "", "options": {}, "cho... | |
1043 | [{"role": "user", "question": "如电图1.3.1 所示，在 $x$ 轴的 $a$ 处分别有两个固定的点电荷，它们的电量均为 $Q(Q>0)$。在 $x$ 轴的 $2a$ 处分别有另外两个固定的点电荷，它们的电量均为 $-8Q$。在原点 $O$ 有一个电量为 $q(q>0)$，质量为 $m$ 的带电质点。设带电质点只能在 $x$ 轴上运动，试分析带电质点所在的平衡位置原点 $O$，是否为稳定平衡位置，若原点 $O$ 为稳定平衡位置，将带电质点移到 $x=A$ 处从静止释放 $(0<A<a)$，试求其振动周期。\n", "answer": "$\\frac{7.4164}{A} \\sqrt{\\frac{... | |
1044 | [{"role": "user", "question": "一无限长均匀带电细线弯成如电图 1.6.1 所示的平面图形，其中 $A B$ 是半圆弧， $A A^{\\prime}$ 和 $B B^{\\prime}$ 是两平行直线，$A^{\\prime}$ 和 $B^{\\prime}$ 向右端无限延伸。试求圆心 $O$ 处的电场强度。", "answer": "0", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【解】本题求电场强度，然而未给出电荷线密度及圆半径等通常必不可少的已知条件，暗示所求$O$点的场强应为零。\n\n均匀带... | |
1045 | [{"role": "user", "question": "如电图 2.16.1 所示，在螺绕环的平均半径 $R$ 处有点源 $P$ ，从 $P$ 点沿磁力线方向注入小孔径角为 $2 \\alpha_{0}\\left(\\alpha_{0} \\ll 1^{\\circ}\\right)$ 的电子束，束中电子都是经电压 $U_{0}$ 加速从 $P$ 点出发的。设绕环中磁场 $B$ 的大小为常量，电子束中各电子间的相互作用可以忽略。为了使电子束沿环形磁场运动，需要另外加一个使电子束偏转的均匀磁场 $B_{1}$ ，对于在环内沿半径为 $R$ 的圆形轨道运动的一个电子，试计算所需的 $B_{1}$ 的大小。", "answer":... | |
1046 | [{"role": "user", "question": "如电图 2.16.1 所示，在螺绕环的平均半径 $R$ 处有点源 $P$ ，从 $P$ 点沿磁力线方向注入小孔径角为 $2 \\alpha_{0}\\left(\\alpha_{0} \\ll 1^{\\circ}\\right)$ 的电子束，束中电子都是经电压 $U_{0}$ 加速从 $P$ 点出发的。设绕环中磁场 $B$ 的大小为常量，电子束中各电子间的相互作用可以忽略。当电子束沿环形磁场运动时，为了使电子束绕一圈有四个聚焦点，即如电图 2.16 .1 所示，每绕过 $\\pi / 2$ 的圆周角聚焦一次，试问：$B$ 应为多大？（此处可忽略 $\\boldsymbo... | |
1047 | [{"role": "user", "question": "如电图 2.1.1 所示，电流强度为 $2 I$ 的直流电经半无限长导线到达半径为 $R$ 的金属球面的南极点，由球面流到北极点后，再从另一半无限长直导线流向远处。设想将上述电流分布切成左，右两半。左半部分为半无限长直电流 $I$ 到半球面南极点后，经左半球面（注意，此半球面的电流分布与原全球面左半的电流分布相同）流向北极点，再经半无限长直导线流向远处，并设置 $O x y z$ 坐标，如图 2.1.2 所示。试求球心 $O$ 处的磁感应强度 $B$ 。", "answer": "$\\frac{\\mu_{0} I}{2 \\pi R}$", "hint": "", "... | |
1048 | [{"role": "user", "question": "如电图 2．21．1 所示，半径为 $R$ ，质量为 $m$ 的均质细圆环均匀带电，总电量为 $Q(Q>0)$ ，放在光滑的水平面上．环内，外有垂直环面向上的均匀磁场 $\\boldsymbol{B}$ 。若将圆环以角速度 $\\omega$ 绕着通过圆心的坚直轴均速旋转，试求环内因为这种转动而形成的附加张力，", "answer": "$\\frac{R \\omega}{2 \\pi}(Q B+m \\omega)$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "如电图... | |
