problem_id stringlengths 6 8 | discipline stringclasses 2
values | image stringlengths 17 19 ⌀ | problem dict | answer stringlengths 1 1.36k |
|---|---|---|---|---|
PHY-016 | Physics | null | {
"en": "The refractive index of the glass plates in the glass plate stack is \\(n = 1.48\\), and it is placed in air. Natural light is incident at the Brewster angle. In order to make the degree of polarization of the polarized light emerging from the glass plate stack reach 98%, how many glass plates are required a... | 18 |
PHY-017 | Physics | null | {
"en": "A particle starts from the origin (-a, 0) and is only acted upon by a central potential field V(r). Now, we hope to construct an orbit such that the particle can reach the point (0, b) in the shortest time on this orbit. If the parametric equation of this observed orbit is \\(r = \\frac{p}{1 + e\\cos\\theta}... | V(r) =$ V_1 - \frac{(V_1-V_2)*(1-e^2)}{1-e^2+\frac{2*r*e^2}{p}}$ |
PHY-018 | Physics | null | {
"en": "A mixture of \\(n\\)-dimensional ideal gases is in an equilibrium state at temperature \\(T\\). The relative velocity between any two molecules with masses \\(m_1\\) and \\(m_2\\) is defined as \\(\\boldsymbol{u}=\\boldsymbol{v}_1 - \\boldsymbol{v}_2\\), where \\(\\boldsymbol{v}_1\\) and \\(\\boldsymbol{v}_2... | \sqrt{\overline{u^2}}=\sqrt{\frac{nkT(m_1+m_2)}{m_1m_2}},\overline{u}=\sqrt{\frac{2kT(m_1+m_2)}{m_1m_2}}\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}. |
PHY-019 | Physics | null | {
"en": "The \\(H_{\\alpha}\\) line in the Balmer series refers to the spectral line emitted when a hydrogen atom makes a transition from the \\(n = 3\\) energy level to the \\(n = 2\\) energy level, and its wavelength is \\(656.280\\ \\text{nm}\\). The hydrogen isotope deuterium also has a similar Balmer series, and... | 3 |
PHY-020 | Physics | null | {
"en": "Consider light passing through a periodic stack of 2N double-layer dielectric films. The thickness of film 1 is \\(d_1\\), and its refractive index is \\(n_{1}\\). The thickness of film 2 is \\(d_2\\), and its refractive index is \\(n_{2}\\). The optical system is arranged periodically as film 1, film 2, fil... | $r= \frac{(a + b*n_3*\cos\theta_3)*n_0*\cos\theta_0-(c + d*n_3*\cos\theta_3)}{(a + b*n_3*\cos\theta_3)*n_0*\cos\theta_0+(c + d*n_3*\cos\theta_3)}$
$t = \frac{2*n_0*\cos\theta_0}{(a + b*n_3*\cos\theta_3)*n_0*\cos\theta_0+(c + d*n_3*\cos\theta_3)}$
where,
$$
\begin{cases}
a = \left[ \cos\beta_1 \cos\beta_2 - \frac{... |
PHY-021 | Physics | null | {
"en": "Two coherent parallel beams with the same wavelength \\(\\lambda\\) and an amplitude ratio of \\(\\alpha>1\\) are incident at an angle \\(\\theta\\). The observation screen is perpendicular to the angular bisector of the \\(\\theta\\) angle. Try to find: 1. The contrast \\(\\gamma\\) of the interference frin... | \[\gamma=\frac{2\alpha}{\alpha^2+1},\Delta x=\frac{\lambda}{2\sin\frac{\theta}{2}}\] |
PHY-022 | Physics | null | {
"en": "There is a thin spherical shell with radius \\(R\\) and uniform surface charge density \\(\\sigma\\), which rotates with a constant angular velocity \\(\\omega\\) about a fixed axis. Try to find the following under the condition that the divergence of the gauge \\(\\mathbf{A}\\) is \\(0\\):\n\n(1) The surfac... | (1)j(θ) = σωRsinθ\hat{φ}
If r>R then A(r,θ) = $\frac{μ_0*\pi*σωR^4*sinθ}{3r^2}\hat{φ}$
If r<R then A(r,θ) = $\frac{μ_0*\pi*σω*R*rsinθ}{3}\hat{φ}$
(2) If r>R, then B = $\frac{μ_0*\pi*σωR^4}{3r^2}(2cosθ\hat{r}+sinθ\hat{θ})$
E = $\frac{σR^2}{ε_0*r^2}\hat{r}$
If r<R, then B = $\frac{μ_0*\pi*σωR^4}{3r^2}(cosθ\hat{r}-sinθ\ha... |
PHY-023 | Physics | null | {
"en": "A ring with radius \\(R_{1}\\) is uniformly charged with a total charge of \\(Q\\). To its right, there is a grounded conducting sphere with radius \\(R_{2}\\). The distance between the center of the sphere and the center of the ring is \\(L\\), and the line connecting the two centers is perpendicular to the... | \(Q'=-\frac{R_2}{\sqrt{R_1^{2}+L^{2}}}Q\) |
PHY-024 | Physics | null | {
"en": "A stepped grating is composed of $N$ stepped optical media on an opaque screen. The refractive index of the optical medium is $n$, the horizontal spacing between adjacent steps is $d$, and the vertical spacing is $h$. Given that the wavelength of parallel light incident perpendicularly to the plane of the st... | $I = I_0 \times \left(\frac{\sin\left(N\pi(nd - h\sin\theta - d\cos\theta)/\lambda\right)}{\sin\left(\pi(nd - h\sin\theta - d\cos\theta)/\lambda\right)}\right)^2\times\left(\frac{\sin\left(\pi\times h\sin\theta/\lambda\right)}{\pi\times h\sin\theta/\lambda}\right)^2$ |
PHY-025 | Physics | null | {
"en": "In a $k$-dimensional closed container, there is stored a $k$-dimensional ideal gas with molecular mass $m$. Suppose at equilibrium the temperature is $T$, and both the molecular velocity and speed follow the Maxwell distribution. The Boltzmann constant is $k_{B}$.\n\\subsection*{(1)}\nDenote the number densi... | (1)n_k*\sqrt{\frac{k_B*T}{2*\pi*m}}
(2)$\overline{v}$ = $\sqrt{\frac{2k_{B}T}{\pi m}} \cdot \frac{\Gamma\left(\frac{k + 1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}
$
$\overline{v^2}$ = k*k_{B}*T/m
(3)
(3.1)
f_k(v) = $v^k*exp(\frac{-m v^2}{2 k_{B} T}*\frac{2*m^((k+1)/2)}{\Gamma(\frac{k+1}{2})*(2*k_{B}*T)^((k+1)/2)}$
$... |
PHY-026 | Physics | null | {
"en": "Moist air flows adiabatically and continuously over a hill, going up the slope, crossing the peak, and then flowing down the slope. Meteorological stations H0 and H3 measure the atmospheric pressure as both being \\(p_0\\). At H0, the air temperature is \\(T_0\\); as the air rises to H1, the measured pressur... | (1)$T_{1} = T_{0} \left( \frac{p_{1}}{p_{0}} \right)^{1 - \frac{1}{\gamma}}$
(2)$h_{1} = \frac{2(\rho_{0} - \rho_{1})}{\rho_{0}g\left(1 + \frac{p_{1}T_{0}}{p_{0}T_{1}}\right)}$
(3)$T_{2} = T_{1} \left( \frac{p_{2}}{p_{1}} \right)^{1 - \frac{1}{\gamma}}+\frac{L*m_rain}{C_p}$
(4)$T_3 = T_{2} \left( \frac{p_{3}}{p_{2}}... |
PHY-027 | Physics | null | {
"en": "Two long straight wires, carrying opposite uniform line free charges ±λ, are situated on either side of a long dielectric cylinder . The cylinder (which carries no free net charge) has radius R, and the wires are a distance a (a>R) from the axis.The permittivity inside the cylinder is $ε_{1}$ and the permitt... | When r<R,
V(r,θ) = $\frac{λ)}{2\pi*(\varepsilon_{1}+\varepsilon_{2})}*ln(\frac{r^2+a^2-2racosθ}{r^2+a^2+2racosθ})$
When r>R
V(r,θ) = $\sum_{n = 1}^{\infty}(\frac{λ(\varepsilon_{1}-\varepsilon_{2})}{\varepsilon_{2}*\pi*(\varepsilon_{1}+\varepsilon_{2})}*cosnθ*(\frac{R^2}{a*r})^n*(1-(-1)^n)/n)+\frac{λ)}{4\pi*\varepsilon_... |
PHY-028 | Physics | null | {
"en": "An ideal electric dipole is situated at the origin, and points in the z direction. An electric charge is released from rest at a point in the x y plane. suppose the mass of electric charge is m,the charge is q,the dipole is p,the distance is R at the begin.\n(1)Show that it swings back and forth in a semi-ci... | (1)Use energy is constant and force analyze we can get:
$\frac{(d^2)r}{dt^2}+\frac{(dr/dt)^2}{r}=0$
solve this differential equation we get r=R is the only solution which satisfies both differential equation and beginning state.
