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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

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

Step by step solution

01

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.
02

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, \( \mathbf{A} = \frac{1}{2} B \rho \hat{\phi} \).
03

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 \( L = \frac{1}{2}m(\dot{\rho}^2 + \rho^2 \dot{\phi}^2 + \dot{z}^2) + \frac{1}{2}qB \rho^2 \dot{\phi} \).
04

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).
05

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 \( m \ddot{z} = 0 \).
06

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a horizontal plane with constant angular velocity \(\omega\). Write down the Lagrangian for a bead threaded on the rod, using \(r\) as your generalized coordinate, where \(r, \phi\) are the polar coordinates of the bead. (Notice that \(\phi\) is not an independent variable since it is fixed by the rotation of the rod to be \(\phi=\omega t\).) Solve Lagrange's equation for \(r(t) .\) What happens if the bead is initially at rest at the origin? If it is released from any point \(r_{\mathrm{o}}>0,\) show that \(r(t)\) eventually grows exponentially. Explain your results in terms of the centrifugal force \(m \omega^{2} r\).

Write down the Lagrangian for a projectile (subject to no air resistance) in terms of its Cartesian coordinates \((x, y, z),\) with \(z\) measured vertically upward. Find the three Lagrange equations and show that they are exactly what you would expect for the equations of motion.

A mass \(m\) is suspended from a massless string, the other end of which is wrapped several times around a horizontal cylinder of radius \(R\) and moment of inertia \(I\), which is free to rotate about a fixed horizontal axle. Using a suitable coordinate, set up the Lagrangian and the Lagrange equation of motion, and find the acceleration of the mass \(m\). [The kinetic energy of the rotating cylinder is \(\frac{1}{2} I \omega^{2} .\)]

Prove that the potential energy of a central force \(\mathbf{F}=-k r^{n} \hat{\mathbf{r}}(\text { with } n \neq-1)\) is \(U=k r^{n+1} /(n+1)\). In particular, if \(n=1,\) then \(\mathbf{F}=-k \mathbf{r}\) and \(U=\frac{1}{2} k r^{2}\).

Write down the Lagrangian for a cylinder (mass \(m\), radius \(R\), and moment of inertia \(I\) ) that rolls without slipping straight down an inclined plane which is at an angle \(\alpha\) from the horizontal. Use as your generalized coordinate the cylinder's distance \(x\) measured down the plane from its starting point. Write down the Lagrange equation and solve it for the cylinder's acceleration \(\ddot{x}\). Remember that \(T=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2},\) where \(v\) is the velocity of the center of mass and \(\omega\) is the angular velocity.

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