1049 | [{"role": "user", "question": "在外磁场中的超导体，平衡后超导体内部的磁感应强度处处为零，超导体表面外侧的磁感应强度与表面平行。在如电图 2.24.1 所示 $O x y z$ 直角坐标系中，$x y$ 平面是水平面，其中有一超导平板，$z$ 轴竖直向上，超导平板在 $z=0$ 处，在 $z=h$ 处有一质量为 $m$ ，半径为 $r$ ，环心在 $z$ 轴上，环平面为水平面的匀质金属圆环，且 $r$ $\\text{gg} h$ 。在圆环内通以稳恒电流，刚好使圆环漂浮在 $z=h$ 处，试求圆环中的电流强度。", "answer": "$\\sqrt{\\frac{2 m g h}{\\mu_{0} ... | |
105 | [{"role": "user", "question": "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 ... | |
1050 | [{"role": "user", "question": "在外磁场中的超导体，平衡后超导体内部的磁感应强度处处为零，超导体表面外侧的磁感应强度与表面平行。在如电图 2.24.1 所示 $O x y z$ 直角坐标系中，$x y$ 平面是水平面，其中有一超导平板，$z$ 轴竖直向上，超导平板在 $z=0$ 处，在 $z=h$ 处有一质量为 $m$ ，半径为 $r$ ，环心在 $z$ 轴上，环平面为水平面的匀质金属圆环，且 $r$ $\\text{gg} h$ 。在圆环内通以稳恒电流，刚好使圆环漂浮在 $z=h$ 处，若使圆环保持水平，从平衡位置稍稍偏上或偏下，则圆环将上，下振动。试求振动周期 $T_{1}$ 。", "answer... | |
1051 | [{"role": "user", "question": "在外磁场中的超导体，平衡后超导体内部的磁感应强度处处为零，超导体表面外侧的磁感应强度与表面平行。在如电图 2.24.1 所示 $O x y z$ 直角坐标系中，$x y$ 平面是水平面，其中有一超导平板，$z$ 轴竖直向上，超导平板在 $z=0$ 处，在 $z=h$ 处有一质量为 $m$ ，半径为 $r$ ，环心在 $z$ 轴上，环平面为水平面的匀质金属圆环，且 $r$ $\\text{gg} h$ 。在圆环内通以稳恒电流，刚好使圆环漂浮在 $z=h$ 处，当圆环处在平衡位置时，其中与 $x$ 轴平行的直径标为 $P_{1} P_{2}$ ，与 $y$ 轴平行的直径标为 ... | |
1052 | [{"role": "user", "question": "磁流体发电机的工作原理如电图 3.10.1 所示。横截面为矩形的管道长为 $l$，宽为 $a$，高为 $b$，上、下两个侧面是绝缘体，相距为 $a$ 的两个侧面是电阻可略的导体，这两个侧面与负载电阻 $R_{L}$ 相连。整个管道处于匀强磁场区域，$\\boldsymbol{B}$ 垂直于上、下侧面指向上。管道内沿长度方向流有电阻率为 $\\rho$ 的电离气体，气体流速处处相同，所受摩擦阻力的大小与流速成正比。现在管的两端维持恒定的压强差 $p$，无磁场存在时气体的流速为 $v_{0}$。试求有磁场存在时，此发电机的电动势 $\\mathcal{E}$。", "answ... | |
1053 | [{"role": "user", "question": "如电图 3.12.1 所示，由铜质细导线弯成的半径为 $a$ 的圆形圈和一内接等边三角形电阻丝构成的电路，电路中各段电阻值已在电图 3.12.1 中标明。在圆形圈所在平面内有垂直纸面向里的铜强磁场 $B, B$ 随时间 $t$ 均匀地减小，其变化率的大小为已知常量 $k$ ，设 $2 r_{1}=3 r_{2}$ 。试求电图 3.12.1 中 $A, B$ 两点之间的电势差 $U_{A B}$ 。", "answer": "$-\\frac{\\sqrt{3}}{32} k a^{2}$", "hint": "", "options": {}, "choices": []... | |
1054 | [{"role": "user", "question": "如电图 3.1.1 所示，在边长为 $a$ 的等边三角形区域内有匀强磁场 $B$，其方向垂直纸面向外。一个边长也为 $a$ 的等边三角形导体框架 $ABC$，在 $t=0$ 时恰好与上述磁场区域的边界重合，尔后以周期 $T$ 绕其中心在纸面内顺时针方向匀速转动，于是在框架 $ABC$ 中产生感应电流。规定电流按 $ABCA$ 方向流动时电流强度取正值，反方向流动时取负值。设框架 $ABC$ 的电阻为 $R$。试求从 $t=0$ 到 $t_{1}=T/6$ 时间内的平均电流强度 $\\bar{I}_{1}$ 和从 $t=0$ 到 $t_{2}=T/2$ 时间内的平均电流强度... | |