And use the energy function V = $\frac{pq}{4\pi*ε_{0}*r^2}$>0 we get θ∈[3\pi/2,\pi/2] is a ... |
PHY-029 | Physics | null | {
"en": "A hemispherical shell-shaped bowl with mass \\(M\\) and inner radius \\(R\\) can slide and roll without friction on a horizontal plane. Inside the bowl, a uniform solid sphere with mass \\(m\\) and radius \\(r\\) undergoes pure rolling. Taking the position where the sphere is at the bottom of the bowl when t... | \[\chi=0.90696\] |
PHY-030 | Physics | null | {
"en": "In a vacuum diode, electrons are “boiled” off a hot cathode, at potential zero, and accelerated across a gap to the anode, which is held at positive\npotential $V_{0}$. The cloud of moving electrons within the gap (called space charge)\nquickly builds up to the point where it reduces the field at the surface... | (a)$\laplacian{V} = -\frac{\rho(x)}{\epsilon_0} \quad \Rightarrow \quad \frac{d^2 V}{dx^2} = -\frac{\rho(x)}{\epsilon_0}
$
(b)$ v(x) = \sqrt{\frac{-2eV(x)}{m}}$
(c)$I = JA = \rho(x)v(x)A \Rightarrow \rho(x) = \frac{I}{Av(x)}
$
(d)$\frac{d^2V}{dx^2} = \frac{I}{A\epsilon_0}\sqrt{\frac{m}{2e}}*V^{-1/2}
$
(e)$V(x) = V_0\le... |
PHY-031 | Physics | null | {
"en": "A spherical hot air balloon has a total mass (including a well-insulated balloon skin, a basket, and other loads) of 350 kg. After heating, the balloon expands to its maximum volume with a diameter of 20 m. Assume the gas composition inside and outside the balloon is the same, and the gas pressure inside the... | 322.8K |
PHY-032 | Physics | null | {
"en": "An infinitely long charged straight line with linear charge density $\\lambda$ is located at $(-a,0)$, and an infinitely long grounded cylindrical conductor with radius $r < a$ whose axis is parallel to the line charge is located at $(a,0)$. Find the electric potential distribution $V(x,y)$ in space.",
"zh... | \frac{-\lambda}{4\pi\varepsilon_0}\ln[\frac{r^2}{4a^2}\frac{(x+a)^2+y^2}{[x-(1-\frac{r^2}{2a^2})a]^2+y^2}] |
PHY-033 | Physics | null | {
"en": "When the distance between two atoms is small, the interaction between them manifests as a strong repulsion; when the distance is large, the interaction manifests as a weak attraction. A large number of atoms can combine into a crystal through this interaction. When the temperature approaches 0K, atoms are ar... | (1)a=(\frac{uA_2u}{vA_2v})^(1/(2u-2V))r_0
(2)(A_n)^1=6,(A_n)^2=12*2^(-n/2)
(A_n)^3 = 8*3^(-n/2)
(3)\sqrt{\frac{2ε*(\frac{u(2u-1)A_(2u+2)(r_0)^2u}{3a^(2u+2)}-\frac{v(2v-1)A_(2v+2)(r_0)^2v}{3a^(2v+2)})}{M}}
(4)\sqrt(2)-1 |
PHY-034 | Physics | null | {
"en": "There are two points in space on the same horizontal plane $y = 0$ with a distance of $l$ between them. A homogeneous catenary with a length of $L>l$ and a linear mass density of $\\lambda$ is hung with these two points as endpoints. Given that the positive direction of the $y$ - axis in space is vertically ... | [\frac{y'^2-1}{L(e^{y/L}-1)}+y''](1+y'^2)-y'y''=0 |
PHY-035 | Physics | null | {
"en": "If an ideal gas with a constant volume heat capacity $C_v$ of $\\frac{1}{\\sqrt{3}-1}R$ undergoes a thermodynamic process that appears as a regular $n$-gon on a $P - V$ diagram, and we denote the vertices of the regular $n$-gon as $\\{A_{i},i = 1,2,\\ldots,n\\}$, then the $P - V$ coordinates of the $k$-th po... | $η_{n}$ = \frac{nsin2*\pi/n}{(\sqrt{3}+1)((N+cos$\frac{2(p+1)\pi}{n})*(N+sin$\frac{2(p+1)\pi}{n})-(N+cos$\frac{2q\pi}{n})*(N+sin$\frac{2q\pi}{n}))+2*(N \sin \left( \frac{2(p+1)\pi}{n} \right) + \frac{p + 1}{2} \sin \left( \frac{2\pi}{n} \right) + \frac{1}{4} \sin \left( \frac{4(p+1)\pi}{n} \right))-2*(N \sin \left( \fr... |
PHY-036 | Physics | null | {
"en": "On the slope, a row of fixed inclined fences is formed by small thin rods that are parallel to each other, with an adjacent horizontal spacing of \\(l\\) and a vertical spacing of \\(h\\). Above the fence, there is a homogeneous circular plate with a mass of \\(m\\) and a radius \\(r\\gg l\\), and the circul... | $mv^2*(h^2+l^2)/(2r^2*l)+mgh/l$ |
PHY-037 | Physics | null | {
"en": "Suppose we have a particle with potential\n\\(V = V_{0}-i\\Gamma,\\)\nwhere \\(V_{0}\\) is the true potential energy and \\(\\Gamma\\) is a positive real constant? Find the lifetime of the particle in terms of \\(\\Gamma\\), you can use $\\hbar$ to simplify your answer.",
"zh": "Suppose we have a particle ... | \frac{\hbar}{2\Gamma} |
PHY-038 | Physics | null | {
"en": "In space, there is an experimental device consisting of a concentric inner sphere and a spherical shell. The space between the inner sphere and the inner surface of the spherical shell is a vacuum. The radius of the inner sphere is \\(r\\), its temperature is kept constant, and its emissivity is \\(e\\). The... | (1)$
T_{2} = \left( \frac{Q}{4\pi R_2^{2}\sigma} + T^{4} \right)^{\frac{1}{4}}
$
(2)$
T_{1} = \frac{(R_{2}-R_{1})) Q}{4\pi κ R_{2} R_{1}} + \left( \frac{Q}{4\pi R_2^{2}\sigma} + T^{4} \right)^{\frac{1}{4}}
$
(3)$
T_{0} = \sqrt[4]{\frac{1 - (1 - e)(1 - E)\dfrac{r^{2}}{R_{1}^{2}}}{4\pi E e \sigma r^{2}} Q + (\frac{(R_{2}... |
PHY-039 | Physics | null | {
"en": "There is an isolated metal sphere with a radius of \\(R\\) and a total charge of \\(0\\). Along two mutually perpendicular diameter directions, at a distance of \\(\\lambda R\\) from the center of the sphere on its upper, lower, left and right sides, there is a point charge \\(q\\) respectively.\n(1) If the ... | (1)λ = 1.4727
(2)W = $\frac{q^2((1+2\sqrt{2})/λ+16/λ^2-4/(λ^2-1)-4/(λ^2+1)-8λ/\sqrt{λ^4+1})}{4\pi*ε_0*R}$
4*π*ε₀*R²*W/q²=0.35923
(3)
ω = $\sqrt{\frac{q^2*((\sqrt{2}/4+1/4)/λ^3+9/λ^4-(λ^2+1)/(λ^2-1)^3-(2λ^2)/(λ^2+1)^3-(4λ^6-2λ^2)/(λ^4+1)^(5/2)}{4\pi*ε_0*R^3m}}$
ω₀ = $\sqrt{\frac{-q^2*0.4514}{4\pi*ε_0*R^3m}}$
Not a stabl... |
PHY-040 | Physics | null | {
"en": "A non-uniformly charged spherical shell of radius R has surface charge density σ(θ) = σ₀(1 + cosθ)，where θ is the polar angle. The shell rotates about its symmetry axis with angular velocity ω. Find the dipole form of A(z) for z > R, and z is lying on the symmetry axis.",
"zh": "A non-uniformly charged sph... | \frac{\mu_0\omega R^2\sigma_0}{4(z/R-1)}B(3/2, 7/2) _2F_1(1/2, 3/2; 5; -\frac{4z/R}{(z/R-1)^2}) |
PHY-041 | Physics | null | {
