Explaining Larmor Frequency: How is w=q*B/2m Derived?

[meadow]

Can someone please explain Larmor frequency to me? (like how the equation w=q*B/2m is derived?) I would greatly appreciate it!

 

[Tide]

The Larmor frequency aka the "gyrofrequency" is the frequency with which charge particles orbit in a magnetic field. It directly from the solution of the equations of motion:

$$\frac {d\vec v}{dt} = \frac {q}{m} \vec v \times \vec B$$

I don't think you meant to include the factor of 1/2.

The term is also used to denote the rate of precession of a magnetic moment in a magnetic field such as an atomic nucleus. I gathered you meant the "gyrofrequency" from the form you displayed.

 

[astrokreng]

The Larmor frequency, denoted as ω, is a fundamental concept in the field of electromagnetism and is commonly used in nuclear magnetic resonance (NMR) and electron spin resonance (ESR) experiments. It is defined as the frequency at which a charged particle, such as an electron or a proton, precesses around the direction of an external magnetic field.

The equation w=q*B/2m is derived from the classical equation of motion for a charged particle in a magnetic field, known as the Lorentz force equation. This equation states that the force experienced by a charged particle moving in a magnetic field is equal to the product of the charge of the particle (q), the magnetic field strength (B), and the velocity of the particle (v). Mathematically, it can be expressed as F=q*v*B.

In order to understand how the Larmor frequency is derived, we need to consider the motion of a charged particle in a uniform magnetic field. The force experienced by the particle due to the magnetic field is always perpendicular to the direction of motion of the particle. This results in the particle moving in a circular path with a constant speed.

Using the equations of circular motion, we can relate the velocity of the particle (v) to its angular velocity (ω) and the radius of its circular path (r). This can be expressed as v=ω*r. Substituting this value into the Lorentz force equation, we get F=q*ω*r*B.

We can also relate the velocity of the particle to its mass (m) and momentum (p). This can be expressed as v=p/m. Substituting this value into the Lorentz force equation, we get F=q*p*B/m.

Since the momentum of the particle is equal to its mass times its velocity (p=m*v), we can substitute this value into the equation above to get F=q*m*v*B/m. Simplifying this equation, we get F=q*B*v.

Finally, we can relate the angular momentum (L) of the particle to its momentum (p) and the radius of its circular path (r). This can be expressed as L=p*r. Substituting this value into the equation above, we get F=q*B*L/r.

Since the force experienced by the particle is always perpendicular to its angular momentum, it causes the particle to precess around the direction of the magnetic field. The frequency of this precession is known as the