1055 | [{"role": "user", "question": "如电图3．3．1 所示，铜制圆环的两个半径分别为 $r_{1}=1 \\text{ cm}$ 和 $r_{2}=0.1 \\text{ mm}$ 。圆环竖直放在地面上，环底部有固定的光滑杆限制，使其不能滑动。圆环周围有竖直向上的均匀的强磁场 $B=1.0 \\text{ T}$ 。如电图3．3．2所示，开始时圆环偏离竖直方向一个小角度 $\\theta_{0}=0.1 \\text{ rad}$ ，尔后圆环倒向地面。已知铜的电导率 $\\rho=8.93 \\times 10^{3} \\text{ kg/m}^{3}$ 。设圆环倒下过程中所受的磁力矩始终等于重力矩，试求... | |
1056 | [{"role": "user", "question": "如电图 3.4.1 所示，在水平桌面上平放着长方形线圈 $a b c d$ 。已知 $a b$ 边长 $l_{1}, b c$边长 $l_{2}$ ，线圈总电阻为 $R, a b$ 边正好指向北方。现将线圈以南北连线为轴翻转 $180^{\\text{o}}$ ，使 $a b$与 $c d$ 边互换位置，在翻转的全过程中测得通过导线的总电量为 $Q_{1}$ 。然后，维持 $a d$ 边（东西方向）不动，将线圈绕 $a d$ 边转动 $90^{\\text{o}}$ ，使之坚直，测得在坚直过程中流过导线的总电量为 $Q_{2}$ 。试求该处地磁场 $B$ 的大小，", "... | |
1057 | [{"role": "user", "question": "如电图 3.5.1 所示为一“日”字形矩形闭合导线框。已知 $a b=b c=c d=d e=e f$ $=f a=0.1 \\text{ m}$ ，已知 $a b, c f, d e$ 段电阻均为 $3 \\text{ }\\text{Ω}, c d, f e$ 段电阻均为 $1.5 \\text{ }\\text{Ω}, b c, a f$ 段电阻均为零。匀强磁场 $B$ 的方向与框面垂直向里，大小为 $B=1 \\text{ T}$ ，磁场的边界与 $d e$ 平行，如电图 3.5.1 中虚线所示。今以图中向右的方向，以 $v=24 \\text{ m/s}$ 的速... | |
1058 | [{"role": "user", "question": "如电图 3.7.1 所示，有一水平方向的匀强磁场，磁感应强度 $B$ 很大。一个位半径为 $R$ ，厚度为 $D(D \\\\ll R)$ 的金属圆盘，在此磁场中竖直下落，盘面始终在坚直平面内并与磁场 $B$ 的方向平行，设金属圆盘的电阻为零，密度为 $\\rho=9 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ ，所受空气阻力可略。为使圆盘在磁场中下落的加速度比没有磁场时减小千分之一，试问 ：$B$ 应为多大？", "answer": "$10^{6} \\mathrm{~T}$", "hint": "", "optio... | |
1059 | [{"role": "user", "question": "如电图 3.8.1 所示，宽为 $L$ 的长薄导体平板沿 $x$ 轴水平放置，平板的电阻可以忽略不计。电图 3.8.1 中的 aebcfd 为圆弧形均匀导线，电阻为 $3 R$ ，圆弧所在平面与 $x$ 轴垂直，圆弧的两端 $a$ 和 $d$ 与导体平板的两侧边相接触，并可沿侧边自由滑动，圆弧 $a e, e b, c f$ ，$f d$ 均为 $1 / 8$ 圆周，圆弧 $b c$ 为 $1 / 4$ 圆周。一个内阻为 $R_{V}=n R$ 的小体积电压表位于圆心 $O$处，电压表的两端分别用理想导线与 $b$ 点和 $c$ 点连接，整个装置处在匀强磁场区域，$B$ ... | |
106 | [{"role": "user", "question": "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?", "answer": "$\\dot{x}_{c}=-\\frac{1}{3}\\omega R$", "hint": "", "options": {}, "choices": [], "n_im... | |
1060 | [{"role": "user", "question": "如电图 3.8.1 所示，宽为 $L$ 的长薄导体平板沿 $x$ 轴水平放置，平板的电阻可以忽略不计。电图 3.8.1 中的 aebcfd 为圆弧形均匀导线，电阻为 $3 R$ ，圆弧所在平面与 $x$ 轴垂直，圆弧的两端 $a$ 和 $d$ 与导体平板的两侧边相接触，并可沿侧边自由滑动，圆弧 $a e, e b, c f$ ，$f d$ 均为 $1 / 8$ 圆周，圆弧 $b c$ 为 $1 / 4$ 圆周。一个内阻为 $R_{V}=n R$ 的小体积电压表位于圆心 $O$处，电压表的两端分别用理想导线与 $b$ 点和 $c$ 点连接，整个装置处在匀强磁场区域，$B$ ... | |
