Larmor radius gyroradius and cyclotron frequency calculator

The Larmor radius calculator (also known as gyroradius or gyration radius) determines the radius of the circular orbit of a charged particle in a uniform magnetic field, per the formulation by Joseph Larmor 1897 and the works of Hendrik Lorentz 1895 on magnetic force F = qv×B. The classic formula: r_L = m·v_⊥ / (|q|·B) = p_⊥ / (|q|·B), where m = particle mass kg, v_⊥ = velocity component perpendicular to field B m/s, q = electric charge in coulombs, B = magnetic field intensity in tesla. The associated cyclotron frequency: ω_c = qB/m (rad/s) and period T_c = 2π/ω_c = 2πm/(qB). For relativistic particles (v near c): r_L = γ·m·v_⊥ / (|q|·B) with γ = 1/√(1 − v²/c²) Lorentz factor, applicable to ultra-high cosmic rays and CERN accelerators. The physics: Lorentz force F = qv×B acts perpendicular to velocity producing helical motion (combination of drift parallel to B + circular rotation around field lines) called gyration motion. The 5 application categories:

• Laboratory fusion plasmas (ITER tokamak 5 T deuterons 10 keV → r_L ≈ 3 mm, Lawson criterion r_L < a/100 confinement)
• Particle accelerators (Lawrence cyclotrons 1932, CERN LHC synchrotrons, K800 INFN-LNS Catania)
• Planetary magnetospheres (Van Allen belts protons 10-100 MeV B=10⁻⁵ T → r_L ≈ km; aurora borealis polar precipitation)
• Solar wind and interplanetary medium (protons 400 km/s B=5 nT → r_L ≈ 836 km, BepiColombo ASI Mercury)
• Cosmic ray astrophysics (galactic confinement break 10¹⁵ eV B=1 nG → r_L ≈ 1 pc; Pierre Auger Argentina ultra-high cosmic rays 10²⁰ eV r_L ≈ 100 Mpc beyond galaxy size, extragalactic AGN sources). The Italian applications:

• ENEA Frascati 1955 + DTT Divertor Tokamak Test 2028 EUROfusion deuterons B=5 T r_L ≈ 3 mm plasma confinement Lawson
• ENEA FTU Frascati Tokamak Upgrade 1989-2020 1.3 MA 8 T compact research tokamak
• INFN-LNS Catania National Southern Labs K800 Superconducting Cyclotron heavy ions energy 80·Q²/A MeV/nucleon hadron therapy + nuclear physics
• INFN Frascati 1957 LNF DAFNE 1999 KLOE-2 e+e− collider Φ-factory
• Carlo Rubbia 1984 Nobel CERN UA1 W/Z bosons proton-antiproton SPS synchrotron
• Pierre Auger Observatory Argentina INFN+INAF Italy 3,000 km² surface cosmic rays 10¹⁹-10²⁰ eV
• AGILE satellite ASI 2007 + AMS-02 ISS 2011 INFN gamma cosmic rays multi-GeV
• Solar Orbiter ESA 2020 ASI 25% magnetometer solar wind interplanetary magnetic field
• BepiColombo ASI 2018 magnetometer Mercury planetary magnetosphere. Typical measurements: electron v=10⁶ m/s B Earth 5×10⁻⁵ T → r_L ≈ 11 cm, proton solar wind 400 km/s B=5 nT IPM → r_L ≈ 836 km, cosmic ray 10¹⁵ eV in 1 nG ISM → r_L ≈ 1 pc galactic break, deuteron ITER 5 T 10 keV → r_L ≈ 3 mm, UHE cosmic ray 10²⁰ eV Pierre Auger → r_L ≈ 100 Mpc

Larmor radius Italian applications table

(Source: Larmor 1897 + Lorentz 1895 + Lawrence 1932 cyclotron Nobel 1939 + ENEA Frascati 1955 + INFN LNF 1957 + LNS Catania + INAF Rome 2002).

