# What is Larmor's theorem

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### Larmor frequency

What is a magnetic dipole?

With every rotating charge (circular current with a circular radius, an enclosed area and a current), a magnetic field is connected at a distance, the field lines of which are shown in (Fig. 1).

The position dependence of this magnetic field corresponds exactly to that of the electric field, which is generated by an electric dipole, whose charges are arranged at a distance, at great distances. The electric dipole is described by a vector, which points from the negative to the positive charge and has the absolute value.

Note
This definition comes from physics. In some chemistry textbooks, the electric dipole is defined in the opposite direction!

Correspondingly, a magnetic dipole vector is assigned to a circulating current. It is perpendicular to the surface described by the circular current, has the absolute value and points to positive values ​​if the circle lies in the plane of a coordinate system and the current of positive charges flows counterclockwise.

The rotating charge of the magnetic dipole is always connected to a rotating mass, with which an angular momentum is connected. The axis of the angular momentum vector coincides with the axis of the magnetic dipole, so that the following proportionality exists:

The constant of proportionality is called the gyromagnetic (also magnetogyric) ratio. It can have positive or negative values. Atomic nuclei that have a spin (pn = ug, gu and uu nuclei number; p = protons, n = neutron number) are to be understood as a circular current. Because of the small size of the circular area, the magnitude () of the nuclear magnetic dipole moment is very small. For the nuclei, () and is positive, for on the other hand negative.

### Larmor precession

Every magnetic dipole that is exposed to a constant, homogeneous magnetic field and forms an angle with it, performs a characteristic precession movement at the angular frequency (Fig. 3). It is the result of the constant effort of the dipole to align itself in the direction of the field. Because of the angular momentum, however, according to the laws of mechanics, a deflection occurs that is always perpendicular to the plane defined by the dipole and field vector. The angular frequency is given by. This frequency is known as the Larmor frequency. Its value is independent of the angle (Larmor theorem). The precession motion of a magnetic moment in a magnetic field is the basis of all NMR experiments.