1061 | [{"role": "user", "question": "如电图 3．9．1 所示，水平与垂直的两根长直导体棒相交，连接成固定不动的十字架形状．由边长为 $a$ 的四根导体棒构成的正方形框架，从电图3．9．1 中的实线框架位置开始，以速率 $v$ 匀速向左运动，在运动过程中框架始终与十字架接触．空间有均匀磁场 $B$ ，其方向如电图 3．9．1 所示，垂直于十字架和框架所在的平面．设所有导体棒单位长度的电阻为 $r=100 \\Omega / \\mathrm{m}, a=0.1 \\mathrm{~m}, v=0.24 \\mathrm{~m} / \\mathrm{s}, B=10^{-4} \\mathrm{~T}$ ．取... | |
1062 | [{"role": "user", "question": "三个相同的均匀金属圆圈两两相交地连接成如电图 4.10.1 所示的网络，已知每一个金属圆圈的电阻都是 $R$ 。试求电图 4.10.1 中 $A, B$ 两点之间的等效电阻 $R_{A B}$ 。", "answer": "$\\frac{5}{48} R$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【分析与解】对于如电图4.10.1所示的以$A$, $B$为端点的两端电阻网络，若电流从$A$点流入，从$B$点流出，由于网络的对称性，不难看出，上、下两半（指包括$A$和... | |
1063 | [{"role": "user", "question": "如电图 4.12.1 所示是由电阻丝连接成的无限电阻网络，已知每一段电阻丝的电阻均为 $r$ 。试求 $A, B$ 两点之间的等效电阻 $R_{A B}$ 。", "answer": "$\\frac{2 \\sqrt{21}}{21} r$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【分析与解】由于电图4.12.1网络的对称性，若电流从$A$点流入，经网络后，从$B$点流出，则电图4.12.1网络中背面那根无限长的电阻丝内应无电流通过，不起作用，可拆去。于是电图4.1... | |
1064 | [{"role": "user", "question": "由 7 个阻值均为 $r$ 的电阻组成的网络元如电图 4.13.1 所示。由这种网络元彼此连接形成的无限梯形网络如电图 4.13.2 所示。试求 $P, Q$ 两点之间的等效电阻，", "answer": "1.32 r", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【解】设 $K$ 个网络元连成的梯形网络的等效电阻为 $R_{K}$，再连一个网络元后等效电阻为 $R_{K+1}$，如电图 4.13.3 所示。它又可简化为如电图 4.13.4 所示的电路，且有\n$$\nR... | |
1065 | [{"role": "user", "question": "用同种均匀金属丝连接成的无限内接等边三角形电阻网络如图 4.14.1 所示，每个外三角形的三边中点为内接三角形的三个顶点。设最外面的等边三角形的边长为 $a$ ，金属丝单位长度的电阻为 $r$ ，试求 $A, B$ 之间的等效电阻 $R_{A B}$ 。", "answer": "$\\frac{1}{3}(\\sqrt{7}-1) a r$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【分析与解】如电图 4.14.2 所示，网络相对图中的虚直线 $M O N$ 具有左右... | |
1066 | [{"role": "user", "question": "10 根电阻均为 $R$ 的电阻丝连接成如电图 4.15.1 所示的电阻网络，试求 $A, B$两点之间的等效电阻 $R_{A B}$ 。", "answer": "$\\frac{15}{8} R$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【分析与解】设电流 $I$ 从 $A$ 点流入，经网络后，由 $B$ 点流出。根据网络的对称性以及电流经交叉点的分流关系，可以确定各支路中的电流如电图 4.15.2 所示。\n\n由对称性，网络中间水平方向的两电流 $I_{2}$... | |
