larmor’s theorem ll2 section 45. system of charges, finite motion, external constant h-field time...
TRANSCRIPT
![Page 1: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/1.jpg)
Larmor’s Theorem
LL2 Section 45
![Page 2: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/2.jpg)
System of charges, finite motion, external constant H-field
Time average force
Time average of time derivative of quantity with finite variations
![Page 3: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/3.jpg)
Time averaged torque
Time average of time derivative of quantity with finite variations
![Page 4: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/4.jpg)
Compare with electric dipole
![Page 5: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/5.jpg)
Lagrangian for charge in a given electro-magnetic field
Free particle termIf no external electric field.
Lagrangian for system of charges in an external constant uniform H-field
For closed systemExtra term due to external H-field,
![Page 6: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/6.jpg)
(19.4) for uniform H-field
Compare
![Page 7: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/7.jpg)
Centrally symmetric electric field.
System of charges, finite motion, v<<c, e.g. electrons of atom
Transform to rotation reference frame
Velocity in lab frame
Velocity in rotating frame
W
r
Suppose v’ = 0,Then v = -W x r
-W x r
![Page 8: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/8.jpg)
Lagrangian of system of charges in lab frame
L = S ½ mv’2 - U
U is a function of the distances from the ea to Q and of the distances between the ea. This function is unchanged by the transform to the rotating frame.
Lagrangian of system of charges in rotating frame
![Page 9: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/9.jpg)
![Page 10: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/10.jpg)
Assume e/m is the same for all particles,e.g. electrons of an atom.
And choose
Neglect for small H
![Page 11: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/11.jpg)
Lagrangian for closed system when v<<c
Lagrangian for external constant uniform H-field
- U
![Page 12: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/12.jpg)
Larmor Theorem:
System of chargesNon-relativisticSame e/m,Finite motionCentral E-field
Weak H field,Coordinates not rotating
No H-field,Coordinates rotating at W = eH/2mc = “Larmor frequency”
These two problems have the same Lagrangian
![Page 13: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/13.jpg)
• For sufficiently weak H, W = eH/2mc << frequencies of finite motion of charges
• Then, average quantities describing the system over t << 2p/W = Larmor period
• Averaged quantities will vary slowly with time at frequency W.
![Page 14: Larmor’s Theorem LL2 Section 45. System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity](https://reader036.vdocuments.net/reader036/viewer/2022082820/56649f435503460f94c62c32/html5/thumbnails/14.jpg)
Time averaged angular momentum <M>t
If e/m is the same for all particles,m = eM/2mc (44.5)
torque
Larmor precession:<M> and <m> rotate around HWithout changing |M|