Chapter 7: Problem 49
Consider a particle of mass \(m\) and charge \(q\) moving in a uniform constant magnetic field \(\mathbf{B}\) in the \(z\) direction. (a) Prove that \(\mathbf{B}\) can be written as \(\mathbf{B}=\nabla \times \mathbf{A}\) with \(\mathbf{A}=\frac{1}{2} \mathbf{B} \times \mathbf{r} .\) Prove equivalently that in cylindrical polar coordinates, \(\mathbf{A}=\frac{1}{2} B \rho \hat{\phi}\). (b) Write the Lagrangian (7.103) in cylindrical polar coordinates and find the three corresponding Lagrange equations. (c) Describe in detail those solutions of the Lagrange equations in which \(\rho\) is a constant.
Short Answer
Step by step solution
Understand the Magnetic Vector Potential
Verify the Vector Potential in Cylindrical Coordinates
Setup the Lagrangian in Cylindrical Coordinates
Derive the Lagrange Equations
Analyze Solutions with Constant \(\rho\)
Conclusion on Motion Behavior with Constant \( \rho \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Magnetic Vector Potential
The beauty of \( \mathbf{A} \) lies in its ability to simplify the representation of the magnetic field and provide insights into the behavior of charged particles in a magnetic environment. One common form of \( \mathbf{A} \) when the magnetic field is constant and pointed in a specific direction is \( \mathbf{A} = \frac{1}{2} \mathbf{B} \times \mathbf{r} \).
This expression holds true because when you calculate the curl of \( \mathbf{A} \), you retrieve \( \mathbf{B} \), thus verifying the correct representation of the magnetic field with respect to vector potential.
Cylindrical Coordinates
These coordinates are particularly handy when dealing with problems involving circular or helical motion, as they naturally match the geometry of such paths. The transformation between Cartesian coordinates (\(x, y, z\)) to cylindrical coordinates is straightforward.
- \( x = \rho \cos(\phi) \)
- \( y = \rho \sin(\phi) \)
- \( z = z \)
Euler-Lagrange Equation
For a system with coordinates \( q_i \), the Euler-Lagrange equation is: \[ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 \] This equation allows us to find the paths that make the action integral stationary (minimum/maximum), thereby determining the actual motion of the particle.
In problems dealing with a charged particle in a magnetic field, the Lagrangian incorporates terms from both the particle's velocity and the magnetic vector potential, leading to complicated but insightful dynamical equations, as we see with the discussed step-by-step solution.
Circular Motion
The force that keeps the particle in circular motion is the magnetic Lorentz force, perpendicular to both the velocity of the particle and the magnetic field. This leads to a constant angular speed or cyclotron frequency \( \omega_c \), given by:
\( \omega_c = \frac{qB}{m} \)
where \( q \) is the charge of the particle, \( B \) is the magnetic field strength, and \( m \) is the mass of the particle.
In a magnetic field, this formula tells us how the particle circulates with a radius dependent on the speed perpendicular to the field, creating a circular path. Understanding this concept helps visualize the paths that charged particles take in a magnetic field.
Helical Path
In the context of a charged particle in a magnetic field, a helical path can arise when the particle has velocity components both parallel and perpendicular to the magnetic field lines.
This occurs because the perpendicular component results in circular motion, while the parallel component allows the particle to move along the direction of the field, forming a helix.
The formula for the pitch or separation between the coils of the helix depends on the component of velocity along the \( z \) direction that is unaffected by the magnetic field. This type of motion is common in plasma physics and helps us understand the behavior of particles trapped in magnetic fields.