1067 | [{"role": "user", "question": "如电图 4.16.1 所示的电阻丝网络包含 $N \\geqslant 3$ 个正方形，网络中每一小段的电阻均为 $R$ 。试求 $A, B$ 之间的等效电阻 $R_{A B}$ 。", "answer": "$\\left[\\frac{N}{2}+\\frac{1}{3-\\frac{\\alpha_{1}^{N-2}-\\alpha_{1}^{2-N}}{\\left(\\alpha_{1}^{N-1}-\\alpha_{1}^{1-N}\\right)-2 \\sqrt{3}}}\\right] R$", "hint": "", "options": {}, "c... | |
1068 | [{"role": "user", "question": "电阻丝网络如电图 4.17.1 所示，每一小段的电阻均为 $R$ 。试求 $A, B$ 之间的等效电阻 $R_{A B}$ 。", "answer": "$\\frac{29}{24} R$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【解】如电图 4.17.2 所示，电流 $I$ 从 $A$ 流入，从 $O$ 流出。因对称地分流，从 $A$ 到 $C$ 的电流应为\n$$\nI_{1}^{\\prime} = \\frac{I}{2}\n$$\n\n又因对称，$B$ 和... | |
1069 | [{"role": "user", "question": "如电图 4.21.1 所示，由电阻丝构成的网络中，每一段电阻丝的电阻均为 $R$ ，试求 $R_{A B}$ 。", "answer": "$\\frac{47}{22} R$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【解】由于所求 $R_{A B}$ 对应的 $A, B$ 两点刚好是网络的对角线，利用对角线 $A B$ 两侧电路的对称性，可将电路以 $A B$ 为边\"折叠\"起来（如同折纸一样），这样做当然不会改变电路的任何性质。折叠后，每一段由原先单一的 $R$... | |
107 | [{"role": "user", "question": "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. ... | |
1070 | [{"role": "user", "question": "如电图 4.22.1 所示的立方体网络中，每一小段直线的电阻均为 $R$ ，试求 $R_{P Q}$ 。", "answer": "$\\frac{24}{35} R$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【解】电图 4.22.1 电路的特点是上下对称，故可\"挤压\"成电图 4.22.2 的等效电路。在电图 4.22.2 中用双线画出的每段的电阻等于两个 $R$ 的并联，即为 $R/2$，用单线画出的每段的电阻仍为 $R$。在电图 4.22.2 中，对于 $P,... | |
1071 | [{"role": "user", "question": "\\n在如电图 4．24．1 所示的电路中，各个电阻的阻值都是 $1 \\Omega$ ，安培计与电源的内阻均可忽略不计，电源的电动势 $\\mathscr{E}=10 \\mathrm{~V}$ 。若电图 4．24．1 中的两个 $r$ 是好电阻，而 $r_{1}, r_{2}, r_{3}$ 是三个坏电阻，它们互不相关地时通，时断，已知 $r_{1}, r_{2}, r_{3}$ 各自通电的概率都是 $1 / 2$ 。试求电源的平均输出功率。", "answer": "$52.5 \\mathrm{~W}$", "hint": "", "options": {}, "... | |
1072 | [{"role": "user", "question": "有若干个电阻构成如电图 4．29．1 所示的电路，其中 $A$ 和 $B$ 两点的接地电阻是固定不动的，输入人电压 $V_{1}, V_{2}, \\\\cdots, V_{n}$ 仅取 1 V 或 0 V 两个值， 0 V 表示接地。试问：$B$ 点的最大输出电压是多少？", "answer": "$\\frac{2}{3}\\left(1-\\frac{1}{2^{n}}\\right) \\mathrm{V}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【分析与解... | |
