Difference between revisions of "Derivation of Larmor frequency equation"
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| − | + | In the following, bold letters are vectors. | |
| − | + | For any rotating body the rate of change of the angular momentum '''J''' equals the applied torque '''T''' | |
| − | + | or '''T''' = ''d'''''J'''/''dt'' | |
| − | + | The torque resulting from subjecting a magnetic moment '''m''' to a magnetic field '''B''' is '''m'''cross'''B''' = mB''sin''φ. | |
| − | + | 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 '''J''' and downwards '''m'''. Replace the gravity with a magnetic flux density B. | |
| + | ''d'''''J'''/''dt'' represents the linear velocity of the pike of the arrow '''J''' along a circle whose radius is J''sin''φ where φ is the angle between '''J''' and the vertical. Hence the angular velocity of the rotation of the spin is: | ||
| − | + | ω=2πf =|''d'''''J'''|/(J*''sin''φ)''dt''=|'''T'''|/(J*''sin''φ)=|'''m'''cross'''B'''|/J''sin''φ=mB''sin''φ/J''sin''φ=(m/J)*B=<span style="font-family:symbol;">g</span>*B. | |
| + | |||
| + | |||
| + | Next, the constant 2π is included in the definition of <span style="font-family:symbol;">g</span>. | ||
| + | |||
| + | Consequently, frequency =<span style="font-family:symbol;">g</span>*B<sub>0</sub> | ||
Latest revision as of 15:33, 25 March 2020
In the following, bold letters are vectors.
For any rotating body the rate of change of the angular momentum J equals the applied torque T
or T = dJ/dt
The torque resulting from subjecting a magnetic moment m to a magnetic field B is mcrossB = mBsinφ.
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 J and downwards m. Replace the gravity with a magnetic flux density B.
dJ/dt represents the linear velocity of the pike of the arrow J along a circle whose radius is Jsinφ where φ is the angle between J and the vertical. Hence the angular velocity of the rotation of the spin is:
ω=2πf =|dJ|/(J*sinφ)dt=|T|/(J*sinφ)=|mcrossB|/Jsinφ=mBsinφ/Jsinφ=(m/J)*B=g*B.
Next, the constant 2π is included in the definition of g.
Consequently, frequency =g*B0
