Derivation of Larmor frequency equation - Revision history

Milllo at 21:33, 25 March 2020

2020-03-25T21:33:15Z

2020-03-25T21:30:48Z

2020-03-25T21:29:41Z

2020-03-25T21:25:35Z

2020-03-25T21:24:33Z

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 or  '''T''' = ''d'''''J'''/''dt''
 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.
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 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>

@@ Line 11: / Line 11: @@
 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.
-Where the torque resulting from subjecting a magnetic moment to a magnetic field is '''m'''cross'''B'''.
+Where the torque resulting from subjecting a magnetic moment to a magnetic field is '''m'''cross'''B''' = mB''sin''φ.
 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>

@@ Line 9: / Line 9: @@
 ''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>

@@ Line 9: / Line 9: @@
 ''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.
+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>

@@ Line 9: / Line 9: @@
 ''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=mB/J=<span style="font-family:symbol;">g</span>*B.
+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>

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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''' which is equal to ''d'''''J'''/''dt''
+For any rotating body the rate of change of the angular momentum '''J''' equals the applied torque '''T'''
 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.

← Older revision		Revision as of 21:19, 25 March 2020
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	Next, the constant 2π is included in the definition of <span style="font-family:symbol;">g</span>.		Next, the constant 2π is included in the definition of <span style="font-family:symbol;">g</span>.

−	Consequently, f= frequency ~~= v<sub>0</sub>~~=<span style="font-family:symbol;">g</span>*B<sub>0</sub>	+	Consequently, frequency =<span style="font-family:symbol;">g</span>*B<sub>0</sub>

← Older revision		Revision as of 21:18, 25 March 2020
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	Next, the constant 2π is included in the definition of <span style="font-family:symbol;">g</span>.		Next, the constant 2π is included in the definition of <span style="font-family:symbol;">g</span>.

−	Consequently, v<sub>0</sub>=<span style="font-family:symbol;">g</span>*B<sub>0</sub>	+	Consequently, f= frequency = v<sub>0</sub>=<span style="font-family:symbol;">g</span>*B<sub>0</sub>

@@ Line 7: / Line 7: @@
 ''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=mB/J=<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>.
+ω=2πf =|''d'''''J'''|/(J*''sin''φ)''dt''=|'''T'''|/(J*''sin''φ)=|'''m'''cross'''B'''|/J''sin''φ=mB''sin''φ/J''sin''φ=mB/J=<span style="font-family:symbol;">g</span>*B.
 Consequently, v<sub>0</sub>=<span style="font-family:symbol;">g</span>*B<sub>0</sub>