"en": "On a $k$-dimensional finite hypercube grid plane, denote the coordinates of the lower-left corner of the grid as the origin of coordinates $(0,0,\\ldots,0)$. For the $i$-th coordinate, there are resistors with resistance value $R_{i}$ connected between adjacent points on the corresponding axis, and there are... | $\boxed{R_{eq}(\mathbf{0}, \mathbf{x}) = \sum_{q_1=0}^{m_1} \dots \sum_{q_k=0}^{m_k} {}^{'} \frac{\left[1 - \prod_{i=1}^k \cos\left(\frac{q_i \pi x_i}{m_i}\right)\right]^2}{N_{\mathbf{q}} \lambda_{\mathbf{q}}} \text{ where } \lambda_{\mathbf{q}} = 4 \sum_{i=1}^k \frac{1}{R_i} \sin^2\left(\frac{q_i \pi}{2m_i}\right), N_... |
PHY-042 | Physics | null | {
"en": "There is a simple pendulum with mass $m$ and string length $l$. Its suspension point undergoes high-frequency vibration with angular frequency $\\gamma\\gg\\sqrt{\\frac{g}{l}}$, an amplitude of $a$, and the direction makes an angle $\\varphi$ with the horizontal direction. Try to find: (1) The new “equilibri... | (1) $\theta = 0$, or $\theta=\pi$,
if $\frac{2gl}{(a\gamma)^2}<\cos2\varphi$, then there is also $\cos^{-1}\left(\frac{2gl}{(a\gamma)^2\cos2\varphi}\right)$
(2) $\theta = 0$ is stable when $\frac{2gl}{(a\gamma)^2}>\cos2\varphi$,
at this time $\omega=\sqrt{\frac{g}{l}-\cos2\varphi*\frac{a^{2}\gamma^{2}}{2l^{2}}}$
$\thet... |
PHY-043 | Physics | null | {
"en": "A homogeneous thin-walled cylinder (mass \\(m\\), radius \\(r\\)) starts to roll down from rest at the top of a semi-cylindrical surface with radius \\(R\\). It is known that the axis of the cylinder is always parallel to the axis of the semi-cylindrical surface, and the acceleration due to gravity is \\(g\\... | (1)\mu_{\min}=\frac{\sqrt{2+\sqrt{2}}}{4}
(2)v_1=\sqrt{\frac{1}{5}g(R+r)} |
PHY-044 | Physics | null | {
"en": "In an infinitely long straight cylindrical conductor of radius \\(a\\), when there is no current, the volume charge density of free electrons is a constant \\(\\rho_{0}\\), and that of positive ions is \\(-\\rho_{0}\\), so the conductor is electrically neutral. When a direct current flows along the length of... | \rho=\frac{\varepsilon_0\mu_0u^2}{1-\varepsilon_0\mu_0u^2}\rho_0, B(r)=\frac{\mu_0a^2u\rho_0}{2(1-\varepsilon_0\mu_0u^2)r} |
PHY-045 | Physics | null | {
"en": "A mass of \\(2.0\\ \\text{kg}\\), temperature of \\(- 13^{\\circ}C\\), and volume of \\(0.19\\ \\text{m}^3\\) of Freon (molecular weight is 121, van der Waals constants \\(a = 1.06\\ \\text{Pa}\\cdot\\text{m}^6/\\text{mol}^2\\), \\(b = 5.42\\times10^{-5}\\ \\text{m}^3/\\text{mol}\\)) is compressed under isot... | 0.79kg |
PHY-046 | Physics | null | {
"en": "Consider following quantum Hamiltonian:\n$$ H_0 = \\frac{p_1^2}{2m} + \\frac{1}{2}m\\omega^2 x_1^2 + \\frac{p_2^2}{2m} + \\frac{1}{2}m\\omega^2 x_2^2 $$\nThis is the Hamiltonian for two decoupled harmonic oscillators.\n\n(a) Calculate the eigenstates and eigenvalues for $H_0$ (an energy eigenstate could be l... | (a)$|n_1, n_2\rangle$=($a_1^\dagger$)^($n_1$)/($n_1$!)^0.5($a_2^\dagger$)^($n_2$)/($n_2$!)^0.5$|0, 0\rangle$,
E = (n_1+n_2+1)hω/(2\pi)
(b)i.use [a_i,$a_i^\dagger$]=1 for i =1,2 to directly verify it
ii.J_z$|n_1, n_2\rangle$=\frac{1}{2}(a_1^\dagger a_1 - a_2^\dagger a_2)($a_1^\dagger$)^($n_1$)/($n_1$!)^0.5($a_2^\dagger$... |
PHY-047 | Physics | null | {
"en": "At the origin \\((0,0)\\) of the polar coordinate system, there is a mass point with mass \\(M\\). In space, there is a homogeneous soft spring with an original length of \\(0\\), linear mass density \\(\\lambda\\), and a uniform spring constant \\(k\\). Its two ends are fixed at \\((r_{1},\\theta_{1})\\) an... | (1) According to the law of conservation of generalized angular momentum \(Tr\sin\beta=\text{Const}\) (where \(\beta\) is defined as \(\tan\beta = r\frac{d\theta}{dr}\)) and the radial force equation \(\cos\beta dT-\sin\beta T(d\beta + d\theta)=\frac{GM\lambda dr}{r^{2}\cos\beta}\), we can obtain through two integratio... |
PHY-048 | Physics | null | {
"en": "Set up a fixed $Oxyz$ coordinate system. There is a glass ellipsoid with the equation $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}+\\frac{z^{2}}{c^{2}} = 1$, which is placed flat on the lower half of a hyperboloid of two sheets - like surface with the equation $- 2\\rho\\sinh(\\frac{xy}{ab})+(\\frac{z}{c})^{2}... | $f(x,y) = \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{2\rho xy}{ab}-(\ 2k+\frac{1}{2})\frac{\lambda}{c}, 0 < |\rho| < 1$ is an ellipse, $\rho=\pm1$ is a parabola, $|\rho| > 1$ is a hyperbola |
PHY-049 | Physics | null | {
"en": "An infinitely long paraboloid of revolution has the equation \\(z = \\frac{r^{2}}{4a}\\) (where \\(r=\\sqrt{x^{2}+y^{2}}\\)). There is a current uniformly distributed along the latitudinal direction on its surface, with a current linear density of \\(\\sigma\\). Try to find the magnetic induction intensity \... | \frac{4}{3}\mu_0\sigma |
PHY-050 | Physics | null | {
"en": "A simple pendulum at a latitude of $\\theta$ has a string length of $l$ and the acceleration due to gravity is $g$, and it undergoes small-angle motion. Assume the angular velocity of the Earth's rotation is $\\omega$, and the air resistance coefficient is $\\eta$ (not necessarily a small quantity). Find the... | r(t) = $αl*\frac{ω_2*exp{iω_1t}-ω_1*exp{iω_2t}}{ω_2-ω_1}$
where, $ω_{1,2}=-η/(2m)-iωsinθ±i\sqrt{g/l+(iωsinθ+η/(2m))^2}$ |
PHY-051 | Physics | null | {
"en": "Inside a vacuum chamber, there are two identical liquid bubbles, each filled with a monatomic ideal gas at a temperature of \\(T_{1}\\). The radius of each liquid bubble is \\(R_{1}\\), and the relationship between the surface tension coefficient and temperature is \\(\\sigma(T)=\\sigma_0\\left(1 - \\frac{T}... | (1)$T_2$=$\frac{T_0}{2}+\frac{c*T_1*T_0}{32*\pi*(R_1)^2*σ_0}+\sqrt{(\frac{T_0}{2}+\frac{c*T_1*T_0}{32*\pi*(R_1)^2*σ_0})^2-\frac{c*(T_1)^2*T_0}{16*\pi*(R_1)^2*σ_0}-T_1*(T_0-T_1)}$
$R_2$ = $\sqrt{\frac{32*T_2}{27*T_1}}*R_1$
(2) T_3 = T_2
R_3 = $\sqrt{\frac{2*T_3}{T_1}}*R_1$ |
PHY-052 | Physics | null | {
"en": "Use the uncertainty relation to estimate the ground-state energy $E_0$ of a quantum harmonic oscillator.",
"zh": "试用不确定性关系，估算基态量子谐振子的能量E_0"
} | hω\(4\pi) |
PHY-053 | Physics | null | {
"en": "There is a horizontal uniform magnetic field with a very large magnetic induction intensity \\(B\\). A metal concentric ring with an inner diameter of \\(R_1\\) and an outer diameter of \\(R_2\\) (\\(R_2 > R_1\\)) and a thickness of \\(D\\) (\\(D\\ll R_1, R_2\\)) falls vertically in this magnetic field. The ... | B=10^6T |