Italian Larmor radius examples

• Electron in Earth magnetic field INAF Trieste 1898 + INGV Rome 1999: m_e = 9.109×10⁻³¹ kg, v_⊥ = 10⁶ m/s, q_e = 1.602×10⁻¹⁹ C, B_Earth = 5×10⁻⁵ T (mid-latitude). r_L = 9.109×10⁻³¹ · 10⁶ / (1.602×10⁻¹⁹ · 5×10⁻⁵) = 0.114 m ≈ 11 cm. ω_c = qB/m = 8.8×10⁶ rad/s ≈ 1.4 MHz. Application: aurora borealis, magnetosphere, ionosphere 100-1000 km altitude, INGV observatories L'Aquila Castello Tesino
• Proton solar wind interplanetary Solar Orbiter ESA 2020 ASI 25% magnetometer: m_p = 1.673×10⁻²⁷ kg, v = 400 km/s = 4×10⁵ m/s, B_IPM = 5 nT = 5×10⁻⁹ T (1 AU). r_L = 836 km. T_c = 2π·m/(qB) = 13 s
• Deuteron ITER France 2025 + DTT Frascati ENEA 2028 EUROfusion Ansaldo Nucleare: m_d = 3.344×10⁻²⁷ kg, T = 10 keV → v_th = √(2kT/m) = 9.8×10⁵ m/s, B = 5 T. r_L = 4.1×10⁻³ m ≈ 4 mm (Lawson r_L < a/100 minor radius a = 2 m ITER). ω_c = 2.4×10⁸ rad/s ≈ 38 MHz (RF Ion Cyclotron Resonance Heating ICRH)
• Heavy ion INFN-LNS Catania K800 Superconducting Cyclotron 2001: ⁴⁰Ca²⁰⁺ E = 80 MeV/A, B = 4.8 T, r_max = 1.8 m. r_L = γm·v/(qB) Q/A=0.5. Hadron therapy CATANA proton therapy 60 MeV oncology Catania Aldo Moro Hospital
• Cosmic ray 10¹⁵ eV ISM galactic break LNGS Gran Sasso 1987 Borexino CUORE CUPID-0: proton E = 10¹⁵ eV → p ≈ E/c relativistic, B_ISM = 1 nG = 10⁻¹³ T. r_L = E/(qB·c) = 3.3×10¹⁶ m ≈ 1.1 pc. Galactic confinement break: above 10¹⁵ eV r_L > ISM turbulence scale → galactic escape (knee spectrum)
• UHE cosmic ray 10²⁰ eV Pierre Auger Argentina Malargüe INFN+INAF Italy: E = 10²⁰ eV (16 J), B_extragal = 10⁻¹³ T. r_L ≈ 3.3×10²¹ m = 100 Mpc beyond galaxy size → extragalactic AGN sources (anisotropy GZK cutoff 5×10¹⁹ eV CMB interaction)
• Proton LHC CERN Carlo Rubbia 1984: E = 7 TeV, B = 8.33 T (NbTi supercond. dipoles). r_L = pc/(qB·c) = 2.8 km (LHC ring radius 4.3 km)

Italian plasma physics and accelerators

Italian plasma and accelerator physics:

• ENEA Frascati Research Center 1955 National Agency New Technologies Energy Sustainable Development (Rome + 11 centers): FTU Frascati Tokamak Upgrade 1989-2020 compact 1.3 MA 8 T research; DTT Divertor Tokamak Test 2028 EUROfusion €600M Ansaldo Nucleare Genoa MITER ITER France; Proto-Sphera spherical pinch; FAST Fusion Advanced Studies Torus
• INFN National Institute Nuclear Physics 4 national labs: LNF Frascati 1957 DAFNE 1999 e+e− Φ-factory 510 MeV KLOE-2 collaboration kaon physics + SPARC_LAB FEL; LNGS Gran Sasso 1987 1,400 m rock underground Borexino solar neutrinos + XENONnT dark matter + CUORE CUPID-0 0νββ + cosmic-ray muon modulation; LNS Catania 1976 National Southern Labs K800 Superconducting Cyclotron 2001 heavy ions hadron therapy CATANA + Tandem 15 MV; LNL Legnaro Padua 1968 Tandem XTU + ALPI superconducting linac + SPES rare isotope 2026
• ICTP Trieste 1964 + SISSA Trieste 1978 theoretical plasma physics, Salam Nobel 1979
• IIT Genoa 2003 Italian Institute Technology quantum tech
• CINECA Bologna 1969 Marconi100 30 PFlops + Leonardo HPC 250 PFlops 2023 (5th TOP500) MHD plasma tokamak gyrokinetic GENE Eagle JOREK simulations
• Italian Physics Nobel Prizes: Marconi 1909, Fermi 1938, Segrè 1959, Carlo Rubbia 1984 Nobel CERN UA1 W/Z bosons proton-antiproton SPS synchrotron, Riccardo Giacconi 2002 X-ray
• Italian space missions ASI Rome 1988: AGILE 2007 gamma cosmic rays 30 MeV-50 GeV; AMS-02 ISS 2011 INFN alpha magnetic spectrometer cosmic rays protons positrons; BepiColombo 2018 Mercury magnetometer; Solar Orbiter ESA 2020 ASI 25% solar wind magnetometer; Pierre Auger Observatory 2004 Argentina 3,000 km² INFN+INAF cosmic rays 10¹⁹-10²⁰ eV
• Pulsars and supernova remnants: Crab Nebula M1 SN1054 INAF Bologna magnetosphere neutron star B=10⁸-10¹² T r_L mm-cm; SNR Bell instability cosmic ray acceleration
• ITER France Cadarache 2025 first plasma: 35 countries €25B, Italy ENEA Ansaldo Nucleare Genoa RFX Padua + Frascati Research Center 12% F4E. Italian R&D tax credits 2026: R&D Tax Credit L. 160/2019 up to 25%; Patent Box L. 178/2020 50% R&D costs uplift; Industry/Transition 5.0 DL 39/2024 art. 38 up to 35% green investments; Hyperdepreciation 250% 4.0 assets; NRRP Mission 2 hydrogen €3.2B + Mission 4 research €6.9B

Updated July 2026