1073 | [{"role": "user", "question": "如电图 4.2.1 所示，圆柱形电阻器的截面积为 $S$ ，长为 $2 L$ ，电导率为\\n\\[\\n\\sigma=\\left\\{\\begin{array}{ll}\\n\\sigma_{0}\\left(1+\\frac{x}{L}\\right), & 0<x<L \\\\\\n\\sigma_{0}\\left(1+\\frac{2 x}{L}\\right), & L<x<2 L\\n\\end{array}\\right.\\n\\]\\n\\n在电阻器两端面之间加电压 $\\Delta U$ ，电流恒定均匀地从一个端面流入，从另一个端面流出。试求：... | |
1074 | [{"role": "user", "question": "如电图4．31．1 所示，左边的三端电容网络为 $\\triangle$ 形网络元，右边的三端电容网络为 $Y$ 形网络元，利用上述等效变换，试计算电图4．31．2 所示电容网络中 $A$ 和 $B$ 两点之间的等效电容 $C_{A B}$ ，电图4．31．2 网络中各电容已用数字标明，单位均为 $\\mu \\mathrm{F}$ 。", "answer": "$6 \\mu \\mathrm{F}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "利用上述$\\mathr... | |
1075 | [{"role": "user", "question": "在如电图4.33.1所示的电容网络中：\n电容值：$C_1=4C_0$，$C_2=2C_0$，$C_3=C_0$；电源电动势$\\mathcal{E}=8$（内阻可忽略）。\n\n操作步骤：\n1. 断开$K_4$，接通$K_1,K_2,K_3$充电\n2. 断开$K_1,K_2,K_3$，接通$K_4$放电\n\n求放电过程中电阻$R$上产生的总热量$Q$。", "answer": "$\\frac{2}{7} C_{0} \\varepsilon^{2}$", "hint": "", "options": {}, "choices": [], "n_images": ... | |
1076 | [{"role": "user", "question": "在如电图4.33.1所示的电容网络中：\n电容值：$C_1=4C_0$，$C_2=2C_0$，$C_3=C_0$；电源电动势$\\mathcal{E}=8$（内阻可忽略）。\n\n操作步骤：\n1. 断开$K_4$，接通$K_1,K_2,K_3$充电\n2. 断开$K_1,K_2,K_3$，接通$K_4$放电\n\n求放电量达总放电量一半时，流经$R$的电流$I$。", "answer": "$\\frac{\\varepsilon}{2 R}$", "hint": "", "options": {}, "choices": [], "n_images": 1, "rea... | |
1077 | [{"role": "user", "question": "由 $n$ 个网络元组成的电容网络如电图4.34.1 所示，每一个网络元由三个电容器连接而成，其中上，下两个电容器的电容均为 $3 C$ ，右侧电容器的电容均为 $2 C$ 。电图 4.34 .1中的 $a$ 和 $b$ 点是网络的输入端，$a^{\\prime}$ 和 $b^{\\prime}$ 点是网络的输出端，在 $a$ 和 $b$ 之间有恒定的输入人电压 $U$ ，在 $a^{\\prime}$ 和 $b^{\\prime}$ 之间连接一个电容为 $C$ 的电容器。试求从第 $k(k<n)$ 个网络元的输入端算起，后面所有电容器贮存的总电能。", "answer"... | |
1078 | [{"role": "user", "question": "由 $n$ 个网络元组成的电容网络如电图4.34.1 所示，每一个网络元由三个电容器连接而成，其中上，下两个电容器的电容均为 $3 C$ ，右侧电容器的电容均为 $2 C$ 。电图 4.34 .1中的 $a$ 和 $b$ 点是网络的输入端，$a^{\\prime}$ 和 $b^{\\prime}$ 点是网络的输出端，在 $a$ 和 $b$ 之间有恒定的输入人电压 $U$ ，在 $a^{\\prime}$ 和 $b^{\\prime}$ 之间连接一个电容为 $C$ 的电容器。先将第 1 个网络元的输出端与其右侧的网络断开，再除去电源，并把它的输入端 $a$和 $b$ 连接起... | |