PHY-054 | Physics | null | {
"en": "At the six vertices of a regular hexagon with side length \\(a\\), there are fixed point charges each with charge \\(Q\\). At the center of the regular hexagon, a free point charge with mass \\(m\\) and charge \\(q\\) (where \\(q\\) has the same sign as \\(Q\\)) is placed. Now, \\(q\\) is moved a very small ... | $2\pi\sqrt{\frac{ma^3}{3kQq}}$ |
PHY-055 | Physics | null | {
"en": "An initially macroscopically uncharged long metal cylinder (radius \\(a\\), length \\(L\\gg a\\)) contains an equal number density \\(n\\) of uniformly distributed electron gas and positive ion gas inside. Now the cylinder rotates at a high angular velocity \\(\\omega\\) about its central axis, and the syste... | ρ(r) = $\frac{-e*n*m*ω^2*R^2*\exp{mω^2*r^2/(2KT)}}{mω^2*R^2*\exp{mω^2*R^2/(2KT)}+1-\exp{mω^2*R^2/(2KT)}}+ne$
E =$ n*e*r/(2ε_0)-\frac{n*e*R^2*(mω^2*r^2*\exp{mω^2*r^2/(2KT)}+1-\exp{mω^2*r^2/(2KT)})}{2*r*ε_{0}mω^2*R^2*\exp{mω^2*R^2/(2KT)}+1-\exp{mω^2*R^2/(2KT)}}$$\hat{r}$
B = $μ_{0}*n*e*ω*r^2/2-\frac{μ_{0}*ω*n*e*R^2*(mω^2... |
PHY-056 | Physics | null | {
"en": "There are two homogeneous rods with masses $m_{1}$, $m_{2}$ and lengths $l_{1}$, $l_{2}$ respectively. The first rod is hinged to the wall, and one end of the second rod is hinged to the other end of the first rod, with the other end free. (1) Find all possible angular frequencies of the system's vibration. ... | (1)ω=$\sqrt{\frac{B±\sqrt{B^2 - 4AC}}{2C}}$
where
A = m_2*g_2*l_2*(m_1*g*l_1/2 + m_2*g*l_1)
B = m_2*m_1*g*l_1*l_2*(l_1/3 + l_2/6)+(m_2)^2*g*l_1*l_2*(l_1 + l_2/3)
C = (m_1*m_2/9+(m_2)^2/12)*(l_1*l_2)^2
(2)
Let ω_{1,2} = $\sqrt{\frac{B'±\sqrt{B'^2 - 4C'}}{2C'}}$
where
$B' = 2g/3l+\frac{4ml^2/3+Mx^2}{3mgl/2+Mgx}$
$C' = \f... |
PHY-057 | Physics | null | {
"en": "A stationary electric dipole **p** = p**ẑ** is situated at the origin. A negative point charge -q (mass m) moves in a uniform magnetic field **B** = B**ẑ**, superimposed with the dipole’s electric field. The charge executes a circular orbit of radius s above x-y plane, and its angular momentum has positive p... | v=\frac{qBs}{2m}(1+\sqrt{1+\frac{\sqrt{2}pm}{3\sqrt{3}\pi\varepsilon_0qB^2s^4}}) |
PHY-058 | Physics | null | {
"en": "A uniform disk of mass has a radius of $R_2$. There is a concentric mirror region with a radius of $R_1 < R_2$ on the disk's surface, while the remaining annular region is a perfect blackbody. The disk is fully illuminated by a uniform parallel laser beam perpendicular to the plane of the disk. The laser is ... | \[\frac{2\pi^2R_2^2}{\lambda}\frac{R_2^2+R_1^2}{R_2^2-R_1^2}\] |
PHY-059 | Physics | null | {
"en": "An object flies vertically upward at a constant speed \\(v_f\\) from a platform at a height \\(h\\) above the ground. A missile is launched from the ground at a horizontal distance \\(L\\) from the take-off point of the flying object. The missile and the flying object are launched simultaneously. The speed o... | t=\frac{\alpha^2-1+\gamma(\alpha^2+1)}{2\alpha\gamma(1-\gamma^2)}\cdot\frac{L}{v}, \alpha=\frac{h}{L}+\sqrt{1+(\frac{h}{L})^2}, \gamma=\frac{v_f}{v} |
PHY-060 | Physics | null | {
"en": "A light rigid rod of length \\(l\\) has two small balls 1 and 2 fixed at its two ends. The mass of each small ball is \\(m\\), and their charges are \\(\\pm q\\) respectively, with the positively charged ball 1 at the upper end and the negatively charged ball 2 at the lower end. Let's call it a “charged barb... | (1)$v_{1}=\frac{qBl}{m}*\sqrt{\frac{(\sin\theta)^2}{2}-\frac{\cos\theta}{2}+\frac{1}{2 + 2(\sin\theta)^2}+\frac{(1 - \cos\theta)\cos\theta\sin\theta}{2\sqrt{1+(\sin\theta)^2}}}$
$v_{2}=\frac{qBl}{m}*\frac{\sin\theta}{\sqrt{1+(\sin\theta)^2}}$
(2)$N=\frac{q^{2}*B^{2}*l}{m}*(1-\frac{(\sin\theta)^2*\cos\theta*(2 + (\sin\t... |
PHY-061 | Physics | null | {
"en": "There are many stones launched from the origin \\((0,0,0)\\) of a three-dimensional rectangular coordinate system with the same magnitude of initial velocity \\(v_{0}\\), different launch angles \\(\\theta\\) (\\(\\arctan(\\sqrt{x^{2}+y^{2}}/z)\\)) and different azimuth angles \\(\\varphi\\) (\\(\\arctan y/x... | The equations are:
\(x^{2}+y^{2}=(-4z^{2}+\frac{2v_{0}^{2}z}{g})\)
and the union of \(z = 0\) and \(0\leq x^{2}+y^{2}\leq\frac{v_{0}^{2}}{g}\)
Offset:
For points \((x,y)\) such that \((x,y)\in x^{2}+y^{2}=(-4z^{2}+\frac{2v_{0}^{2}z}{g})\)
\(\delta(x,y,z)=\omega*\sqrt{\frac{4v_{0}^{2}}{g^{2}}-\frac{6z}{g}}(y\hat{i}-x\h... |
PHY-062 | Physics | null | {
"en": "Inside the inner wall of a hollow spherical shell with radius $R$, there is a small ball that can be regarded as a particle moving at a constant speed along a fixed horizontal circular path. The angle between the line connecting the small ball and the center of the hollow spherical shell and the vertical lin... | $2\pi\sqrt{\frac{R}{g}\frac{\cos\theta}{1 + 3\cos^{2}\theta}}$ |
PHY-063 | Physics | null | {
"en": "A particle 1 collides with a target particle 2 at rest, generating and scattering \\(n - 2\\) new particles \\(3,4,\\cdots,n\\). To achieve this reaction with the minimum kinetic energy of particle 1, find the minimum threshold kinetic energy \\(T\\) of particle 1 under relativistic conditions. Given the spe... | $c^2*\frac{-m_1^2-m_2^2-2m_1*m_2+(\sum_{i=3}^{n} m_i)^2}{2*m_2}$ |
PHY-064 | Physics | null | {
"en": "In a two-dimensional world, inside a rectangular container, there are a large number of identical small balls randomly moving in all directions. The small balls can be regarded as mass points, and elastic collisions occur between the small balls and between the small balls and the end faces of the container ... | TA=const |
PHY-065 | Physics | null | {
"en": "Consider a single electron outside the atomic nucleus, which is in the ground state. In this state, let the mean value of the square of the distance from the electron to the center of the atomic nucleus be denoted by $r_0^2$, and the mean value of the square of the electron's momentum (defined as the square ... | r_0^2p_0^2\geq\frac{9}{4}\hbar^2 |
PHY-066 | Physics | null | {
