{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"f4090519-298d-5a38-a317-5c35d812a711","task_key":"train--Challenge~5f18~5fmain","task_revision_id":"2","upstream_id":"Challenge_18_main","short_description":"Two dielectric nanoparticles are deeply trapped in two Gaussian optical traps…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nepsilon0 = sp.symbols('epsilon0', positive=True)   # vacuum permittivity\\nk, z_R, d0 = sp.symbols('k z_R d0')\\nalpha_1, alpha_2 = sp.symbols('alpha_1 alpha_2')\\nE_1, E_2 = sp.symbols('E_1 E_2')\\nphi_1, phi_2 = sp.symbols('phi_1 phi_2')\\nm, Omega_1, Omega_2 = sp.symbols('m Omega_1 Omega_2')\\n\\ndef answer(epsilon0, k, z_R, d0, alpha_1, alpha_2, E_1, E_2, phi_1, phi_2, m, Omega_1, Omega_2):\\n    r\\\"\\\"\\\"\\n    Return the expression of $k_1$ and $k_2$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    epsilon0:sympy.Symbol, vacuum permittivity $\\\\varepsilon_0$\\n    k:       sympy.Symbol, wave vector, $k$\\n    z_R:     sympy.Symbol, Rayleigh range, $z_R$\\n    d0:      sympy.Symbol, distance between the two spheres at equilibrium, $d_0$\\n    alpha_1: sympy.Symbol, polarizability of nanoparticles 1, $\\\\alpha_1$\\n    alpha_2: sympy.Symbol, polarizability of nanoparticles 2, $\\\\alpha_2$\\n    E_1:     sympy.Symbol, electric-field amplitude of tweezer 1, $E_1$\\n    E_2:     sympy.Symbol, electric-field amplitude of tweezer 2, $E_2$\\n    phi_1:   sympy.Symbol, phase of tweezer 1 at the focal plane, $\\\\phi_1$\\n    phi_2:   sympy.Symbol, phase of tweezer 2 at the focal plane, $\\\\phi_2$\\n    m:       sympy.Symbol, nanoparticle mass, $m$\\n    Omega_1: sympy.Symbol, frequency parameter of nanosphere 1, $\\\\Omega_1$\\n    Omega_2: sympy.Symbol, frequency parameter of nanosphere 2, $\\\\Omega_2$\\n\\n    Outputs\\n    ----------\\n    k1: sympy.Expr, $k_1$ in the equations of motion\\n    k2: sympy.Expr, $k_2$ in the equations of motion\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    k1 = ...  # a SymPy expression of inputs\\n    k2 = ...  # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return k1, k2\",\"problem_description\":\"# Problem Setup\\n\\nTwo dielectric nanoparticles are deeply trapped in two Gaussian optical traps that propagate along the $z$-axis, both characterized by the wave vector $k$ and the Rayleigh range $z_R$. Suppose the focal planes of these traps are located at $z=0$, and the nanoparticles are located at $z=z_1$ and $z=z_2$ respectively, where ${z_1},{z_2} \\\\ll {z_R}$. Let the distance between the two nanoparticles be $d$, satisfying the far-field condition $kd \\\\gg 1$. The polarizabilities of the two nanoparticles are $\\\\alpha_1$ and $\\\\alpha_2$, respectively. Both the tweezers have identical polarization, the electric field amplitudes are $E_1$ and $E_2$, and the phases at the focal planes are $\\\\phi_1$ and $\\\\phi_2$, respectively.\\n\\n# Main problem:\\nAssume that, at equilibrium, the distance vector between the two spheres is $d_0 = (d_0,0,0)$. The angle between the laser polarization and the particle-connecting axis is $\\\\pi/2$. Derive $k_1$ and $k_2$ in the following equations of motion along the $z$-direction for the two nanospheres:\\n\\n$$\\\\begin{aligned}\\nm{{\\\\ddot z}_1} =  - m\\\\Omega _1^2{z_1} - ({k_1} + {k_2}){z_1} + ({k_1} + {k_2}){z_2},\\\\\\\\\\nm{{\\\\ddot z}_2} =  - m\\\\Omega _2^2{z_2} - ({k_1} - {k_2}){z_2} + ({k_1} - {k_2}){z_1}.\\n\\\\end{aligned}$$\\n\\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}