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A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by?
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A particle of mass m and charge e moves in the xy plane in a potential...
Conserved Quantity in Magnetic Fields
In the scenario described, a particle of mass m and charge e moves in a potential V(r) within a uniform magnetic field B directed along the z-axis. The vector potential A is given by A = 1/2(B × r), which influences the dynamics of the charged particle.
Angular Momentum Conservation
- Definition: Angular momentum is a conserved quantity when there is rotational symmetry in the system.
- In Magnetic Fields: The presence of a magnetic field affects the motion of charged particles, leading to the conservation of a modified form of angular momentum.
Modified Angular Momentum
- Expression: The conserved quantity associated with the angular momentum in the presence of a magnetic field is given by L = r × (p - (e/c)A), where p is the linear momentum, and A is the vector potential.
- Components: In polar coordinates (r, θ), the angular momentum can be expressed as L = m r^2 (dθ/dt) + (e/c) (B/2) r^2.
Implications of Conservation
- Physical Meaning: The conservation of angular momentum indicates that the particle's rotational motion around the origin remains constant over time, provided no external torques act on it.
- Energy Considerations: The potential V(r) may influence the energy of the system, but the angular momentum remains conserved due to the symmetrical properties of the magnetic field.
Conclusion
In summary, the conserved quantity for a particle under the influence of a uniform magnetic field and a potential V(r) is the modified angular momentum, which encapsulates the effects of both the particle's motion and the vector potential associated with the magnetic field.
In the scenario described, a particle of mass m and charge e moves in a potential V(r) within a uniform magnetic field B directed along the z-axis. The vector potential A is given by A = 1/2(B × r), which influences the dynamics of the charged particle.
Angular Momentum Conservation
- Definition: Angular momentum is a conserved quantity when there is rotational symmetry in the system.
- In Magnetic Fields: The presence of a magnetic field affects the motion of charged particles, leading to the conservation of a modified form of angular momentum.
Modified Angular Momentum
- Expression: The conserved quantity associated with the angular momentum in the presence of a magnetic field is given by L = r × (p - (e/c)A), where p is the linear momentum, and A is the vector potential.
- Components: In polar coordinates (r, θ), the angular momentum can be expressed as L = m r^2 (dθ/dt) + (e/c) (B/2) r^2.
Implications of Conservation
- Physical Meaning: The conservation of angular momentum indicates that the particle's rotational motion around the origin remains constant over time, provided no external torques act on it.
- Energy Considerations: The potential V(r) may influence the energy of the system, but the angular momentum remains conserved due to the symmetrical properties of the magnetic field.
Conclusion
In summary, the conserved quantity for a particle under the influence of a uniform magnetic field and a potential V(r) is the modified angular momentum, which encapsulates the effects of both the particle's motion and the vector potential associated with the magnetic field.
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A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by?
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A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by?.
A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle of mass m and charge e moves in the xy plane in a potential V(r) and a uniform magnetic field B in the z-direction, with vector potential A =1/2(B×r )(Use polar coordinates). The conserved quantity is given by?.
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