"en": "Two separated thin lenses are combined into a lens group. Their focal lengths are \\(f_1\\) and \\(-f_2\\) respectively, where \\(0 < f_2 < f_1\\). Their average refractive indices within the used light wave band are \\(n_1\\) and \\(n_2\\) respectively, and their average dispersive powers are \\(D_1\\) and ... | \[d=\frac{\frac{f_1D_2}{n_2-1}-\frac{f_2D_1}{n_1-1}}{\frac{D_1}{n_1-1}+\frac{D_2}{n_2-1}}\] |
PHY-067 | Physics | null | {
"en": "An incident light irradiates a light-transmitting surface at an incident angle \\(i\\). The energy flux density of the incident light is \\(I\\). The complex reflectivity of the surface to light is \\(r\\) (including phase), and the complex transmittance to light is \\(t\\), with an exit angle \\(\\varphi\\)... | P = I((1+rr*)cosi-tt*cosφ)/c
F_切 = I((1-rr*)sini-tt*sinφ)/c |
PHY-068 | Physics | null | {
"en": "The volume of the gas storage tank is \\(V\\), and the gas pressure inside the tank is \\(p\\). The gas storage tank is connected to a vacuum chamber with volume \\(V_0\\) via a valve. The valve is opened to charge the vacuum chamber. After reaching equilibrium, the valve is closed. Then, the gas storage tan... | \left\lceil \frac{\ln\left( \dfrac{p}{p_0} \right)}{\ln\left( 1 + \dfrac{V_0}{V} \right)} \right\rceil |
PHY-069 | Physics | null | {
"en": "On the Oxy plane, there are two spheres with centers at $(-2R,0,0)$ and $(2R,0,0)$ respectively, both with a radius of $R$. The two spheres are uniformly charged with volume charge densities $\\pm\\rho$ respectively, and they have the same angular velocity $\\omega(t)$ pointing in the positive $z$-axis direc... | If $(x + 2R)^2 + y^2+z^2>R^2$ and $(x - 2R)^2 + y^2+z^2>R^2$
then
$E=\frac{\mu_0*(\pi)^2\rho\beta R^6}{15}*(\frac{y}{((x + 2R)^2 + y^2+z^2)^{\frac{3}{2}}}-\frac{y}{((x - 2R)^2 + y^2+z^2)^{\frac{3}{2}}})\hat{i}+\frac{\mu_0*(\pi)^2\rho\beta R^6}{15}*(\frac{(x - 2R)}{((x - 2R)^2 + y^2+z^2)^{\frac{3}{2}}}-\frac{(x + 2R)}{(... |
PHY-070 | Physics | null | {
"en": "An interface is formed by medium 1 and medium 2, with the refractive indices of the two media being \\(n_1\\) and \\(n_2\\). Introduce \\(n = n_2/n_1\\), and the normal of the interface is aligned with the \\(x\\)-axis of the \\(S\\) frame. Now, the interface moves at a uniform velocity \\(\\beta c\\) along ... | \[\theta_f=\arcsin\frac{n}{\sqrt{n^2\beta^2+\frac{1-\beta^2}{1-n^2\beta^2}}}-\arcsin\frac{n\beta}{\sqrt{n^2\beta^2+\frac{1-\beta^2}{1-n^2\beta^2}}}.\] |
PHY-071 | Physics | null | {
"en": "Take the rectangular coordinate system xyz. There is a conduction current in an infinite layer region where \\(-d\\leqslant x\\leqslant d\\). The direction of the current density is along the positive z-axis, and its magnitude varies linearly with x, that is \\( j(x) = j_0 \\frac{|x|}{d} \\). Try to find the... | B=\frac{\mu_0j_0}{2d}\min\{x^2,d^2\} |
PHY-072 | Physics | null | {
"en": "It is known that the refractive index \\(n\\) of an optical fiber has a radial distribution given by\n\\(n = n_{0}(1-\\alpha^{2}r^{2})\\)\nwhere \\(n_0\\) is the refractive index at the center, and \\(\\alpha\\) is a positive number much smaller than 1. Try to find the trajectory of a light ray propagating i... | \[r(z)=\frac{r_0'}{\sqrt{2}\alpha}\sin(\sqrt{2}\alpha)\] |
PHY-073 | Physics | null | {
"en": "The electric charge of a muon \\(\\mu\\) is \\(q = -e\\) (\\(e = 1.6\\times10^{-19}\\ C\\)), its rest mass is \\(m_0 = 100\\ MeV/c^2\\), and its lifetime at rest is \\(\\tau_0 = 10^{-6}\\ s\\). Suppose there is a muon at a height \\(h = 10^{4}\\ m\\) above the Earth's equator moving vertically downwards at a... | (1) $3.3\times10^3\text{MeV}$
(2) Deflect westward by $0.023\text{ rad}$ |
PHY-074 | Physics | null | {
"en": "For an electron gas obeying the Fermi distribution with a level degeneracy of 2, given the electron number density \\(n\\), electron mass \\(m\\), Planck constant \\(\\hbar\\), speed of light \\(c\\), and gas volume \\(V\\), at \\(T = 0\\ K\\):\n(1) Find the magnitude distribution \\(f(p)\\) of the electron ... | (1)f(p) = \frac{V}{\pi^{2}\hbar^{3}}*p^2*I_{0,(3\pi^{2})^{1/3}\left(\frac{N}{V}\right)^{1/3}\hbar}(p)
where \(I\) is the indicator function
(2)
\(U=\frac{cV}{8\pi^{2}\hbar^{3}} \left[ p_{F} (2p_{F}^{2} + m^{2}c^{2}) \sqrt{p_{F}^{2} + m^{2}c^{2}} - (mc)^{4} \mathrm{arsinh}\left(\frac{p_{F}}{mc}\right) \right]\)
\(P=\fra... |
PHY-075 | Physics | null | {
"en": null,
"zh": null
} | None |
PHY-76 | Physics | images/PHY-76.png | {
"en": "Three homogeneous rigid rods are hinged together as shown in the figure. The angular coordinates are denoted as $\\theta_1$, $\\theta_2$, $\\theta_3$ as shown in the figure. The range of values for all three angular coordinates is $[0, 2\\pi]$. The initial hinge coordinate of the first rod is $(0,0)$, and th... | (1)
Three equations determine $\theta_{10}$, $\theta_{20}$, $\theta_{30}$:
$
l_1*\cos\theta_{10}+
l_2*\cos\theta_{20}+
l_3*\cos\theta_{30}=d
l_1*\sin\theta_{10}+
l_2*\sin\theta_{20}+
l_3*\sin\theta_{30}=h
(m_1 + m_2+M)*\cos\theta_{10}*(\sin(\theta_{30}-\theta_{20}))+(m_2 + M + 2M*x/l_2)*\sin(\theta_1-\theta_3)+m_3*\sin... |
PHY-77 | Physics | images/PHY-77.png | {
"en": "The crank \\(OA = r\\) rotates with a constant angular acceleration \\(\\alpha\\) about the fixed axis \\(O\\). The connecting rod \\(AB\\) is connected to the end point \\(A\\) of the crank by a hinge and can slide inside a hinged sleeve \\(N\\). When \\(\\varphi=\\pi\\), the angular velocity of the crank \... | v=(l/2-r)\omega,
a=\sqrt{(r-l/4)^2\omega^4+(l/2-r)^2\alpha^2},
\rho=\frac{l/2-r}{l/4-r}. |
PHY-78 | Physics | images/PHY-78.png | {
"en": "Consider a symmetric ellipsoid that is in pure rolling on the XOZ plane with center $O'$. The angle between the major axis of the ellipsoid and the $y$ axis is $\\theta$. The equation of the ellipsoid is: $\\frac{x'^{2}+z'^{2}}{b^{2}}+\\frac{y'^{2}}{a^{2}} = 1$. The relationship between the coordinate axes $... | v = $ω(a^2 \cos^2 θ + b^2 \sin^2 θ)^1/2 \hat{x}-ω \frac{(a^2-b^2)cosθsinθ} {\sqrt{a^2 \cos^2 θ+b^2\sin^2 θ} }\hat{y}$
a = $β \sqrt{a^2 \cos^2{θ} + b^2 \sin ^2{θ}} - ω^2\frac{(a^2-b^2)cosθsinθ} {\sqrt{a^2 \cos^2 θ+b^2\sin^2 θ} }\hat{x}-( β \frac{(a^2-b^2)sinθcosθ}{\sqrt{(a^2 \cos^2 θ + b^2 \sin^2 θ) }}+ ω^2 \frac{(a^2-b... |
PHY-79 | Physics | images/PHY-79.png | {