1079 | [{"role": "user", "question": "如电图 4.35.1 所示， 12 根电阻均为 $R$ 的电阻丝连接成正六面体（立方体）。在两根电阻丝中连接有电动势分别为 $\\theta_{1}$ 和 $\\theta_{2}$ 的电源，另 5 根电阻丝中连接有 5 个相同的电容器 $C$ 。设电源正、负极之间的距离均可忽略，内阻亦可略，且 $\\theta_{1}=2 I_{0} R, \\theta_{2}=I_{0} R$ 。试求： $A B$ 中的电流 $I_{A B}$ 。", "answer": "$\\frac{3}{5} I_{0}$", "hint": "", "options": {}, "cho... | |
108 | [{"role": "user", "question": "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 $I$... | |
1080 | [{"role": "user", "question": "如电图4.35.1所示，12根电阻均为$R$的电阻丝连接成正六面体（立方体）。在两根电阻丝中连接有电动势分别为$\\mathscr{E}_{1}$和$\\mathscr{E}_{2}$的电源，另5根电阻丝中连接有5个相同的电容器$C$。设电源正、负极之间的距离均可忽略，内阻亦可略，且$\\mathscr{E}_{1}=2I_{0}R$, $\\mathscr{E}_{2}=I_{0}R$。试求：$A^{\\prime}B^{\\prime}$中电容器极板上的电量$Q_{A^{\\prime}B^{\\prime}}$。", "answer": "$\\frac{3}{5}... | |
1081 | [{"role": "user", "question": "如电图4.4.1所示介质分布：\n\n$0\\leq r<R_1$区域：介电常数$\\varepsilon_1$，电导率$\\sigma_1$；\n\n$R_1<r<R_2$区域：介电常数$\\varepsilon_2$，电导率$\\sigma_2$；\n\n$r>R_2$区域：真空。\n\n初始自由电荷分布：\n$$\n\\rho_e(r,0)=\\begin{cases}\n\\rho_{c0}, & 0\\leq r<R_1 \\\\\n0, & R_1<r<R_2\n\\end{cases}\n$$\n且球面$R_1$、$R_2$处$\\sigma_{e1}=\\... | |
1082 | [{"role": "user", "question": "将阻值分别为 $1 \\Omega, 2 \\Omega, 3 \\Omega, 4 \\Omega, 5 \\Omega, 6 \\Omega$ 的 6 个电阻连接成如电图 4.7.1 所示的电路。若已测得 $R_{X Y}=7 \\frac{3}{13} \\Omega, R_{Y Z}=10 \\frac{1}{13} \\Omega, R_{Z X}=6 \\frac{9}{13} \\Omega$ 。试求电图 4.7.1 中 6 个电阻 $a, b, c, d, e, f$ 各自的阻值。", "answer": "$1 \\Omega; 3 \\Omega; 4... | |
1083 | [{"role": "user", "question": "正四面体框架形电阻网络如电图 4.9.1 所示，其中每一小段的电阻均为 $R$ 。试求： $A, B$ 两端点之间的等效电阻 $R_{A B}$ 。", "answer": "$\\frac{3}{4} R$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "【解】1.如电图 4.9.1 所示，设电流 $i$ 从 $A$ 点流入，从 $B$ 点流出。鉴于网络的对称性，在电图 4.9.1 所示 $D$ 点聚集的 6 条支路的电流之间应满足下述关系：\n$$\ni_{1}=i_{2... | |
1084 | [{"role": "user", "question": "正四面体框架形电阻网络如电图 4.9.1 所示，其中每一小段的电阻均为 $R$ 。试求： $C, D$ 两端点之间的等效电阻 $R_{C D}$ 。", "answer": "$\\frac{3}{8} R$", "hint": "", "options": {}, "choices": [], "n_images": 1, "reasoning": "2.如电图 4.9.1 所示，设电流从 $C$ 点流入，从 $D$ 点流出，则网络相对 $A B D$ 平面具有对称性，与 $A B$ 栏平行的中间小正方形四个顶点等势，故此正方形的四条边都可拆去，余下部分相对 $A B ... |
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