in hertz), that is proportional to the external field: For this reason, values of , in units of hertz per tesla (Hz/T), are often quoted instead of . Heuristic derivation The derivation of this relation is as follows: First we must prove that the torque resulting from subjecting a magnetic moment to a magnetic field is The identity of the functional form of the stationary electric and magnetic fields has led to defining the magnitude of the magnetic dipole moment equally well as , or in the following way, imitating the moment of an electric dipole: The magnetic dipole can be represented by a needle of a compass with fictitious magnetic charges on the two poles and vector distance between the poles under the influence of the magnetic field of earth By classical mechanics the torque on this needle is But as previously stated so the desired formula comes up. is the unit distance vector. The model of the spinning electron we use in the derivation has an evident analogy with a gyroscope. For any rotating body the rate of change of the angular momentum equals the applied torque : Note as an example the precession of a gyroscope. The earth's gravitational attraction applies a force or torque to the gyroscope in the vertical direction, and the angular momentum vector along the axis of the gyroscope rotates slowly about a vertical line through the pivot. In the place of the gyroscope imagine a sphere spinning around the axis and with its center on the pivot of the gyroscope, and along the axis of the gyroscope two oppositely directed vectors both originated in the center of the sphere, upwards and downwards Replace the gravity with a magnetic flux density represents the linear velocity of the pike of the arrow along a circle whose radius is where is the angle between and the vertical. Hence the angular velocity of the rotation of the spin is Consequently, This relationship also explains an apparent contradiction between the two equivalent terms, gyromagnetic ratio versus magnetogyric ratio: whereas it is a ratio of a magnetic property (i.e. dipole moment) to a gyric (rotational, from , "turn") property (i.e. angular momentum), it is also, at the same time, a ratio between the angular precession frequency (another gyric property) and the magnetic field. The angular precession frequency has an important physical meaning: It is the angular cyclotron frequency,