"en": "As shown in the figure, there are two points A and B in the spherical coordinate system. Using the subscripts $\\theta$ and $\\varphi$, their coordinates are recorded as $(0,0)$ and $(\\frac{\\pi}{2},0)$ respectively, and the radius of the sphere is denoted as $R$. Now a missile is launched from point A, and... | $
dθ/dt = v/(R \sqrt{1+\tan^2 (u t/R-φ)}
dφ/dt =\tan(u*t/R-φ)*v/(R*\tanθ* \sqrt{1+\tan^2 (u t/R-φ)}
$ |
PHY-80 | Physics | images/PHY-80.png | {
"en": "Consider a symmetric ellipsoid that is in pure rolling on the XOZ plane with its center at O′. The angle between the principal axis of the ellipsoid and the y-axis is θ, and the equation of the ellipsoid is: \\(\\frac{x'^{2}+z'^{2}}{b^{2}}+\\frac{y'^{2}}{a^{2}} = 1\\). The relationship between the coordinate... | a_C = $(\begin{aligned} =\beta\left[\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}+\frac{ab(\cos\psi \cos\theta-\sin\theta \sin\psi \sin\varphi)}{\sqrt{a^{2}\sin^{2}\psi+\cos^{2}\psi b^{2}}}\right] \\ & -w^{2}\left[\frac{\left(a^{2}-b^{2}\right)\cos\theta\sin\theta}{\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}}+\frac... |
PHY-81 | Physics | images/PHY-81.png | {
"en": "Three homogeneous rigid rods are hinged together as shown in the figure. The subscripts are denoted as $\\theta_1$, $\\theta_2$, $\\theta_3$ as shown in the figure, and the value range of all three subscript angles is $[0, 2\\pi]$. The lengths of the three rods are denoted as $l_1$, $l_2$, $l_3$ respectively... | ρ = $\frac{(sin^2 (θ_2-θ_3)+sin^2 (θ_3-θ_1)/4+sin(θ_2-θ_3)sin (θ_3-θ_1)cos(θ_1-θ_2))^{3/2}}{sin^3 (θ_2-θ_3)/(l_1)+cos(θ_1-θ_2)sin(θ_2-θ_3)sin (θ_3-θ_1)(sin(θ_2-θ_3)/(l_1)+sin (θ_3-θ_1)/(l_2))+sin^3 (θ_3-θ_1)/(4*l_2)+sin(θ_1-θ_2)*(sin^2 (θ_1-θ_2)/l_3+sin^2 (θ_3-θ_1)cos(θ_2-θ_3)/l_2+sin^2 (θ_2-θ_3)cos(θ_3-θ_1)/l_1)/2}$ |
PHY-82 | Physics | images/PHY-82.png | {
"en": "As shown in the figure is a binary star system. Two stars with masses $M_1$ and $M_2$ are located at points $\\alpha_1$ and $\\alpha_2$ in the figure respectively, and the distance between them is $2\\sigma$. The gravitational constant is $G$. Now an object with mass $m$ moves in the above field.\n(1) Now as... | (1)$\begin{aligned} & \beta=\sigma^{2}\left(p_{\rho}^{2}+\frac{p_{\varphi}^{2}}{\rho^{2}}\right)-M^{2}-2m\sigma(G M_{1}\mathrm{cos}\theta_{1}+G M_{2}\mathrm{cos}\theta_{2}), \\ & \text{where} \\ & M^{2}=(r\times p)^{2}=p_{\rho}^{2}z^{2}+p_{z}^{2}\rho^{2}+\frac{r^{2}p_{\varphi}^{2}}{\rho^{2}}-2z\rho p_{z}p_{\rho}, \end{... |
PHY-83 | Physics | images/PHY-83.png | {
"en": "As shown in the figure, an orbital station with mass \\(M\\) and a spaceship with mass \\(m\\) docked to it are moving together in a circular orbit around the Earth. The orbital radius is \\(n\\) times the Earth's radius \\(R\\), where \\(n = 1.5\\). At a certain instant, the orbital station launches the spa... | \frac{m}{M}=0.074或0.240 |
PHY-84 | Physics | images/PHY-84.png | {
"en": "As shown in the figure, for a horizontally placed parallel-plate capacitor, one plate is below the dielectric liquid surface and the other is above the liquid surface. The relative permittivity of the liquid is $\\epsilon_r$, and its density is $\\rho$. After delivering surface charge densities $\\sigma$ and... | (2) Solution: $h = (\varepsilon _{\mathrm{r} }^{2}- 1) \sigma ^{2}/ 2\varepsilon _{\mathrm{r} }^{2}\varepsilon _{0}\rho g$,
(1)(3) Solution: $h=(\varepsilon_r-1)\sigma^2/2\varepsilon_r\varepsilon_0\rho g.$ |
PHY-85 | Physics | images/PHY-85.png | {
"en": "An infinitely long coaxial and co - apical double cone made of thin sheets are insulated from each other at the vertex \\(O\\). The angles between their generatrices and the cone axis are \\(\\theta_1\\) and \\(\\theta_2\\) respectively, as shown in the figure.\n\nIf both thin sheets are conductors, with the... | $V(r,\theta,t)=\frac{\exp{-\sigma t/\varepsilon}(\varphi_1 - \varphi_2)\ln\left(\tan\frac{\theta}{2}\right)-\varphi_2\ln\left(\tan\frac{\theta_2}{2}\right)+\varphi_2\ln\left(\tan\frac{\theta_1}{2}\right)}{\ln\left(\tan\frac{\theta_1}{2}\right)-\ln\left(\tan\frac{\theta_2}{2}\right)}$ |
PHY-86 | Physics | images/PHY-86.png | {
"en": "As shown in the figure, at a distance \\(h\\) from an infinitely large uncharged conducting plane, there is an uncharged conducting sphere with a radius of \\(R\\). A uniform electric field \\(E\\) exists throughout space, directed from top to bottom. Using the infinite method of images, assume that the new ... | (1)$p_0 = 4 \pi ε_0 E R^3$
$x_0 = 0$
$q_0 = 0$
$x_{n+1} = \frac{R^2}{2h-x_n}$
$p_{n+1} = (\frac{x_{n+1}}{R})^3*p_n$
$q_{n+1} =q_n*\frac{x_{n+1}}{R}+p_n*(\frac{x_{n+1}}{R})^2$
(2)
$x_n = R^2*\frac{(h+\sqrt{h^2-R^2})^n-(h-\sqrt{h^2-R^2})^n}{(h+\sqrt{h^2-R^2}）^{n+1}-(h-\sqrt{h^2-R^2})^{n+1}}$
$p_n = \frac{8 \pi ε_0 E R^{... |
PHY-87 | Physics | images/PHY-87.png | {
"en": "Two parallel infinite conducting planes are separated by a distance $d$. At a point $P$ which is at a distance $a$ from the surface of one of the conductors, there is a point charge with charge $q$. Given that the electric potential of both conductors is zero, find the induced charges on the two conducting p... | Q_2 = -aq/d
Q_1 = -(d-a)q/d |
PHY-88 | Physics | images/PHY-88.png | {
"en": "As shown in the figure, there is an electric dipole $p_1$ at the origin, and another non-fixed electric dipole $p_2$ at a distance $r$ from this point. The relative angle of the position is denoted as $\\theta$, and the deflection angles of the electric dipole vectors $p_1$ and $p_2$ are denoted as $\\varphi... | (1) r = $(\frac{3p_1 p_2}{2\pi ε_0 k})^{(1/5)}$
2θ - φ_1 - φ_2 = 0 or $\pi$
φ_1 - φ_2 = 0 or $\pi$
(2)
The angular frequency of r's vibration is $\sqrt{\frac{5 k m_1 m_2}{m_1 + m_2}}$
The three angular frequencies of θ, φ_1, φ_2 are
ω_1 = 0
ω_23^2 is the solution of the following quadratic equation
$abc x^2+(2ab + 2ac... |
PHY-89 | Physics | images/PHY-89.png | {
"en": "As shown in the figure is the generation process of a Koch curve, and the figure shows the boundary of the Koch curve. Now, connect the bases of the sub-triangles generated at each step to obtain a fractal resistor network. Assume that the resistance per unit length is $\\lambda$, and the side length of the ... | d=$\frac{ln(4)}{ln(3)}$
R= 11λa/36 |
PHY-90 | Physics | images/PHY-90.png | {
