Science
Cyclotron Frequency, Larmor Gyroradius & MRI Proton Larmor Frequency
Calculate charged-particle Cyclotron Frequency f_c = |q| * B / (2 * PI() * m) (MHz), magnetic Larmor Gyroradius r_L = (m * v_perp) / (|q| * B) (mm), and 1H MRI nuclear Larmor resonance (42.577 MHz/T).
Design particle cyclotrons, tokamak plasma confinement, and 3T MRI RF coils!
When a particle of charge |q| and mass m enters a uniform magnetic field B (Tesla) with perpendicular velocity v_perp, the Lorentz force q * (v x B) bends its path into a helix of Larmor Gyroradius r_L = (m * v_perp) / (|q| * B)! Dividing orbital speed v_perp by circumference 2 * PI() * r_L cancels velocity completely—giving Ernest Lawrence's velocity-independent Cyclotron Frequency: f_c = (|q| * B) / (2 * PI() * m)!
In clinical Magnetic Resonance Imaging (MRI), proton spin magnetic moments precess at the Nuclear Larmor Frequency f_0 = (gamma / 2*PI) * B_0, where the shielded 1H proton gyromagnetic ratio is 42.57748 MHz/T—so a 1.5 T scanner excites hydrogen at 63.87 MHz and a 3.0 T scanner operates at 127.73 MHz!
| Magnetic System (Field B = 3.0 Tesla) | Formula | Calculated Value | Engineering Role |
|---|---|---|---|
| Proton Orbital Cyclotron f_c | (|q| * B / (2*PI()*m_p)) / 1e6 | 45.736 MHz | RF dee accelerator frequency |
| Proton Larmor Radius (v = 2e6 m/s) | (m_p * v / (|q| * B)) * 1000 | 6.960 mm | Magnetic confinement radius |
| 1H Spin MRI Larmor Resonance | 42.57748 * B | 127.73 MHz (63.87 MHz at 1.5 T) | MRI RF transmit/receive tune |
Charged-Particle Cyclotron Frequency f_c (MHz)
Computes f_c = (|q| * B) / (2 * PI() * m) / 1e6 from elementary charge B2 (C), mass B3 (kg), and magnetic field B4 (Tesla).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Charge, Mass & Field Must Be > 0", ROUND(((B2 * B4) / (2 * PI() * B3)) / 1000000, 3))Larmor Gyroradius r_L (mm) & 1H MRI Spin Resonance (MHz)
Evaluates r_L = (m * v_perp) / (|q| * B) * 1000 in millimeters from perpendicular speed B5 (m/s).
=IF(OR(B2 <= 0, B4 <= 0), "Charge & Field Must Be > 0", ROUND(((B3 * B5) / (B2 * B4)) * 1000, 3))Try the example
Use the original values or change them to test your understanding.