Conserved quantity for a particle in a homogeneous and static magnetic field

In summary, the equation of motion for a charged particle in a static magnetic field shows that the quantity ##m\dot{\vec{r}}-q \vec{r}\times \vec{B}## is conserved. This can be derived through Noether's theorem by using a simultaneous spatial and gauge transformation for a static and homogeneous magnetic field, resulting in the conserved quantity ##m\dot{\vec{r}}+q\vec{A}##. However, this quantity is not the canonical momentum of the particle and does not match the equation of motion, so further exploration is needed to arrive at the desired conserved quantity.
  • #1
Joker93
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The equation of motion for a charged particle with mass ##m## and charge ##q## in a static magnetic field is:
##\frac{d}{dt}[m{\dot{\vec{r}}}]=q\ \dot{\vec{r}}\times \vec{B}##
From this, we can see that ##\frac{d}{dt}[m\dot{\vec{r}}-q \vec{r}\times \vec{B}]=0##
and so the following quantity is conserved:
##m\dot{\vec{r}}-q \vec{r}\times \vec{B}##
My question is how can I derive the above conserved quantity through Noether's theorem?

I attempted to do this by a simultaneous spatial transformation and a gauge transformation for a static and homogeneous magnetic field, so that the Lagrangian remains invariant. Through doing this, and finding this more "generalized" translation symmetry (generalized due to the presence of the magnetic field, or just a gauge field), I found that the quantity ##m\dot{\vec{r}}+q\vec{A}## , where ##\vec{A}## is the vector potential, is conserved. But that is the canonical momentum of the particle, which we know that it is not conserved and this is evident by the fact that its (total) time derivative is not zero, as can be seen from the equation of motion of the particle (As given above).

So, how does one arrive at the desirable conserved quantity through Noether's theorem?

Thanks in advance.
 
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Related to Conserved quantity for a particle in a homogeneous and static magnetic field

1. What is a conserved quantity for a particle in a homogeneous and static magnetic field?

A conserved quantity is a physical quantity that remains constant over time in a system. For a particle in a homogeneous and static magnetic field, the conserved quantity is its magnetic moment, which is the product of its magnetic dipole moment and the strength of the magnetic field.

2. How does a homogeneous and static magnetic field affect the motion of a particle?

A homogeneous and static magnetic field exerts a force on a moving charged particle, causing it to experience a change in direction. This force is perpendicular to both the direction of motion of the particle and the direction of the magnetic field.

3. What is the relationship between the magnetic moment and the angular momentum of a particle in a homogeneous and static magnetic field?

The magnetic moment and angular momentum of a particle in a homogeneous and static magnetic field are directly proportional to each other. This means that as the magnetic moment of the particle changes, so does its angular momentum.

4. How is the conserved quantity of a particle in a homogeneous and static magnetic field related to its energy?

The conserved quantity for a particle in a homogeneous and static magnetic field is related to its energy through the principle of conservation of energy. As the particle's magnetic moment changes, its energy also changes, but the total energy of the system remains constant.

5. What is the significance of the conserved quantity for a particle in a homogeneous and static magnetic field?

The conserved quantity for a particle in a homogeneous and static magnetic field is important because it allows us to make predictions about the behavior of the particle in the magnetic field. It also helps us understand fundamental principles of electromagnetism and conservation laws in physics.

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