"en": "As shown in the figure, it is a resistance network of a regular N-sided prism. There are \\(n\\) resistors with resistance \\(R\\) on each edge, and there are also \\(n\\) regular N-sided cross-sections located between every two resistors, all with a resistance of \\(R\\). (The figure shows the case where \\... | r = $\sum_{p=0}^{n} \sum_{q=0}^{N-1} \frac{2 R \cos(p\pi/(n+1))(1-\cos(2 k q\pi/N))\cos((p \pi*(2z+1)/(n+1))}{N(n+1)(2-\cos(2p\pi/(n+1))-\cos(2q\pi/N))}$ |
PHY-91 | Physics | images/PHY-91.png | {
"en": "As shown in the figure, it is one iteration of a resistor network, that is, the original resistor network composed of nine small triangles has its internal six small triangles refined into smaller nine triangles. What is the resistance value between points B and C after n such processes? Given that the side ... | r = $\frac{2*λa*5^(n+1)}{3*7^(n+1)}$ |
PHY-92 | Physics | images/PHY-92.png | {
"en": "As shown in the figure is a source-containing circuit. The relationship between each power source and time is a periodic function $\\epsilon(t)=at^{2}\\ (-T/2 < t < T/2)$, $\\epsilon(t + T)=\\epsilon(t)$. Try to find the steady-state current $I$ in each branch.",
"zh": "如图为一含源电路，各个电源与时间的关系为周期函数ε(t) =$at^(2... | ["$\\begin{cases}i_1=\\sum_{n=1}^{∞}\\frac{4\\left(3Z_L+2Z_C\\right)}{\\left(3Z_L+Z_C\\right)\\left(4Z_L+3Z_C\\right)}\\widetilde{u}, \\\\ \\\\ i_2=\\sum_{n=1}^{∞}\\frac{4Z_L}{\\left(3Z_L+Z_C\\right)\\left(4Z_L+3Z_C\\right)}\\widetilde{u}. \u0026 \\end{cases}$\nwhere $Z_L = i*L*2n\\pi/T$\n$Z_C = 1/(i*C*2n\\pi/T)$\n\n$\... |
PHY-93 | Physics | images/PHY-93.png | {
"en": "As shown in the figure is a finite-length solenoid coil with a current magnitude of \\(I\\) and the number of turns per unit length of \\(n\\). Use the parameters given in the figure to calculate the magnitude of the magnetic field at this location.",
"zh": "如图是一个有限长螺线圈，电流大小为I，单位长度的匝数为n，利用图中所给参数计算该处磁场大小"
} | B= μ_0*n*I(cosθ_1+cosθ_2)/2 |
PHY-94 | Physics | images/PHY-94.png | {
"en": "As shown in the figure, in the $Oxyz$ coordinate space of a certain inertial frame, the distribution of the steady current is\n\\(\\boldsymbol{j}=\n\\begin{cases}\nj_0\\cdot(x/a)^2n\\boldsymbol{e}_j, & \\vert x\\vert\\leqslant a, \\\\\n0, & \\vert x\\vert>a, & & \n\\end{cases}\\) \nwhere \\(j_0\\) is a const... | t = $\frac{a*\Gamma(1/2)*\Gamma(1/(4n+4))}{v*(2n+2)*\Gamma(1/2+1/(4n+4))}$ |
PHY-95 | Physics | images/PHY-95.png | {
"en": "The refractive index of a semi-elliptical cylindrical glass is \\(n = \\sqrt{2}\\), and the eccentricity \\(e = 3/4\\). It is placed in air. In the plane perpendicular to the axis of the semi-elliptical cylinder, light is incident at an angle of \\(45^{\\circ}\\) on the planar surface of the major axis of th... | 121.4^{\circ}\leq\phi\lea 176.2^{\circ} |
PHY-96 | Physics | images/PHY-96.png | {
"en": "As shown in the figure, a plano-convex thin lens made of the same glass is attached to the outer wall of a thin-walled glass liquid tank filled with kerosene. The focal length of the lens in air is \\(f_0\\). Given that the refractive indices of kerosene, glass, and air are \\(n_1=\\frac{7}{5}\\), \\(n = \\f... | 1. s'=-\frac{5}{9}f_0, M=14/9
2. s'=-\frac{15}{37}f_0, M=42/37 |
PHY-97 | Physics | images/PHY-97.png | {
"en": "As shown in the figure, a converging lens is placed behind the double holes \\(S_1\\) and \\(S_2\\) with a spacing of \\(d\\), and a screen is placed on the back focal plane of the lens. This interference device faces a distant extended light source. The extended light source is rectangular, with a total len... | \(\gamma=\left|\frac{\cos(\frac{\pi db}{\lambda R})}{1-(\frac{2db}{\lambda R})^2}\right|\) |
PHY-98 | Physics | images/PHY-98.png | {
"en": "As shown in the figure, there is a magnetic dipole inside a spherical magnetic medium with a magnetic moment magnitude of \\(m\\). The permeability inside the sphere is \\(\\mu_1\\), and outside it is \\(\\mu_2\\). The radius of the sphere is \\(R\\). Find the magnetic field distribution \\(B(r,\\theta)\\) i... | If \(r > R\)
\(B(r,\theta)=\frac{3m\mu_1\mu_2\cos\theta}{2\pi r^3(\mu_1 + 2\mu_2)}\hat{r}+\frac{3m\mu_1\mu_2\sin\theta}{4\pi r^3(\mu_1 + 2\mu_2)}\hat{\theta}\)
If \(r < R\)
\((\frac{m\mu_1\cos\theta}{2\pi r^3}-\frac{3m\mu_1(\mu_1 - \mu_2)\cos\theta}{2\pi R^3(\mu_1 + 2\mu_2)})\hat{r}+(\frac{m\mu_1\sin\theta}{4\pi r^3}+\... |
PHY-99 | Physics | images/PHY-99.png | {
"en": "As shown in the figure, it is a spiral whose polar coordinate equation can be written as \\(r = a\\exp{\\theta}\\) (\\(0\\leq\\theta<\\infty\\)).\nTry to calculate the magnetic vector potential at a point directly above the origin and at a distance \\(z < a\\). Express the answer in the form of a series. Use... | A = $\sum_{n=0}^{∞} \frac{(2n-1)(-1)^n \Gamma{1/2+n}z^{2n}}{a^{2n} n!*\sqrt{\pi}(4n^2+1)}\hat{i}+\sum_{n=0}^{∞} \frac{(2n+1)(-1)^n \Gamma{1/2+n}z^{2n}}{a^{2n} n!*\sqrt{\pi}(4n^2+1)}\hat{j}$ |
PHY-100 | Physics | images/PHY-100.png | {
"en": "As shown in the figure, it is a five-pointed star frame composed of five metal rods with length \\(l\\) and resistance \\(r\\) connected end-to-end. The metal rods intersect but do not make contact. Now, this frame rotates around one of its left-right axes of symmetry with an angular velocity \\(\\omega\\). ... | (1)
q = $\frac{B_0 l^2 sin(\pi/10) cos(\pi/10)}{r(1+sin(\pi/10))}$
(2)
M = $\frac{5B_0^2 ω l^4 sin^2(\pi/10) cos^2(\pi/10) sin^2(2ωt)}{16r(1+sin(\pi/10))^2}$ |
PHY-101 | Physics | images/PHY-101.png | {
"en": "As it is an electromagnetic system, there is a charge of \\(q/4\\) at the origin, and there are two charges of \\(-q\\) with mass \\(m\\) above and below the point at a distance \\(a\\) from the origin. They are connected to the charge at the origin by springs with a spring constant of \\(k\\). Initially, th... | P = $\frac{q^2 a^4 k^3 sin^2(2θ)}{2(\pi)^2 ε_0 c^5 r^2 m^3}$ |
PHY-102 | Physics | images/PHY-102.png | {
"en": "As shown in the figure, the coordinates of a point in the original spherical coordinate system are denoted as $(r', \\theta', \\varphi')$, and the metric in this coordinate system is the normal metric $(ds)^2 = c^2(dt)^2-(dx)^2-(dy)^2-(dz)^2$. Now, relative to the original reference frame, a rotating referen... | (1) The new metric is $g_{00}=c^2 - \omega^2 r^2\sin^2(\theta), g_{11}=-1, g_{22}=-r^2, g_{33}=-r^2\sin^2(\theta), g_{03}=-\omega r^2\sin^2(\theta)$, and the rest are 0.
