Larmor's Theorem and Lagrangian Formulation in Electromagnetic Fields
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 with finite variations
Time averaged torque
Time average of
time derivative of
quantity with finite
variations
Compare with electric dipole
Lagrangian for charge in a given electro-magnetic field
Free particle term
If no external electric field.
Lagrangian for system of charges in an external constant uniform
H
-field
For closed system
Extra term due
to external H-
field,
(19.4) for uniform
H
-field
Compare
Centrally
symmetric electric
field.
System of charges, finite motion,
v<<c, e.g. electrons of atom
Transform to rotating reference frame
Velocity in
lab frame
Velocity in
rotating frame
r
Suppose v’ = 0,
Then
v
= -
x r
-
x r
No magnetic field.
Lagrangian of system of charges in lab frame
L =
½ mv’
2
- U
U is a function of the distances from the
e
a
to Q and of the distances between the
e
a
.
This function is unchanged by the transform to the rotating frame.
Lagrangian of system of charges in
rotating
frame and
without an applied
magnetic field
Assume e/m is the same for all particles, e.g. electrons of an atom.
And choose
Neglect for small H
Lagrangian for closed
system when v<<c
Same as the Lagrangian
for chargres in an
external constant
uniform H-field, but it
appears just because of a
particular choice of
.
- U
Larmor Theorem:
System of charges, Non-relativistic, Same e/m, Finite motion, Central E-field
System of identical chargres in a
weak applied magnetic field H and
centrally symmetric E-field,
Coordinates not rotating
Same system without an
H-field, but now
coordinates rotating at
= eH/2mc
= “Larmor frequency”
These two problems have the same Lagrangian, and hence the same equations of motion.
•
For sufficiently weak H,
= eH/2mc is much smaller than the frequencies
of finite motion for the charges.
•
Then, average quantities describing the system over times t << 2
/
=
Larmor period
•
Averaged quantities will vary slowly with time at frequency
.
Time averaged angular momentum <
M
>
t
If e/m is the same for all particles,
m
= e
M
/2mc (44.5)
torque
Larmor precession:
<
M
> and <
m
> rotate around H
Without changing |M|
Explore Larmor's Theorem, time-averaged forces, torques, and Lagrangian formulations for systems of charges in electromagnetic fields. Dive into the comparison with electric dipoles, transformation to rotating frames, and Lagrangian analysis for closed systems with finite motions. Uncover the intricate relationships between external constant H-fields, electric charges, and particle dynamics in varying reference frames.
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Presentation Transcript
Larmors Theorem LL2 Section 45
System of charges, finite motion, external constant H-field Time average force Time average of time derivative of quantity with finite variations
Time averaged torque Time average of time derivative of quantity with finite variations
Lagrangian for charge in a given electro-magnetic field If no external electric field. Free particle term Lagrangian for system of charges in an external constant uniform H-field Extra term due to external H- field, For closed system
(19.4) for uniform H-field Compare
Centrally symmetric electric field. System of charges, finite motion, v<<c, e.g. electrons of atom No magnetic field. Transform to rotating reference frame - x r r Velocity in lab frame Velocity in rotating frame Suppose v = 0, Then v = - x r
Lagrangian of system of charges in lab frame L = 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 and without an applied magnetic field
Assume e/m is the same for all particles, e.g. electrons of an atom. And choose Neglect for small H
- U Same as the Lagrangian for chargres in an external constant uniform H-field, but it appears just because of a particular choice of . Lagrangian for closed system when v<<c
System of charges, Non-relativistic, Same e/m, Finite motion, Central E-field Larmor Theorem: Same system without an H-field, but now coordinates rotating at = eH/2mc = Larmor frequency System of identical chargres in a weak applied magnetic field H and centrally symmetric E-field, Coordinates not rotating These two problems have the same Lagrangian, and hence the same equations of motion.
For sufficiently weak H, = eH/2mc is much smaller than the frequencies of finite motion for the charges. Then, average quantities describing the system over times t << 2 / = Larmor period Averaged quantities will vary slowly with time at frequency .
Time averaged angular momentum <M>t torque If e/m is the same for all particles, m = eM/2mc (44.5) Larmor precession: <M> and <m> rotate around H Without changing |M|
