Question

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

The particle moves in a helical path with constant \( \rho \) and \( z \)-axis velocity.

Step-by-step solution

Understand the Magnetic Vector Potential

The magnetic field \( \mathbf{B} \) is related to the vector potential \( \mathbf{A} \) by the equation \( \mathbf{B} = abla \times \mathbf{A} \). In this problem, the vector potential is given as \( \mathbf{A} = \frac{1}{2} \mathbf{B} \times \mathbf{r} \). We need to verify this expression in cylindrical polar coordinates.

Verify the Vector Potential in Cylindrical Coordinates

In Cartesian coordinates \( \mathbf{r} = (x, y, z) \). For a magnetic field aligned in the \( z \)-direction, \( \mathbf{B} = (0, 0, B) \). Calculate \( \mathbf{A} = \frac{1}{2} \mathbf{B} \times \mathbf{r} \): \[ \mathbf{A} = \frac{1}{2} (0, 0, B) \times (x, y, z) = \frac{1}{2}(-By, Bx, 0) \]. In cylindrical coordinates, \( x = \rho \cos(\phi) \), \( y = \rho \sin(\phi) \). Therefore, \( \mathbf{A} = \frac{1}{2} B \rho \hat{\phi} \).

Setup the Lagrangian in Cylindrical Coordinates

The Lagrangian under a magnetic field is given by: \[ L = \frac{1}{2} m ( \dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2 ) + q \mathbf{v} \cdot \mathbf{A} \]. Substitute \( \mathbf{A} = \frac{1}{2} B \rho \hat{\phi} \) and \( \mathbf{v} = (\dot{\rho}, \rho \dot{\phi}, \dot{z}) \). This gives \( L = \frac{1}{2}m(\dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2) + \frac{1}{2}qB \rho^2 \dot{\phi} \).

Derive the Lagrange Equations

Using the Euler-Lagrange equation \( \frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i}) - \frac{\partial L}{\partial q_i} = 0 \), derive equations for \( \rho \), \( \phi \), and \( z \): 1. For \( \rho \): \[ m \ddot{\rho} - m \rho \dot{\phi}^2 = -qB\rho \dot{\phi} \]2. For \( \phi \): \[ \frac{d}{dt}(m \rho^2 \dot{\phi} + \frac{1}{2} qB \rho^2) = 0 \]3. For \( z \): \[ m \ddot{z} = 0 \] (no magnetic field effect in the z direction).

Analyze Solutions with Constant \(\rho\)

If \( \rho \) is constant, then \( \ddot{\rho} = 0 \) and equation 1 becomes \( m\rho \dot{\phi}^2 = qB\rho \dot{\phi} \) leading to \( \dot{\phi} = \frac{qB}{m} \). Equation 2 implies \( \dot{\phi} \) is constant if \( \rho \) is constant. The motion of \( z(t) \) is linear because \( m \ddot{z} = 0 \).

Conclusion on Motion Behavior with Constant \( \rho \)

The particle moves in a circular trajectory in the xy-plane with \( \dot{\phi} = \frac{qB}{m} \) and maintains a constant velocity along the z-axis, which describes a helical path.

Key concepts

Magnetic Vector Potential

The Magnetic Vector Potential, denoted as \( \mathbf{A} \), is a fundamental concept in electromagnetism. It acts behind the scenes to help describe magnetic fields. This potential is related to the magnetic field \( \mathbf{B} \) through the curl operation, expressed as \( \mathbf{B} = abla \times \mathbf{A} \).
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

Cylindrical Coordinates provide a convenient way to describe points in 3D space, especially where there is rotational symmetry. The system uses three parameters: radius \( \rho \), angle \( \phi \), and height \( z \).
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.

• \( x = \rho \cos(\phi) \)
• \( y = \rho \sin(\phi) \)
• \( z = z \)

This aligns with the intuitive understanding of using angled radius (\(\rho\)), rotation (\(\phi\)), and vertical height (\(z\)). Using these coordinates to express a vector potential simplifies the calculations necessary for describing motion in magnetic fields.

Euler-Lagrange Equation

The Euler-Lagrange Equation is a powerful mathematical tool used to derive the equations of motion for a system from its Lagrangian. The Lagrangian \( L \), typically expressed in terms of coordinates and velocities, is given by the difference between kinetic and potential energy.
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

Circular Motion occurs when an object moves along the perimeter of a circle. For a charged particle in a magnetic field, such motion is typical in the plane perpendicular to the field lines.
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

A Helical Path is a three-dimensional trajectory resembling a spring or coil, often described as a combination of circular motion in one plane and linear motion in another direction.
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.