(2) $\Delta t=\frac{2\pi\omega R^2}{c^2 - \omega^2 R^2}$
(3) $f=\frac{m\omega^2 r\sin\theta}{\sqrt{(1 - v^2/c^2)*(1-\omega^2 r^2\sin^2(\theta)/c^2)}}*... |
PHY-103 | Physics | images/PHY-103.png | {
"en": "As shown in the figure, a high-energy particle with an impact parameter \\(b\\) and velocity \\(v_0\\) is incident from infinity towards a target particle. The high-energy particle has a mass of \\(m\\) and a charge of \\(Z_1e\\), while the target particle has a mass of \\(M\\) and a charge of \\(Z_2e\\). De... |
\(\theta=\frac{2Lc\arccos\left(\frac{kZ_1Z_2e^{2}E}{\sqrt{L^{2}c^{2}E^{2}-m^{2}L^{2}c^{6}+k^{2}Z_1^{2}Z_2^{2}e^{4}m^{2}c^{4}}}\right)}{\sqrt{L^{2}c^{2}-k^{2}e^{4}Z_1^{2}Z_2^{2}}}\)
|
PHY-104 | Physics | images/PHY-104.png | {
"en": "As shown in the figure is the unit cell structure diagram of the NaCl crystal. Assume that the lattice constants $a = b = c$, $\\alpha=\\beta=\\gamma=\\frac{\\pi}{2}$, then at equilibrium, ions are arranged in turn at the vertices of the cubic lattice, that is: $R_{l_{1}l_{2}l_{3}}=(l_{1}a, l_{2}a, l_{3}a, l... | (1)\sum_{l_1,l_2,l_3≠0,0,0}^{∞} \frac{e^2 (-1)^{l_1+l_2+l_3+1}}{4 \pi ε_0 a^2 \sqrt{l_1^2+l_2^2+l_3^2}}-\frac{A \sqrt{l_1^2+l_2^2+l_3^2}e^{-\sqrt{l_1^2+l_2^2+l_3^2}a/ρ}}{ρ}=0
(2)T_1=$2\pi*\sqrt{\frac{m}{\sum_{l_1,l_2,l_3≠0,0,0}^{∞} {2e^{-\sqrt{l_1^2+l_2^2+l_3^2}a/ρ}/(3ρ^2)-5e^{-\sqrt{l_1^2+l_2^2+l_3^2}a/ρ}/(3ρ a \sqrt{... |
PHY-105 | Physics | images/PHY-105.png | {
"en": "As shown in the figure is a right-angle potential barrier.\nIf an incident particle with energy \\(E > U_0\\) is incident from left to right, let \\(k_1=\\frac{\\sqrt{2mE}}{\\hbar}\\), \\(k_2 = \\frac{\\sqrt{2m(E - U_0)}}{\\hbar}\\) Try to find the expression of the system wave function.",
"zh": "如图为一直角势垒，... | If \(x < 0\),
\(\psi(x)=e^{i k_1 x}+\frac{(k_1 - k_2)(1 - e^{i k_2 a})e^{-i k_1 x}}{k_1 + k_2+(k_1 - k_2)e^{i k_2 a}}\)
If \(0 < x < a\),
\(\psi(x)=\frac{2 k_1 (k_1 + k_2)e^{i k_2 x}}{(k_1 + k_2)^2-(k_1 - k_2)^2 e^{2 i k_2 a}}+\frac{2 k_1 (-k_1 + k_2)e^{-i k_2 x}}{(k_1 + k_2)^2-(k_1 - k_2)^2 e^{2 i k_2 a}}\)
If \(x > a... |
PHY-106 | Physics | images/PHY-106.png | {
"en": "As shown in the figure, between two infinitely long concentric grounded conducting cylinders with radii \\( R_1 \\) and \\( R_2 \\) respectively, there is an infinitely long charged concentric elliptical cylinder with the equation \\( \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\). The surface charge density of... | λ_1 = -λ*$\frac{2\pi ln(\frac{2ab}{(b+a)R_1})}{ln(R_2/R_1)}$
λ_2 = λ*$\frac{2\pi ln(\frac{2ab}{(b+a)R_2})}{ln(R_2/R_1)}$ |
PHY-107 | Physics | images/PHY-107.png | {
"en": "\n\nAs shown in the figure, an infinitely long straight wire carries a current \\( I \\). Beside it is a uniformly charged ring with radius \\( r \\) and charge linear density \\( \\lambda \\), whose center is at a distance \\( d > r \\) from the long straight wire. In the current state, the ring is stationa... |
The ring still maintains periodic translation in the plane and periodic rotation about an axis perpendicular to itself, with the normal frequency being
\[
\omega = \frac{\mu_0 I \lambda \left( \frac{d^2 + r^2}{d^2 - r^2} - \frac{2d}{\sqrt{d^2 - r^2}} + 1 \right)}{m r^2}
\] |
PHY-108 | Physics | images/PHY-108.png | {
"en": "\n\nIn a constant-temperature environment at temperature \\( T \\), there is a vertical infinite long straight cylinder containing an ideal gas separated by \\( K \\) smooth pistons with masses \\( M_1 = M \\), \\( M_2 = 2M \\), ..., \\( M_K = KM \\). The thickness of the pistons is negligible, and there is ... | \frac{K(K+1)}{2}NkT |
PHY-109 | Physics | images/PHY-109.png | {
"en": "\n\nAs shown in the figure, there is a vertical three-chamber cylinder with two pistons of masses \\( M_1, M_2 \\) and cross-sectional areas \\( S_1, S_2 \\), connected by a spring with a spring constant \\( k \\). Initially, all three chambers are in equilibrium at a temperature \\( T_0 \\). Chamber \\( C_1... | (1).$p_2=\frac{M_1 g +M_2 g +p_1 S_1 - p_3 S_2}{S_1-S_2}$
$Δx = \frac{(p_3-p_1) S_2 S_1 -M_2 g S_1 -M_1 g S_2}{k (S_1-S_2)}$
(2).$P V (1+\frac{L}{R T_0}-ln \frac{P}{P_1})^5 (\frac{L}{R T_0}-ln \frac{P}{P_1})^{-3}=P_1 V_1 (1+\frac{L}{R T_0})^5 (\frac{L}{R T_0})^{-3}$
(3)$M_1 M_2 ω^4 - (A M_2+B M_1) ω^2 + A B -C^2 = 0
其中... |
PHY-110 | Physics | images/PHY-110.png | {
"en": "\n\nA device deployed by a spacecraft falls vertically at a constant speed, approaching and reaching the surface of a certain planet. The curve of pressure \\( p \\) versus time \\( t \\) recorded by the device in conventional units is shown in the figure. When landing on the planet's surface, the device als... | Approximately 17.7m/s |
PHY-111 | Physics | images/PHY-111.png | {
"en": "\n\n\"Since the piston in a heat engine may undergo small vibrations, i.e., \\( P \\) and \\( V \\) fluctuate relative to their equilibrium positions, studying the thermodynamic cycle of this process has practical significance. Shown in the figure is the \\( P\\text{-}V \\) diagram of an off-axis elliptical ... | (1)η=$\frac{\pi a b }{2 \sqrt{(3cosθ+4sinθ)^2 b^2 + (3sinθ-4cosθ)^2 a^2}}$
(2)η= $\frac{\pi^2 a b}{5 C(a,b)}$ |
PHY-112 | Physics | images/PHY-112.png | {
"en": "\n\n$\\nu$ moles of an ideal gas undergo a thermodynamic process as shown in the $S$-$T$ diagram. In the first half, $S$ is a quadratic function of $T$, with a horizontal tangent at $T_1 = T_0$. At $T' = 2T_0$, the process transitions to a linear segment along the tangent line and stops at $T_2 = 3T_0$. Deri... |
When \( T_1 < T < T' \),
\[
\ln TV^{\gamma-1} - \frac{S_2 - S_1}{3\nu C_V T_0^2}(T - T_0)^2 = \text{const.}
\]
When \( T' < T < T_2 \),
\[
\ln TV^{\gamma-1} - \frac{2(S_2 - S_1)}{3\nu C_V T_0}T = \text{const.}
\] |
PHY-113 | Physics | images/PHY-113.png | {
"en": "As shown in the figure, near the center of a uniform gaseous spherical planet O with mass volume density ρ, there is a celestial explorer of mass m captured by the planet's gravity. The explorer is at a distance d from the center, and the planet has a radius r. At a distance R from the planet's center, there... |
Taking \( x \) as the distance from the explorer to \( O \) and \( \varphi \) as the angle between the explorer's position vector to \( O \) and the initial position vector, a polar coordinate system is established, yielding:
If \( x < r \), then
\[
\frac{d\varphi}{dx} = \frac{1}{x \sqrt{\frac{x^2 \left( E - \fra... |
PHY-114 | Physics | images/PHY-114.png | {
"en": "As shown in the figure, it is a schematic diagram of the first two iterations of a fractal pattern generated from an equilateral triangle with side length a (i.e., each iteration decomposes 6 small triangles into 9 smaller triangles, and fills the topmost small triangle among the remaining 3 small triangles ... | I(x,y)= $I_0 \frac{t^2+4 cos \frac{\pi a x}{3 λ d}^2+4 t cos \frac{\pi a x}{3 λ d} cos \frac{\sqrt{3} \pi a y}{9 λ d}}{ (t+2)^2 (9+4 cos \frac{\pi a x}{3 λ d}^2+4(cos \frac{\pi a x}{3 λ d}+cos \frac{\sqrt{3} \pi a y}{3 λ d})^2-4 cos \frac{\pi a x}{3 λ d} cos \frac{2 \sqrt{3} \pi a y}{9 λ d}-4 (cos \frac{\pi a x}{3 λ d}... |
PHY-115 | Physics | images/PHY-115.png | {
"en": "As shown in the figure, it is a negative uniaxial crystal with a sphere of radius R removed. The optical axis of the crystal makes an angle θ with the normal to the crystal's bottom surface. The refractive indices for o-light and e-light are nₒ and nₑ, respectively. Light with wavelength λ is incident downwa... | (1)$x^2 + y^2 = \frac{2 k λ R}{n_0-\frac{1}{\sqrt{cos^2 θ/n_e^2 + sin^2 θ/n_o^2}}}$
(2)$(x^2+y^2)^2 = \frac{4 k λ R^3}{(n_o^2/n_e^2-1) (1-1/n_o)^2}$ |
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